LMFDB - Embedded newform 1680.3.n.c.1471.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,3,Mod(1471,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.1471"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1680.n (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,0,0,48,0,0,0,-96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$45.7766844125$
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} - 7 x^{15} + 6 x^{14} + 115 x^{13} - 535 x^{12} + 605 x^{11} + 2453 x^{10} - 10997 x^{9} + \cdots + 92164 $ x^16 - 7x^15 + 6x^14 + 115x^13 - 535x^12 + 605x^11 + 2453x^10 - 10997x^9 + 23723x^8 - 49514x^7 + 98500x^6 - 121640x^5 + 87351x^4 - 81896x^3 + 155185x^2 - 191418*x + 92164
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{36} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.16
Root			$1.69504 + 2.74011i$ of defining polynomial
Character	$\chi$	$=$	1680.1471
Dual form			1680.3.n.c.1471.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +2.23607 q^{5} +2.64575i q^{7} -3.00000 q^{9} +0.994379i q^{11} +11.7318 q^{13} +3.87298i q^{15} -14.5812 q^{17} -31.6374i q^{19} -4.58258 q^{21} -17.8342i q^{23} +5.00000 q^{25} -5.19615i q^{27} +49.5599 q^{29} -46.7254i q^{31} -1.72232 q^{33} +5.91608i q^{35} +47.1423 q^{37} +20.3200i q^{39} -5.75058 q^{41} +65.1579i q^{43} -6.70820 q^{45} -2.11270i q^{47} -7.00000 q^{49} -25.2554i q^{51} -2.48676 q^{53} +2.22350i q^{55} +54.7975 q^{57} +47.9819i q^{59} -31.0102 q^{61} -7.93725i q^{63} +26.2331 q^{65} -76.9087i q^{67} +30.8897 q^{69} -33.2482i q^{71} -48.1952 q^{73} +8.66025i q^{75} -2.63088 q^{77} -42.7205i q^{79} +9.00000 q^{81} -6.74079i q^{83} -32.6046 q^{85} +85.8403i q^{87} +150.734 q^{89} +31.0394i q^{91} +80.9307 q^{93} -70.7433i q^{95} +155.531 q^{97} -2.98314i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} + 48 q^{13} - 96 q^{17} + 80 q^{25} + 112 q^{29} + 48 q^{33} - 224 q^{37} + 160 q^{41} - 112 q^{49} - 96 q^{53} + 80 q^{61} - 80 q^{65} + 48 q^{69} - 48 q^{73} + 144 q^{81} - 160 q^{89}+ \cdots + 592 q^{97}+O(q^{100}) $ 16 * q - 48 * q^9 + 48 * q^13 - 96 * q^17 + 80 * q^25 + 112 * q^29 + 48 * q^33 - 224 * q^37 + 160 * q^41 - 112 * q^49 - 96 * q^53 + 80 * q^61 - 80 * q^65 + 48 * q^69 - 48 * q^73 + 144 * q^81 - 160 * q^89 + 592 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	0.994379i	0.0903981i	0.998978	+	0.0451991i	$0.0143922\pi$
$11$	0.994379i	0.0903981i	−0.998978	+	0.0451991i	$0.985608\pi$
$12$	0	0
$13$	11.7318	0.902445	0.451222	−	0.892412i	$-0.350988\pi$
$13$	11.7318	0.902445	0.451222	+	0.892412i	$0.350988\pi$
$14$	0	0
$15$	3.87298i	0.258199i
$16$	0	0
$17$	−14.5812	−0.857718	−0.428859	−	0.903372i	$-0.641084\pi$
$17$	−14.5812	−0.857718	−0.428859	+	0.903372i	$0.641084\pi$
$18$	0	0
$19$	− 31.6374i	− 1.66512i	−0.553932	−	0.832562i	$-0.686873\pi$
$19$	− 31.6374i	− 1.66512i	0.553932	−	0.832562i	$-0.313127\pi$
$20$	0	0
$21$	−4.58258	−0.218218
$22$	0	0
$23$	− 17.8342i	− 0.775399i	−0.921786	−	0.387699i	$-0.873270\pi$
$23$	− 17.8342i	− 0.775399i	0.921786	−	0.387699i	$-0.126730\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	49.5599	1.70896	0.854482	−	0.519481i	$-0.173875\pi$
$29$	49.5599	1.70896	0.854482	+	0.519481i	$0.173875\pi$
$30$	0	0
$31$	− 46.7254i	− 1.50727i	−0.657293	−	0.753635i	$-0.728299\pi$
$31$	− 46.7254i	− 1.50727i	0.657293	−	0.753635i	$-0.271701\pi$
$32$	0	0
$33$	−1.72232	−0.0521914
$34$	0	0
$35$	5.91608i	0.169031i
$36$	0	0
$37$	47.1423	1.27412	0.637058	−	0.770816i	$-0.280151\pi$
$37$	47.1423	1.27412	0.637058	+	0.770816i	$0.280151\pi$
$38$	0	0
$39$	20.3200i	0.521027i
$40$	0	0
$41$	−5.75058	−0.140258	−0.0701290	−	0.997538i	$-0.522341\pi$
$41$	−5.75058	−0.140258	−0.0701290	+	0.997538i	$0.522341\pi$
$42$	0	0
$43$	65.1579i	1.51530i	0.652662	+	0.757650i	$0.273652\pi$
$43$	65.1579i	1.51530i	−0.652662	+	0.757650i	$0.726348\pi$
$44$	0	0
$45$	−6.70820	−0.149071
$46$	0	0
$47$	− 2.11270i	− 0.0449511i	−0.999747	−	0.0224755i	$-0.992845\pi$
$47$	− 2.11270i	− 0.0449511i	0.999747	−	0.0224755i	$-0.00715479\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 25.2554i	− 0.495204i
$52$	0	0
$53$	−2.48676	−0.0469199	−0.0234600	−	0.999725i	$-0.507468\pi$
$53$	−2.48676	−0.0469199	−0.0234600	+	0.999725i	$0.507468\pi$
$54$	0	0
$55$	2.22350i	0.0404273i
$56$	0	0
$57$	54.7975	0.961360
$58$	0	0
$59$	47.9819i	0.813253i	0.913594	+	0.406627i	$0.133295\pi$
$59$	47.9819i	0.813253i	−0.913594	+	0.406627i	$0.866705\pi$
$60$	0	0
$61$	−31.0102	−0.508364	−0.254182	−	0.967156i	$-0.581806\pi$
$61$	−31.0102	−0.508364	−0.254182	+	0.967156i	$0.581806\pi$
$62$	0	0
$63$	− 7.93725i	− 0.125988i
$64$	0	0
$65$	26.2331	0.403586
$66$	0	0
$67$	− 76.9087i	− 1.14789i	−0.818893	−	0.573946i	$-0.805412\pi$
$67$	− 76.9087i	− 1.14789i	0.818893	−	0.573946i	$-0.194588\pi$
