LMFDB - Embedded newform 1728.3.g.a.703.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(703,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, 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("1728.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1728.703
Dual form			1728.3.g.a.703.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.00000 q^{5} -8.66025i q^{7} +O(q^{10})$	$q-7.00000 q^{5} -8.66025i q^{7} -8.66025i q^{11} -20.0000 q^{13} -8.00000 q^{17} -10.3923i q^{19} -3.46410i q^{23} +24.0000 q^{25} -10.0000 q^{29} -53.6936i q^{31} +60.6218i q^{35} +10.0000 q^{37} -50.0000 q^{41} -17.3205i q^{43} +86.6025i q^{47} -26.0000 q^{49} +47.0000 q^{53} +60.6218i q^{55} +34.6410i q^{59} +64.0000 q^{61} +140.000 q^{65} -86.6025i q^{67} -55.0000 q^{73} -75.0000 q^{77} +6.92820i q^{79} -29.4449i q^{83} +56.0000 q^{85} +10.0000 q^{89} +173.205i q^{91} +72.7461i q^{95} -25.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{5}+O(q^{10}) $ 2 * q - 14 * q^5	$ 2 q - 14 q^{5} - 40 q^{13} - 16 q^{17} + 48 q^{25} - 20 q^{29} + 20 q^{37} - 100 q^{41} - 52 q^{49} + 94 q^{53} + 128 q^{61} + 280 q^{65} - 110 q^{73} - 150 q^{77} + 112 q^{85} + 20 q^{89} - 50 q^{97}+O(q^{100}) $ 2 * q - 14 * q^5 - 40 * q^13 - 16 * q^17 + 48 * q^25 - 20 * q^29 + 20 * q^37 - 100 * q^41 - 52 * q^49 + 94 * q^53 + 128 * q^61 + 280 * q^65 - 110 * q^73 - 150 * q^77 + 112 * q^85 + 20 * q^89 - 50 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$1$

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	0
$3$	0	0
$4$	0	0
$5$	−7.00000	−1.40000	−0.700000	−	0.714143i	$-0.746817\pi$
$5$	−7.00000	−1.40000	−0.700000	+	0.714143i	$0.746817\pi$
$6$	0	0
$7$	− 8.66025i	− 1.23718i	−0.785714	−	0.618590i	$-0.787704\pi$
$7$	− 8.66025i	− 1.23718i	0.785714	−	0.618590i	$-0.212296\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 8.66025i	− 0.787296i	−0.919261	−	0.393648i	$-0.871213\pi$
$11$	− 8.66025i	− 0.787296i	0.919261	−	0.393648i	$-0.128787\pi$
$12$	0	0
$13$	−20.0000	−1.53846	−0.769231	−	0.638971i	$-0.779360\pi$
$13$	−20.0000	−1.53846	−0.769231	+	0.638971i	$0.779360\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−8.00000	−0.470588	−0.235294	−	0.971924i	$-0.575605\pi$
$17$	−8.00000	−0.470588	−0.235294	+	0.971924i	$0.575605\pi$
$18$	0	0
$19$	− 10.3923i	− 0.546963i	−0.961877	−	0.273482i	$-0.911825\pi$
$19$	− 10.3923i	− 0.546963i	0.961877	−	0.273482i	$-0.0881753\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.46410i	− 0.150613i	−0.997160	−	0.0753066i	$-0.976006\pi$
$23$	− 3.46410i	− 0.150613i	0.997160	−	0.0753066i	$-0.0239935\pi$
$24$	0	0
$25$	24.0000	0.960000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−10.0000	−0.344828	−0.172414	−	0.985025i	$-0.555157\pi$
$29$	−10.0000	−0.344828	−0.172414	+	0.985025i	$0.555157\pi$
$30$	0	0
$31$	− 53.6936i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$31$	− 53.6936i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	60.6218i	1.73205i
$36$	0	0
$37$	10.0000	0.270270	0.135135	−	0.990827i	$-0.456853\pi$
$37$	10.0000	0.270270	0.135135	+	0.990827i	$0.456853\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−50.0000	−1.21951	−0.609756	−	0.792589i	$-0.708733\pi$
$41$	−50.0000	−1.21951	−0.609756	+	0.792589i	$0.708733\pi$
$42$	0	0
$43$	− 17.3205i	− 0.402803i	−0.979509	−	0.201401i	$-0.935450\pi$