$68$	0	0
$69$	30.8897	0.447677
$70$	0	0
$71$	− 33.2482i	− 0.468285i	−0.972202	−	0.234142i	$-0.924772\pi$
$71$	− 33.2482i	− 0.468285i	0.972202	−	0.234142i	$-0.0752282\pi$
$72$	0	0
$73$	−48.1952	−0.660208	−0.330104	−	0.943945i	$-0.607084\pi$
$73$	−48.1952	−0.660208	−0.330104	+	0.943945i	$0.607084\pi$
$74$	0	0
$75$	8.66025i	0.115470i
$76$	0	0
$77$	−2.63088	−0.0341673
$78$	0	0
$79$	− 42.7205i	− 0.540766i	−0.962753	−	0.270383i	$-0.912850\pi$
$79$	− 42.7205i	− 0.540766i	0.962753	−	0.270383i	$-0.0871503\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 6.74079i	− 0.0812143i	−0.999175	−	0.0406071i	$-0.987071\pi$
$83$	− 6.74079i	− 0.0812143i	0.999175	−	0.0406071i	$-0.0129292\pi$
$84$	0	0
$85$	−32.6046	−0.383583
$86$	0	0
$87$	85.8403i	0.986671i
$88$	0	0
$89$	150.734	1.69365	0.846823	−	0.531875i	$-0.178512\pi$
$89$	150.734	1.69365	0.846823	+	0.531875i	$0.178512\pi$
$90$	0	0
$91$	31.0394i	0.341092i
$92$	0	0
$93$	80.9307	0.870222
$94$	0	0
$95$	− 70.7433i	− 0.744666i
$96$	0	0
$97$	155.531	1.60342	0.801708	−	0.597716i	$-0.203925\pi$
$97$	155.531	1.60342	0.801708	+	0.597716i	$0.203925\pi$
$98$	0	0
$99$	− 2.98314i	− 0.0301327i
$100$	0	0
$101$	190.584	1.88697	0.943485	−	0.331415i	$-0.107526\pi$
$101$	190.584	1.88697	0.943485	+	0.331415i	$0.107526\pi$
$102$	0	0
$103$	35.7042i	0.346643i	0.984865	+	0.173321i	$0.0554500\pi$
$103$	35.7042i	0.346643i	−0.984865	+	0.173321i	$0.944550\pi$
$104$	0	0
$105$	−10.2470	−0.0975900
$106$	0	0
$107$	− 102.527i	− 0.958195i	−0.877762	−	0.479098i	$-0.840964\pi$
$107$	− 102.527i	− 0.958195i	0.877762	−	0.479098i	$-0.159036\pi$
$108$	0	0
$109$	−30.2467	−0.277493	−0.138746	−	0.990328i	$-0.544307\pi$
$109$	−30.2467	−0.277493	−0.138746	+	0.990328i	$0.544307\pi$
$110$	0	0
$111$	81.6528i	0.735611i
$112$	0	0
$113$	−190.193	−1.68312	−0.841561	−	0.540161i	$-0.818363\pi$
$113$	−190.193	−1.68312	−0.841561	+	0.540161i	$0.818363\pi$
$114$	0	0
$115$	− 39.8784i	− 0.346769i
$116$	0	0
$117$	−35.1953	−0.300815
$118$	0	0
$119$	− 38.5782i	− 0.324187i
$120$	0	0
$121$	120.011	0.991828
$122$	0	0
$123$	− 9.96029i	− 0.0809780i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	197.210i	1.55284i	0.630218	+	0.776418i	$0.282965\pi$
$127$	197.210i	1.55284i	−0.630218	+	0.776418i	$0.717035\pi$
$128$	0	0
$129$	−112.857	−0.874858
$130$	0	0
$131$	81.7886i	0.624340i	0.950026	+	0.312170i	$0.101056\pi$
$131$	81.7886i	0.624340i	−0.950026	+	0.312170i	$0.898944\pi$
$132$	0	0
$133$	83.7046	0.629358
$134$	0	0
$135$	− 11.6190i	− 0.0860663i
$136$	0	0
$137$	93.2266	0.680486	0.340243	−	0.940338i	$-0.389491\pi$
$137$	93.2266	0.680486	0.340243	+	0.940338i	$0.389491\pi$
$138$	0	0
$139$	− 164.448i	− 1.18308i	−0.806277	−	0.591538i	$-0.798521\pi$
$139$	− 164.448i	− 1.18308i	0.806277	−	0.591538i	$-0.201479\pi$
$140$	0	0
$141$	3.65930	0.0259525
$142$	0	0
$143$	11.6658i	0.0815793i
$144$	0	0
$145$	110.819	0.764272
$146$	0	0
$147$	− 12.1244i	− 0.0824786i
$148$	0	0
$149$	−105.038	−0.704953	−0.352476	−	0.935821i	$-0.614660\pi$
$149$	−105.038	−0.704953	−0.352476	+	0.935821i	$0.614660\pi$
$150$	0	0
$151$	39.2119i	0.259681i	0.991535	+	0.129841i	$0.0414466\pi$
$151$	39.2119i	0.259681i	−0.991535	+	0.129841i	$0.958553\pi$
$152$	0	0
$153$	43.7436	0.285906
$154$	0	0
$155$	− 104.481i	− 0.674071i
$156$	0	0
$157$	−36.9481	−0.235338	−0.117669	−	0.993053i	$-0.537542\pi$
$157$	−36.9481	−0.235338	−0.117669	+	0.993053i	$0.537542\pi$
$158$	0	0
$159$	− 4.30719i	− 0.0270892i
$160$	0	0
$161$	47.1848	0.293073
$162$	0	0
$163$	70.6697i	0.433557i	0.976221	+	0.216778i	$0.0695549\pi$
$163$	70.6697i	0.433557i	−0.976221	+	0.216778i	$0.930445\pi$
$164$	0	0
$165$	−3.85121	−0.0233407
$166$	0	0
$167$	251.441i	1.50563i	0.658231	+	0.752816i	$0.271305\pi$
$167$	251.441i	1.50563i	−0.658231	+	0.752816i	$0.728695\pi$
$168$	0	0
$169$	−31.3653	−0.185594
$170$	0	0
$171$	94.9121i	0.555041i
$172$	0	0
$173$	75.1537	0.434415	0.217207	−	0.976125i	$-0.430305\pi$
$173$	75.1537	0.434415	0.217207	+	0.976125i	$0.430305\pi$
$174$	0	0
$175$	13.2288i	0.0755929i
$176$	0	0
$177$	−83.1072	−0.469532
$178$	0	0
$179$	136.861i	0.764584i	0.924042	+	0.382292i	$0.124865\pi$
$179$	136.861i	0.764584i	−0.924042	+	0.382292i	$0.875135\pi$
$180$	0	0
$181$	−59.7855	−0.330307	−0.165153	−	0.986268i	$-0.552812\pi$
$181$	−59.7855	−0.330307	−0.165153	+	0.986268i	$0.552812\pi$
$182$	0	0
$183$	− 53.7112i	− 0.293504i
$184$	0	0
$185$	105.413	0.569802
$186$	0	0
$187$	− 14.4992i	− 0.0775361i
$188$	0	0
$189$	13.7477	0.0727393
$190$	0	0
$191$	73.8672i	0.386739i	0.981126	+	0.193370i	$0.0619417\pi$
$191$	73.8672i	0.386739i	−0.981126	+	0.193370i	$0.938058\pi$
$192$	0	0
$193$	347.940	1.80280	0.901400	−	0.432987i	$-0.142540\pi$
$193$	347.940	1.80280	0.901400	+	0.432987i	$0.142540\pi$
$194$	0	0
$195$	45.4370i	0.233010i
$196$	0	0
$197$	272.209	1.38177	0.690885	−	0.722965i	$-0.257221\pi$
$197$	272.209	1.38177	0.690885	+	0.722965i	$0.257221\pi$
$198$	0	0
$199$	− 196.370i	− 0.986785i	−0.869807	−	0.493392i	$-0.835757\pi$
$199$	− 196.370i	− 0.986785i	0.869807	−	0.493392i	$-0.164243\pi$
$200$	0	0
$201$	133.210	0.662735
$202$	0	0
$203$	131.123i	0.645928i
$204$	0	0
$205$	−12.8587	−0.0627253
$206$	0	0
$207$	53.5025i	0.258466i
$208$	0	0
$209$	31.4595	0.150524
$210$	0	0
$211$	− 325.848i	− 1.54430i	−0.635439	−	0.772151i	$-0.719181\pi$
$211$	− 325.848i	− 1.54430i	0.635439	−	0.772151i	$-0.280819\pi$
$212$	0	0
$213$	57.5876	0.270364
$214$	0	0
$215$	145.697i	0.677662i
$216$	0	0
$217$	123.624	0.569694
$218$	0	0
$219$	− 83.4765i	− 0.381171i
$220$	0	0
$221$	−171.063	−0.774043
$222$	0	0
$223$	− 319.693i	− 1.43360i	−0.697277	−	0.716801i	$-0.745605\pi$
$223$	− 319.693i	− 1.43360i	0.697277	−	0.716801i	$-0.254395\pi$
$224$	0	0
$225$	−15.0000	−0.0666667
$226$	0	0
$227$	− 71.1456i	− 0.313417i	−0.987645	−	0.156708i	$-0.949912\pi$
$227$	− 71.1456i	− 0.313417i	0.987645	−	0.156708i	$-0.0500883\pi$
$228$	0	0
$229$	−52.6230	−0.229795	−0.114897	−	0.993377i	$-0.536654\pi$
$229$	−52.6230	−0.229795	−0.114897	+	0.993377i	$0.536654\pi$
$230$	0	0
$231$	− 4.55682i	− 0.0197265i
$232$	0	0
$233$	268.024	1.15032	0.575158	−	0.818042i	$-0.304940\pi$
$233$	268.024	1.15032	0.575158	+	0.818042i	$0.304940\pi$
$234$	0	0
$235$	− 4.72414i	− 0.0201027i
$236$	0	0
$237$	73.9941	0.312211
$238$	0	0
$239$	53.3789i	0.223343i	0.993745	+	0.111671i	$0.0356204\pi$
$239$	53.3789i	0.223343i	−0.993745	+	0.111671i	$0.964380\pi$
$240$	0	0
$241$	447.919	1.85859	0.929293	−	0.369343i	$-0.120417\pi$
$241$	447.919	1.85859	0.929293	+	0.369343i	$0.120417\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−15.6525	−0.0638877
$246$	0	0
$247$	− 371.163i	− 1.50268i
$248$	0	0
$249$	11.6754	0.0468891
$250$	0	0
$251$	211.604i	0.843046i	0.906818	+	0.421523i	$0.138504\pi$
$251$	211.604i	0.843046i	−0.906818	+	0.421523i	$0.861496\pi$
$252$	0	0
$253$	17.7339	0.0700946
$254$	0	0
$255$	− 56.4727i	− 0.221462i
$256$	0	0
$257$	289.044	1.12468	0.562342	−	0.826905i	$-0.309900\pi$
$257$	289.044	1.12468	0.562342	+	0.826905i	$0.309900\pi$
$258$	0	0
$259$	124.727i	0.481571i
$260$	0	0
$261$	−148.680	−0.569655
$262$	0	0
$263$	487.110i	1.85213i	0.377364	+	0.926065i	$0.376830\pi$
$263$	487.110i	1.85213i	−0.377364	+	0.926065i	$0.623170\pi$
$264$	0	0
$265$	−5.56056	−0.0209832
$266$	0	0
$267$	261.080i	0.977827i
$268$	0	0
$269$	−210.980	−0.784313	−0.392157	−	0.919898i	$-0.628271\pi$
$269$	−210.980	−0.784313	−0.392157	+	0.919898i	$0.628271\pi$
$270$	0	0
$271$	99.8457i	0.368434i	0.982886	+	0.184217i	$0.0589750\pi$
$271$	99.8457i	0.368434i	−0.982886	+	0.184217i	$0.941025\pi$
$272$	0	0
$273$	−53.7618	−0.196930
$274$	0	0
$275$	4.97190i	0.0180796i
$276$	0	0
$277$	256.365	0.925504	0.462752	−	0.886488i	$-0.346862\pi$
$277$	256.365	0.925504	0.462752	+	0.886488i	$0.346862\pi$
$278$	0	0
$279$	140.176i	0.502423i
$280$	0	0
$281$	−307.325	−1.09368	−0.546842	−	0.837236i	$-0.684170\pi$
$281$	−307.325	−1.09368	−0.546842	+	0.837236i	$0.684170\pi$
$282$	0	0
$283$	− 141.756i	− 0.500904i	−0.968129	−	0.250452i	$-0.919421\pi$
$283$	− 141.756i	− 0.500904i	0.968129	−	0.250452i	$-0.0805793\pi$
$284$	0	0
$285$	122.531	0.429933
$286$	0	0
$287$	− 15.2146i	− 0.0530125i
$288$	0	0
$289$	−76.3886	−0.264320
$290$	0	0
$291$	269.388i	0.925733i
$292$	0	0
$293$	−253.145	−0.863977	−0.431989	−	0.901879i	$-0.642188\pi$
$293$	−253.145	−0.863977	−0.431989	+	0.901879i	$0.642188\pi$
$294$	0	0
$295$	107.291i	0.363698i
$296$	0	0
$297$	5.16695	0.0173971
$298$	0	0
$299$	− 209.227i	− 0.699754i
$300$	0	0
$301$	−172.391	−0.572729
$302$	0	0
$303$	330.101i	1.08944i
$304$	0	0
$305$	−69.3409	−0.227347
$306$	0	0
$307$	− 225.087i	− 0.733184i	−0.930382	−	0.366592i	$-0.880525\pi$
$307$	− 225.087i	− 0.733184i	0.930382	−	0.366592i	$-0.119475\pi$
$308$	0	0
$309$	−61.8415	−0.200134
$310$	0	0
$311$	− 480.911i	− 1.54634i	−0.634200	−	0.773169i	$-0.718670\pi$
$311$	− 480.911i	− 1.54634i	0.634200	−	0.773169i	$-0.281330\pi$
$312$	0	0
$313$	−336.694	−1.07570	−0.537849	−	0.843041i	$-0.680763\pi$
$313$	−336.694	−1.07570	−0.537849	+	0.843041i	$0.680763\pi$
$314$	0	0
$315$	− 17.7482i	− 0.0563436i
$316$	0	0
$317$	82.9923	0.261805	0.130903	−	0.991395i	$-0.458212\pi$
$317$	82.9923	0.261805	0.130903	+	0.991395i	$0.458212\pi$
$318$	0	0
$319$	49.2814i	0.154487i
$320$	0	0
$321$	177.582	0.553214
$322$	0	0
$323$	461.311i	1.42821i
$324$	0	0
$325$	58.6589	0.180489
$326$	0	0
$327$	− 52.3889i	− 0.160211i
$328$	0	0
$329$	5.58968	0.0169899
$330$	0	0
$331$	− 352.607i	− 1.06528i	−0.846343	−	0.532638i	$-0.821201\pi$
$331$	− 352.607i	− 1.06528i	0.846343	−	0.532638i	$-0.178799\pi$
$332$	0	0
$333$	−141.427	−0.424705
$334$	0	0
$335$	− 171.973i	− 0.513352i
$336$	0	0
$337$	−205.523	−0.609860	−0.304930	−	0.952375i	$-0.598633\pi$
$337$	−205.523	−0.609860	−0.304930	+	0.952375i	$0.598633\pi$
$338$	0	0
$339$	− 329.424i	− 0.971751i
$340$	0	0
$341$	46.4627	0.136254
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	69.0714	0.200207
$346$	0	0
$347$	7.65489i	0.0220602i	0.999939	+	0.0110301i	$0.00351106\pi$
$347$	7.65489i	0.0220602i	−0.999939	+	0.0110301i	$0.996489\pi$
$348$	0	0
$349$	12.6019	0.0361086	0.0180543	−	0.999837i	$-0.494253\pi$
$349$	12.6019	0.0361086	0.0180543	+	0.999837i	$0.494253\pi$
$350$	0	0
$351$	− 60.9601i	− 0.173676i
$352$	0	0
$353$	−189.511	−0.536858	−0.268429	−	0.963300i	$-0.586504\pi$
$353$	−189.511	−0.536858	−0.268429	+	0.963300i	$0.586504\pi$
$354$	0	0
$355$	− 74.3452i	− 0.209423i
$356$	0	0
$357$	66.8195	0.187169
$358$	0	0
$359$	− 34.5577i	− 0.0962609i	−0.998841	−	0.0481304i	$-0.984674\pi$
$359$	− 34.5577i	− 0.0962609i	0.998841	−	0.0481304i	$-0.0153263\pi$
$360$	0	0
$361$	−639.923	−1.77264
$362$	0	0
$363$	207.866i	0.572632i
$364$	0	0
$365$	−107.768	−0.295254
$366$	0	0
$367$	− 269.436i	− 0.734158i	−0.930190	−	0.367079i	$-0.880358\pi$
$367$	− 269.436i	− 0.734158i	0.930190	−	0.367079i	$-0.119642\pi$
$368$	0	0
$369$	17.2517	0.0467526
$370$	0	0
$371$	− 6.57934i	− 0.0177341i
$372$	0	0
$373$	−320.910	−0.860349	−0.430175	−	0.902746i	$-0.641548\pi$
$373$	−320.910	−0.860349	−0.430175	+	0.902746i	$0.641548\pi$
$374$	0	0
$375$	19.3649i	0.0516398i
$376$	0	0
$377$	581.426	1.54225
$378$	0	0
$379$	− 524.801i	− 1.38470i	−0.721562	−	0.692350i	$-0.756575\pi$
$379$	− 524.801i	− 1.38470i	0.721562	−	0.692350i	$-0.243425\pi$
$380$	0	0
$381$	−341.578	−0.896530
$382$	0	0
$383$	67.9966i	0.177537i	0.996052	+	0.0887684i	$0.0282931\pi$
$383$	67.9966i	0.177537i	−0.996052	+	0.0887684i	$0.971707\pi$
$384$	0	0
$385$	−5.88283	−0.0152801
$386$	0	0
$387$	− 195.474i	− 0.505100i
$388$	0	0
$389$	−653.486	−1.67991	−0.839957	−	0.542653i	$-0.817420\pi$
$389$	−653.486	−1.67991	−0.839957	+	0.542653i	$0.817420\pi$
$390$	0	0
$391$	260.044i	0.665073i
$392$	0	0
$393$	−141.662	−0.360463
$394$	0	0
$395$	− 95.5260i	− 0.241838i
$396$	0	0
$397$	171.786	0.432712	0.216356	−	0.976315i	$-0.430583\pi$
$397$	171.786	0.432712	0.216356	+	0.976315i	$0.430583\pi$
$398$	0	0
$399$	144.981i	0.363360i
$400$	0	0
$401$	664.507	1.65712	0.828562	−	0.559897i	$-0.189159\pi$
$401$	664.507	1.65712	0.828562	+	0.559897i	$0.189159\pi$
$402$	0	0
$403$	− 548.172i	− 1.36023i
$404$	0	0
$405$	20.1246	0.0496904
$406$	0	0
$407$	46.8773i	0.115178i
$408$	0	0
$409$	−430.284	−1.05204	−0.526020	−	0.850472i	$-0.676316\pi$
$409$	−430.284	−1.05204	−0.526020	+	0.850472i	$0.676316\pi$
$410$	0	0
$411$	161.473i	0.392879i
$412$	0	0
$413$	−126.948	−0.307381
$414$	0	0
$415$	− 15.0729i	− 0.0363201i
$416$	0	0
$417$	284.832	0.683049
$418$	0	0
$419$	− 40.8289i	− 0.0974437i	−0.998812	−	0.0487218i	$-0.984485\pi$
$419$	− 40.8289i	− 0.0974437i	0.998812	−	0.0487218i	$-0.0155148\pi$
$420$	0	0
$421$	−548.381	−1.30257	−0.651284	−	0.758834i	$-0.725769\pi$
$421$	−548.381	−1.30257	−0.651284	+	0.758834i	$0.725769\pi$
$422$	0	0
$423$	6.33810i	0.0149837i
$424$	0	0
$425$	−72.9060	−0.171544
$426$	0	0
$427$	− 82.0453i	− 0.192143i
$428$	0	0
$429$	−20.2058	−0.0470998
$430$	0	0
$431$	519.249i	1.20475i	0.798212	+	0.602377i	$0.205779\pi$
$431$	519.249i	1.20475i	−0.798212	+	0.602377i	$0.794221\pi$
$432$	0	0
$433$	26.0559	0.0601752	0.0300876	−	0.999547i	$-0.490421\pi$
$433$	26.0559	0.0601752	0.0300876	+	0.999547i	$0.490421\pi$
$434$	0	0
$435$	191.945i	0.441253i
$436$	0	0
$437$	−564.226	−1.29114
$438$	0	0
$439$	− 779.712i	− 1.77611i	−0.459739	−	0.888054i	$-0.652057\pi$
$439$	− 779.712i	− 1.77611i	0.459739	−	0.888054i	$-0.347943\pi$
$440$	0	0
$441$	21.0000	0.0476190
$442$	0	0
$443$	− 559.509i	− 1.26300i	−0.775376	−	0.631500i	$-0.782440\pi$
$443$	− 559.509i	− 1.26300i	0.775376	−	0.631500i	$-0.217560\pi$
$444$	0	0
$445$	337.053	0.757421
$446$	0	0
$447$	− 181.931i	− 0.407005i
$448$	0	0
$449$	−138.772	−0.309070	−0.154535	−	0.987987i	$-0.549388\pi$
$449$	−138.772	−0.309070	−0.154535	+	0.987987i	$0.549388\pi$
$450$	0	0
$451$	− 5.71825i	− 0.0126791i
$452$	0	0
$453$	−67.9170	−0.149927
$454$	0	0
$455$	69.4062i	0.152541i
$456$	0	0
$457$	154.391	0.337837	0.168918	−	0.985630i	$-0.445973\pi$
$457$	154.391	0.337837	0.168918	+	0.985630i	$0.445973\pi$
$458$	0	0
$459$	75.7661i	0.165068i
$460$	0	0
$461$	48.2176	0.104594	0.0522968	−	0.998632i	$-0.483346\pi$
$461$	48.2176	0.104594	0.0522968	+	0.998632i	$0.483346\pi$
$462$	0	0
$463$	236.896i	0.511654i	0.966723	+	0.255827i	$0.0823477\pi$
$463$	236.896i	0.511654i	−0.966723	+	0.255827i	$0.917652\pi$
$464$	0	0
$465$	180.967	0.389175
$466$	0	0
$467$	549.126i	1.17586i	0.808912	+	0.587929i	$0.200057\pi$
$467$	549.126i	1.17586i	−0.808912	+	0.587929i	$0.799943\pi$
$468$	0	0
$469$	203.481	0.433862
$470$	0	0
$471$	− 63.9960i	− 0.135873i
$472$	0	0
$473$	−64.7916	−0.136980
$474$	0	0
$475$	− 158.187i	− 0.333025i
$476$	0	0
$477$	7.46027	0.0156400
$478$	0	0
$479$	563.264i	1.17592i	0.808892	+	0.587958i	$0.200068\pi$
$479$	563.264i	1.17592i	−0.808892	+	0.587958i	$0.799932\pi$
$480$	0	0
$481$	553.063	1.14982
$482$	0	0
$483$	81.7264i	0.169206i
$484$	0	0
$485$	347.779	0.717070
$486$	0	0
$487$	195.019i	0.400450i	0.979750	+	0.200225i	$0.0641673\pi$
$487$	195.019i	0.400450i	−0.979750	+	0.200225i	$0.935833\pi$
$488$	0	0
$489$	−122.404	−0.250314
$490$	0	0
$491$	829.328i	1.68906i	0.535509	+	0.844530i	$0.320120\pi$
$491$	829.328i	1.68906i	−0.535509	+	0.844530i	$0.679880\pi$
$492$	0	0
$493$	−722.644	−1.46581
$494$	0	0
$495$	− 6.67050i	− 0.0134758i
$496$	0	0
$497$	87.9665	0.176995
$498$	0	0
$499$	− 117.062i	− 0.234593i	−0.993097	−	0.117296i	$-0.962577\pi$
$499$	− 117.062i	− 0.234593i	0.993097	−	0.117296i	$-0.0374227\pi$
$500$	0	0
$501$	−435.508	−0.869277
$502$	0	0
$503$	− 570.038i	− 1.13328i	−0.823966	−	0.566639i	$-0.808244\pi$
$503$	− 570.038i	− 1.13328i	0.823966	−	0.566639i	$-0.191756\pi$
$504$	0	0
$505$	426.159	0.843879
$506$	0	0
$507$	− 54.3263i	− 0.107152i
$508$	0	0
$509$	−903.167	−1.77439	−0.887197	−	0.461390i	$-0.847351\pi$
$509$	−903.167	−1.77439	−0.887197	+	0.461390i	$0.847351\pi$
$510$	0	0
$511$	− 127.512i	− 0.249535i
$512$	0	0
$513$	−164.393	−0.320453
$514$	0	0
$515$	79.8371i	0.155023i
$516$	0	0
$517$	2.10083	0.00406349
$518$	0	0
$519$	130.170i	0.250809i
$520$	0	0
$521$	−910.593	−1.74778	−0.873890	−	0.486124i	$-0.838410\pi$
$521$	−910.593	−1.74778	−0.873890	+	0.486124i	$0.838410\pi$
$522$	0	0
$523$	− 460.293i	− 0.880101i	−0.897973	−	0.440051i	$-0.854961\pi$
$523$	− 460.293i	− 0.880101i	0.897973	−	0.440051i	$-0.145039\pi$
$524$	0	0
$525$	−22.9129	−0.0436436
$526$	0	0
$527$	681.312i	1.29281i
$528$	0	0
$529$	210.942	0.398757
$530$	0	0
$531$	− 143.946i	− 0.271084i
$532$	0	0
$533$	−67.4645	−0.126575
$534$	0	0
$535$	− 229.257i	− 0.428518i
$536$	0	0
$537$	−237.049	−0.441433
$538$	0	0
$539$	− 6.96066i	− 0.0129140i
$540$	0	0
$541$	−628.342	−1.16144	−0.580722	−	0.814102i	$-0.697230\pi$
$541$	−628.342	−1.16144	−0.580722	+	0.814102i	$0.697230\pi$
$542$	0	0
$543$	− 103.552i	− 0.190703i
$544$	0	0
$545$	−67.6337	−0.124099
$546$	0	0
$547$	− 408.278i	− 0.746395i	−0.927752	−	0.373197i	$-0.878261\pi$
$547$	− 408.278i	− 0.746395i	0.927752	−	0.373197i	$-0.121739\pi$
$548$	0	0
$549$	93.0306	0.169455
$550$	0	0
$551$	− 1567.95i	− 2.84564i
$552$	0	0
$553$	113.028	0.204390
$554$	0	0
$555$	182.581i	0.328975i
$556$	0	0
$557$	−338.823	−0.608299	−0.304149	−	0.952624i	$-0.598372\pi$
$557$	−338.823	−0.608299	−0.304149	+	0.952624i	$0.598372\pi$
$558$	0	0
$559$	764.418i	1.36747i
$560$	0	0
$561$	25.1134	0.0447655
$562$	0	0
$563$	− 109.818i	− 0.195058i	−0.995233	−	0.0975289i	$-0.968906\pi$
$563$	− 109.818i	− 0.195058i	0.995233	−	0.0975289i	$-0.0310938\pi$
$564$	0	0
$565$	−425.284	−0.752715
$566$	0	0
$567$	23.8118i	0.0419961i
$568$	0	0
$569$	−472.870	−0.831054	−0.415527	−	0.909581i	$-0.636403\pi$
$569$	−472.870	−0.831054	−0.415527	+	0.909581i	$0.636403\pi$
$570$	0	0
$571$	528.480i	0.925534i	0.886480	+	0.462767i	$0.153143\pi$
$571$	528.480i	0.925534i	−0.886480	+	0.462767i	$0.846857\pi$
$572$	0	0
$573$	−127.942	−0.223284
$574$	0	0
$575$	− 89.1709i	− 0.155080i
$576$	0	0
$577$	783.886	1.35855	0.679277	−	0.733882i	$-0.262293\pi$
$577$	783.886	1.35855	0.679277	+	0.733882i	$0.262293\pi$
$578$	0	0
$579$	602.651i	1.04085i
$580$	0	0
$581$	17.8344	0.0306961
$582$	0	0
$583$	− 2.47278i	− 0.00424148i
$584$	0	0
$585$	−78.6992	−0.134529
$586$	0	0
$587$	1128.14i	1.92187i	0.276776	+	0.960934i	$0.410734\pi$
$587$	1128.14i	1.92187i	−0.276776	+	0.960934i	$0.589266\pi$
$588$	0	0
$589$	−1478.27	−2.50979
$590$	0	0
$591$	471.479i	0.797765i
$592$	0	0
$593$	−286.239	−0.482696	−0.241348	−	0.970439i	$-0.577590\pi$
$593$	−286.239	−0.482696	−0.241348	+	0.970439i	$0.577590\pi$
$594$	0	0
$595$	− 86.2636i	− 0.144981i
$596$	0	0
$597$	340.123	0.569720
$598$	0	0
$599$	− 510.217i	− 0.851781i	−0.904775	−	0.425891i	$-0.859961\pi$
$599$	− 510.217i	− 0.851781i	0.904775	−	0.425891i	$-0.140039\pi$
$600$	0	0
$601$	−425.581	−0.708122	−0.354061	−	0.935222i	$-0.615199\pi$
$601$	−425.581	−0.708122	−0.354061	+	0.935222i	$0.615199\pi$
$602$	0	0
$603$	230.726i	0.382630i
$604$	0	0
$605$	268.353	0.443559
$606$	0	0
$607$	− 611.987i	− 1.00822i	−0.863641	−	0.504108i	$-0.831821\pi$
$607$	− 611.987i	− 1.00822i	0.863641	−	0.504108i	$-0.168179\pi$
$608$	0	0
$609$	−227.112	−0.372926
$610$	0	0
$611$	− 24.7857i	− 0.0405659i
$612$	0	0
$613$	−713.054	−1.16322	−0.581610	−	0.813468i	$-0.697577\pi$
$613$	−713.054	−1.16322	−0.581610	+	0.813468i	$0.697577\pi$
$614$	0	0
$615$	− 22.2719i	− 0.0362144i
$616$	0	0
$617$	883.052	1.43120	0.715601	−	0.698509i	$-0.246153\pi$
$617$	883.052	1.43120	0.715601	+	0.698509i	$0.246153\pi$
$618$	0	0
$619$	− 1011.23i	− 1.63365i	−0.576888	−	0.816823i	$-0.695733\pi$
$619$	− 1011.23i	− 1.63365i	0.576888	−	0.816823i	$-0.304267\pi$
$620$	0	0
$621$	−92.6691	−0.149226
$622$	0	0
$623$	398.806i	0.640138i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	54.4895i	0.0869051i
$628$	0	0
$629$	−687.391	−1.09283
$630$	0	0
$631$	797.511i	1.26388i	0.775016	+	0.631942i	$0.217742\pi$
$631$	797.511i	1.26388i	−0.775016	+	0.631942i	$0.782258\pi$
$632$	0	0
$633$	564.385	0.891603
$634$	0	0
$635$	440.975i	0.694449i
$636$	0	0
$637$	−82.1225	−0.128921
$638$	0	0
$639$	99.7446i	0.156095i
$640$	0	0
$641$	930.805	1.45211	0.726057	−	0.687634i	$-0.241351\pi$
$641$	930.805	1.45211	0.726057	+	0.687634i	$0.241351\pi$
$642$	0	0
$643$	823.306i	1.28041i	0.768202	+	0.640207i	$0.221152\pi$
$643$	823.306i	1.28041i	−0.768202	+	0.640207i	$0.778848\pi$
$644$	0	0
$645$	−252.355	−0.391249
$646$	0	0
$647$	− 1090.05i	− 1.68477i	−0.538873	−	0.842387i	$-0.681150\pi$
$647$	− 1090.05i	− 1.68477i	0.538873	−	0.842387i	$-0.318850\pi$
$648$	0	0
$649$	−47.7122	−0.0735166
$650$	0	0
$651$	214.122i	0.328913i
$652$	0	0
$653$	−488.199	−0.747626	−0.373813	−	0.927504i	$-0.621950\pi$
$653$	−488.199	−0.747626	−0.373813	+	0.927504i	$0.621950\pi$
$654$	0	0
$655$	182.885i	0.279214i
$656$	0	0
$657$	144.585	0.220069
$658$	0	0
$659$	989.416i	1.50139i	0.660649	+	0.750695i	$0.270281\pi$
$659$	989.416i	1.50139i	−0.660649	+	0.750695i	$0.729719\pi$
$660$	0	0
$661$	488.091	0.738413	0.369206	−	0.929347i	$-0.379630\pi$
$661$	488.091	0.738413	0.369206	+	0.929347i	$0.379630\pi$
$662$	0	0
$663$	− 296.291i	− 0.446894i
$664$	0	0
$665$	187.169	0.281457
$666$	0	0
$667$	− 883.861i	− 1.32513i
$668$	0	0
$669$	553.725	0.827691
$670$	0	0
$671$	− 30.8359i	− 0.0459551i
$672$	0	0
$673$	−244.295	−0.362994	−0.181497	−	0.983391i	$-0.558094\pi$
$673$	−244.295	−0.362994	−0.181497	+	0.983391i	$0.558094\pi$
$674$	0	0
$675$	− 25.9808i	− 0.0384900i
$676$	0	0
$677$	794.825	1.17404	0.587020	−	0.809573i	$-0.300301\pi$
$677$	794.825	1.17404	0.587020	+	0.809573i	$0.300301\pi$
$678$	0	0
$679$	411.497i	0.606034i
$680$	0	0
$681$	123.228	0.180951
$682$	0	0
$683$	321.002i	0.469989i	0.971997	+	0.234994i	$0.0755072\pi$
$683$	321.002i	0.469989i	−0.971997	+	0.234994i	$0.924493\pi$
$684$	0	0
$685$	208.461	0.304323
$686$	0	0
$687$	− 91.1457i	− 0.132672i
$688$	0	0
$689$	−29.1741	−0.0423427
$690$	0	0
$691$	− 82.7256i	− 0.119719i	−0.998207	−	0.0598593i	$-0.980935\pi$
$691$	− 82.7256i	− 0.119719i	0.998207	−	0.0598593i	$-0.0190652\pi$
$692$	0	0
$693$	7.89264	0.0113891
$694$	0	0
$695$	− 367.716i	− 0.529088i
$696$	0	0
$697$	83.8503	0.120302
$698$	0	0
$699$	464.231i	0.664136i
$700$	0	0
$701$	110.691	0.157905	0.0789523	−	0.996878i	$-0.474843\pi$
$701$	110.691	0.157905	0.0789523	+	0.996878i	$0.474843\pi$
$702$	0	0
$703$	− 1491.46i	− 2.12156i
$704$	0	0
$705$	8.18245	0.0116063
$706$	0	0
$707$	504.238i	0.713208i
$708$	0	0
$709$	−857.208	−1.20904	−0.604519	−	0.796590i	$-0.706635\pi$
$709$	−857.208	−1.20904	−0.604519	+	0.796590i	$0.706635\pi$
$710$	0	0
$711$	128.162i	0.180255i
$712$	0	0
$713$	−833.308	−1.16873
$714$	0	0
$715$	26.0856i	0.0364834i
$716$	0	0
$717$	−92.4550	−0.128947
$718$	0	0
$719$	437.760i	0.608845i	0.952537	+	0.304423i	$0.0984635\pi$
$719$	437.760i	0.608845i	−0.952537	+	0.304423i	$0.901536\pi$
$720$	0	0
$721$	−94.4645	−0.131019
$722$	0	0
$723$	775.819i	1.07306i
$724$	0	0
$725$	247.800	0.341793
$726$	0	0
$727$	1046.25i	1.43913i	0.694426	+	0.719564i	$0.255658\pi$
$727$	1046.25i	1.43913i	−0.694426	+	0.719564i	$0.744342\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 950.080i	− 1.29970i
$732$	0	0
$733$	−30.8518	−0.0420898	−0.0210449	−	0.999779i	$-0.506699\pi$
$733$	−30.8518	−0.0420898	−0.0210449	+	0.999779i	$0.506699\pi$
$734$	0	0
$735$	− 27.1109i	− 0.0368856i
$736$	0	0
$737$	76.4764	0.103767
$738$	0	0
$739$	281.630i	0.381097i	0.981678	+	0.190548i	$0.0610266\pi$
$739$	281.630i	0.381097i	−0.981678	+	0.190548i	$0.938973\pi$
$740$	0	0
$741$	642.873	0.867574
$742$	0	0
$743$	883.156i	1.18863i	0.804231	+	0.594317i	$0.202578\pi$
$743$	883.156i	1.18863i	−0.804231	+	0.594317i	$0.797422\pi$
$744$	0	0
$745$	−234.872	−0.315264
$746$	0	0
$747$	20.2224i	0.0270714i
$748$	0	0
$749$	271.261	0.362164
$750$	0	0
$751$	− 1165.59i	− 1.55205i	−0.630705	−	0.776023i	$-0.717234\pi$
$751$	− 1165.59i	− 1.55205i	0.630705	−	0.776023i	$-0.282766\pi$
$752$	0	0
$753$	−366.510	−0.486733
$754$	0	0
$755$	87.6805i	0.116133i
$756$	0	0
$757$	−240.068	−0.317131	−0.158566	−	0.987348i	$-0.550687\pi$
$757$	−240.068	−0.317131	−0.158566	+	0.987348i	$0.550687\pi$
$758$	0	0
$759$	30.7161i	0.0404691i
$760$	0	0
$761$	−88.8965	−0.116815	−0.0584077	−	0.998293i	$-0.518602\pi$
$761$	−88.8965	−0.116815	−0.0584077	+	0.998293i	$0.518602\pi$
$762$	0	0
$763$	− 80.0253i	− 0.104882i
$764$	0	0
$765$	97.8137	0.127861
$766$	0	0
$767$	562.914i	0.733916i
$768$	0	0
$769$	−1048.03	−1.36285	−0.681427	−	0.731886i	$-0.738640\pi$
$769$	−1048.03	−1.36285	−0.681427	+	0.731886i	$0.738640\pi$
$770$	0	0
$771$	500.638i	0.649337i
$772$	0	0
$773$	328.781	0.425331	0.212666	−	0.977125i	$-0.431786\pi$
$773$	328.781	0.425331	0.212666	+	0.977125i	$0.431786\pi$
$774$	0	0
$775$	− 233.627i	− 0.301454i
$776$	0	0
$777$	−216.033	−0.278035
$778$	0	0
$779$	181.933i	0.233547i
$780$	0	0
$781$	33.0613	0.0423320
$782$	0	0
$783$	− 257.521i	− 0.328890i
$784$	0	0
$785$	−82.6185	−0.105247
$786$	0	0
$787$	− 757.713i	− 0.962787i	−0.876505	−	0.481393i	$-0.840131\pi$
$787$	− 757.713i	− 0.962787i	0.876505	−	0.481393i	$-0.159869\pi$
$788$	0	0
$789$	−843.700	−1.06933
$790$	0	0
$791$	− 503.203i	− 0.636161i
$792$	0	0
$793$	−363.805	−0.458770
$794$	0	0
$795$	− 9.63117i	− 0.0121147i
$796$	0	0
$797$	−315.973	−0.396453	−0.198227	−	0.980156i	$-0.563518\pi$
$797$	−315.973	−0.396453	−0.198227	+	0.980156i	$0.563518\pi$
$798$	0	0
$799$	30.8057i	0.0385553i
$800$	0	0
$801$	−452.203	−0.564549
$802$	0	0
$803$	− 47.9243i	− 0.0596815i
$804$	0	0
$805$	105.508	0.131066
$806$	0	0
$807$	− 365.429i	− 0.452823i
$808$	0	0
$809$	−686.413	−0.848471	−0.424235	−	0.905552i	$-0.639457\pi$
$809$	−686.413	−0.848471	−0.424235	+	0.905552i	$0.639457\pi$
$810$	0	0
$811$	− 480.889i	− 0.592958i	−0.955039	−	0.296479i	$-0.904187\pi$
$811$	− 480.889i	− 0.592958i	0.955039	−	0.296479i	$-0.0958125\pi$
$812$	0	0
$813$	−172.938	−0.212716
$814$	0	0
$815$	158.022i	0.193892i
$816$	0	0
$817$	2061.42	2.52316
$818$	0	0
$819$	− 93.1181i	− 0.113697i
$820$	0	0
$821$	−1147.09	−1.39719	−0.698593	−	0.715519i	$-0.746190\pi$
$821$	−1147.09	−1.39719	−0.698593	+	0.715519i	$0.746190\pi$
$822$	0	0
$823$	509.637i	0.619243i	0.950860	+	0.309621i	$0.100202\pi$
$823$	509.637i	0.619243i	−0.950860	+	0.309621i	$0.899798\pi$
$824$	0	0
$825$	−8.61158	−0.0104383
$826$	0	0
$827$	15.6299i	0.0188995i	0.999955	+	0.00944974i	$0.00300799\pi$
$827$	15.6299i	0.0188995i	−0.999955	+	0.00944974i	$0.996992\pi$
$828$	0	0
$829$	133.845	0.161454	0.0807268	−	0.996736i	$-0.474276\pi$
$829$	133.845	0.161454	0.0807268	+	0.996736i	$0.474276\pi$
$830$	0	0
$831$	444.036i	0.534340i
$832$	0	0
$833$	102.068	0.122531
$834$	0	0
$835$	562.238i	0.673339i
$836$	0	0
$837$	−242.792	−0.290074
$838$	0	0
$839$	818.045i	0.975024i	0.873116	+	0.487512i	$0.162095\pi$
$839$	818.045i	0.975024i	−0.873116	+	0.487512i	$0.837905\pi$
$840$	0	0
$841$	1615.19	1.92056
$842$	0	0
$843$	− 532.303i	− 0.631439i
$844$	0	0
$845$	−70.1350	−0.0830000
$846$	0	0
$847$	317.520i	0.374876i
$848$	0	0
$849$	245.528	0.289197
$850$	0	0
$851$	− 840.744i	− 0.987948i
$852$	0	0
$853$	1142.17	1.33901	0.669503	−	0.742810i	$-0.266507\pi$
$853$	1142.17	1.33901	0.669503	+	0.742810i	$0.266507\pi$
$854$	0	0
$855$	212.230i	0.248222i
$856$	0	0
$857$	1259.18	1.46929	0.734645	−	0.678452i	$-0.237349\pi$
$857$	1259.18	1.46929	0.734645	+	0.678452i	$0.237349\pi$
$858$	0	0
$859$	535.939i	0.623910i	0.950097	+	0.311955i	$0.100984\pi$
$859$	535.939i	0.623910i	−0.950097	+	0.311955i	$0.899016\pi$
$860$	0	0
$861$	26.3524	0.0306068
$862$	0	0
$863$	− 1375.51i	− 1.59387i	−0.604065	−	0.796935i	$-0.706453\pi$
$863$	− 1375.51i	− 1.59387i	0.604065	−	0.796935i	$-0.293547\pi$
$864$	0	0
$865$	168.049	0.194276
$866$	0	0
$867$	− 132.309i	− 0.152605i
$868$	0	0
$869$	42.4804	0.0488842
$870$	0	0
$871$	− 902.276i	− 1.03591i
$872$	0	0
$873$	−466.594	−0.534472
$874$	0	0
$875$	29.5804i	0.0338062i
$876$	0	0
$877$	445.165	0.507600	0.253800	−	0.967257i	$-0.418319\pi$
$877$	445.165	0.507600	0.253800	+	0.967257i	$0.418319\pi$
$878$	0	0
$879$	− 438.461i	− 0.498817i
$880$	0	0
$881$	−518.052	−0.588027	−0.294014	−	0.955801i	$-0.594991\pi$
$881$	−518.052	−0.588027	−0.294014	+	0.955801i	$0.594991\pi$
$882$	0	0
$883$	549.847i	0.622703i	0.950295	+	0.311351i	$0.100782\pi$
$883$	549.847i	0.622703i	−0.950295	+	0.311351i	$0.899218\pi$
$884$	0	0
$885$	−185.833	−0.209981
$886$	0	0
$887$	− 92.7547i	− 0.104571i	−0.998632	−	0.0522856i	$-0.983349\pi$
$887$	− 92.7547i	− 0.104571i	0.998632	−	0.0522856i	$-0.0166506\pi$
$888$	0	0
$889$	−521.769	−0.586917
$890$	0	0
$891$	8.94941i	0.0100442i
$892$	0	0
$893$	−66.8403	−0.0748491
$894$	0	0
$895$	306.030i	0.341932i
$896$	0	0
$897$	362.391	0.404003
$898$	0	0
$899$	− 2315.71i	− 2.57587i
$900$	0	0
$901$	36.2599	0.0402441
$902$	0	0
$903$	− 298.591i	− 0.330665i
$904$	0	0
$905$	−133.684	−0.147718
$906$	0	0
$907$	− 468.980i	− 0.517067i	−0.966002	−	0.258533i	$-0.916761\pi$
$907$	− 468.980i	− 0.517067i	0.966002	−	0.258533i	$-0.0832392\pi$
$908$	0	0
$909$	−571.752	−0.628990
$910$	0	0
$911$	− 988.736i	− 1.08533i	−0.839949	−	0.542665i	$-0.817415\pi$
$911$	− 988.736i	− 1.08533i	0.839949	−	0.542665i	$-0.182585\pi$
$912$	0	0
$913$	6.70290	0.00734162
$914$	0	0
$915$	− 120.102i	− 0.131259i
$916$	0	0
$917$	−216.392	−0.235978
$918$	0	0
$919$	− 90.9640i	− 0.0989815i	−0.998775	−	0.0494908i	$-0.984240\pi$
$919$	− 90.9640i	− 0.0989815i	0.998775	−	0.0494908i	$-0.0157598\pi$
$920$	0	0
$921$	389.863	0.423304
$922$	0	0
$923$	− 390.061i	− 0.422601i
$924$	0	0
$925$	235.711	0.254823
$926$	0	0
$927$	− 107.113i	− 0.115548i
$928$	0	0
$929$	148.918	0.160299	0.0801497	−	0.996783i	$-0.474460\pi$
$929$	148.918	0.160299	0.0801497	+	0.996783i	$0.474460\pi$
$930$	0	0
$931$	221.462i	0.237875i
$932$	0	0
$933$	832.963	0.892779
$934$	0	0
$935$	− 32.4213i	− 0.0346752i
$936$	0	0
$937$	671.087	0.716208	0.358104	−	0.933682i	$-0.383423\pi$
$937$	671.087	0.716208	0.358104	+	0.933682i	$0.383423\pi$
$938$	0	0
$939$	− 583.171i	− 0.621055i
$940$	0	0
$941$	−1521.48	−1.61688	−0.808439	−	0.588580i	$-0.799687\pi$
$941$	−1521.48	−1.61688	−0.808439	+	0.588580i	$0.799687\pi$
$942$	0	0
$943$	102.557i	0.108756i
$944$	0	0
$945$	30.7409	0.0325300
$946$	0	0
$947$	1390.42i	1.46823i	0.679024	+	0.734116i	$0.262403\pi$
$947$	1390.42i	1.46823i	−0.679024	+	0.734116i	$0.737597\pi$
$948$	0	0
$949$	−565.415	−0.595801
$950$	0	0
$951$	143.747i	0.151153i
$952$	0	0
$953$	−987.802	−1.03652	−0.518259	−	0.855224i	$-0.673420\pi$
$953$	−987.802	−1.03652	−0.518259	+	0.855224i	$0.673420\pi$
$954$	0	0
$955$	165.172i	0.172955i
$956$	0	0
$957$	−85.3579	−0.0891932
$958$	0	0
$959$	246.654i	0.257200i
$960$	0	0
$961$	−1222.26	−1.27186
$962$	0	0
$963$	307.581i	0.319398i
$964$	0	0
$965$	778.019	0.806237
$966$	0	0
$967$	740.025i	0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$967$	740.025i	0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$968$	0	0
$969$	−799.014	−0.824576
$970$	0	0
$971$	− 58.6445i	− 0.0603960i	−0.999544	−	0.0301980i	$-0.990386\pi$
$971$	− 58.6445i	− 0.0603960i	0.999544	−	0.0301980i	$-0.00961378\pi$
$972$	0	0
$973$	435.087	0.447161
$974$	0	0
$975$	101.600i	0.104205i
$976$	0	0
$977$	1644.71	1.68343	0.841714	−	0.539923i	$-0.181547\pi$
$977$	1644.71	1.68343	0.841714	+	0.539923i	$0.181547\pi$
$978$	0	0
$979$	149.887i	0.153102i
$980$	0	0
$981$	90.7402	0.0924976
$982$	0	0
$983$	738.274i	0.751042i	0.926814	+	0.375521i	$0.122536\pi$
$983$	738.274i	0.751042i	−0.926814	+	0.375521i	$0.877464\pi$
$984$	0	0
$985$	608.677	0.617946
$986$	0	0
$987$	9.68161i	0.00980913i
$988$	0	0
$989$	1162.04	1.17496
$990$	0	0
$991$	838.030i	0.845640i	0.906214	+	0.422820i	$0.138960\pi$
$991$	838.030i	0.845640i	−0.906214	+	0.422820i	$0.861040\pi$
$992$	0	0
$993$	610.733	0.615038
$994$	0	0
$995$	− 439.097i	− 0.441303i
$996$	0	0
$997$	1206.20	1.20983	0.604913	−	0.796291i	$-0.293208\pi$
$997$	1206.20	1.20983	0.604913	+	0.796291i	$0.293208\pi$
$998$	0	0
$999$	− 244.959i	− 0.245204i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.n.c.1471.16	yes	16
4.3	odd	2	inner	1680.3.n.c.1471.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.3.n.c.1471.5	✓	16	4.3	odd	2	inner
1680.3.n.c.1471.16	yes	16	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1680.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +2.23607 q^{5} +2.64575i q^{7} -3.00000 q^{9} +0.994379i q^{11} +11.7318 q^{13} +3.87298i q^{15} -14.5812 q^{17} -31.6374i q^{19} -4.58258 q^{21} -17.8342i q^{23} +5.00000 q^{25} -5.19615i q^{27} +49.5599 q^{29} -46.7254i q^{31} -1.72232 q^{33} +5.91608i q^{35} +47.1423 q^{37} +20.3200i q^{39} -5.75058 q^{41} +65.1579i q^{43} -6.70820 q^{45} -2.11270i q^{47} -7.00000 q^{49} -25.2554i q^{51} -2.48676 q^{53} +2.22350i q^{55} +54.7975 q^{57} +47.9819i q^{59} -31.0102 q^{61} -7.93725i q^{63} +26.2331 q^{65} -76.9087i q^{67} +30.8897 q^{69} -33.2482i q^{71} -48.1952 q^{73} +8.66025i q^{75} -2.63088 q^{77} -42.7205i q^{79} +9.00000 q^{81} -6.74079i q^{83} -32.6046 q^{85} +85.8403i q^{87} +150.734 q^{89} +31.0394i q^{91} +80.9307 q^{93} -70.7433i q^{95} +155.531 q^{97} -2.98314i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} + 48 q^{13} - 96 q^{17} + 80 q^{25} + 112 q^{29} + 48 q^{33} - 224 q^{37} + 160 q^{41} - 112 q^{49} - 96 q^{53} + 80 q^{61} - 80 q^{65} + 48 q^{69} - 48 q^{73} + 144 q^{81} - 160 q^{89}+ \cdots + 592 q^{97}+O(q^{100}) \) 16 * q - 48 * q^9 + 48 * q^13 - 96 * q^17 + 80 * q^25 + 112 * q^29 + 48 * q^33 - 224 * q^37 + 160 * q^41 - 112 * q^49 - 96 * q^53 + 80 * q^61 - 80 * q^65 + 48 * q^69 - 48 * q^73 + 144 * q^81 - 160 * q^89 + 592 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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