$43$	− 17.3205i	− 0.402803i	0.979509	−	0.201401i	$-0.0645495\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	86.6025i	1.84261i	0.388844	+	0.921304i	$0.372875\pi$
$47$	86.6025i	1.84261i	−0.388844	+	0.921304i	$0.627125\pi$
$48$	0	0
$49$	−26.0000	−0.530612
$50$	0	0
$51$	0	0
$52$	0	0
$53$	47.0000	0.886792	0.443396	−	0.896326i	$-0.353773\pi$
$53$	47.0000	0.886792	0.443396	+	0.896326i	$0.353773\pi$
$54$	0	0
$55$	60.6218i	1.10221i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	34.6410i	0.587136i	0.955938	+	0.293568i	$0.0948427\pi$
$59$	34.6410i	0.587136i	−0.955938	+	0.293568i	$0.905157\pi$
$60$	0	0
$61$	64.0000	1.04918	0.524590	−	0.851355i	$-0.324219\pi$
$61$	64.0000	1.04918	0.524590	+	0.851355i	$0.324219\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	140.000	2.15385
$66$	0	0
$67$	− 86.6025i	− 1.29258i	−0.763094	−	0.646288i	$-0.776321\pi$
$67$	− 86.6025i	− 1.29258i	0.763094	−	0.646288i	$-0.223679\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−55.0000	−0.753425	−0.376712	−	0.926330i	$-0.622945\pi$
$73$	−55.0000	−0.753425	−0.376712	+	0.926330i	$0.622945\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−75.0000	−0.974026
$78$	0	0
$79$	6.92820i	0.0876988i	0.999038	+	0.0438494i	$0.0139622\pi$
$79$	6.92820i	0.0876988i	−0.999038	+	0.0438494i	$0.986038\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 29.4449i	− 0.354757i	−0.984143	−	0.177379i	$-0.943238\pi$
$83$	− 29.4449i	− 0.354757i	0.984143	−	0.177379i	$-0.0567617\pi$
$84$	0	0
$85$	56.0000	0.658824
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.0000	0.112360	0.0561798	−	0.998421i	$-0.482108\pi$
$89$	10.0000	0.112360	0.0561798	+	0.998421i	$0.482108\pi$
$90$	0	0
$91$	173.205i	1.90335i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	72.7461i	0.765749i
$96$	0	0
$97$	−25.0000	−0.257732	−0.128866	−	0.991662i	$-0.541134\pi$
$97$	−25.0000	−0.257732	−0.128866	+	0.991662i	$0.541134\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	155.000	1.53465	0.767327	−	0.641256i	$-0.221586\pi$
$101$	155.000	1.53465	0.767327	+	0.641256i	$0.221586\pi$
$102$	0	0
$103$	138.564i	1.34528i	0.739969	+	0.672641i	$0.234840\pi$
$103$	138.564i	1.34528i	−0.739969	+	0.672641i	$0.765160\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	129.904i	1.21405i	0.794681	+	0.607027i	$0.207638\pi$
$107$	129.904i	1.21405i	−0.794681	+	0.607027i	$0.792362\pi$
$108$	0	0
$109$	−134.000	−1.22936	−0.614679	−	0.788777i	$-0.710714\pi$
$109$	−134.000	−1.22936	−0.614679	+	0.788777i	$0.710714\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−74.0000	−0.654867	−0.327434	−	0.944874i	$-0.606184\pi$
$113$	−74.0000	−0.654867	−0.327434	+	0.944874i	$0.606184\pi$
$114$	0	0
$115$	24.2487i	0.210858i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	69.2820i	0.582202i
$120$	0	0
$121$	46.0000	0.380165
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.00000	0.0560000
$126$	0	0
$127$	− 25.9808i	− 0.204573i	−0.994755	−	0.102286i	$-0.967384\pi$
$127$	− 25.9808i	− 0.204573i	0.994755	−	0.102286i	$-0.0326158\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	164.545i	1.25607i	0.778186	+	0.628034i	$0.216140\pi$
$131$	164.545i	1.25607i	−0.778186	+	0.628034i	$0.783860\pi$
$132$	0	0
$133$	−90.0000	−0.676692
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−62.0000	−0.452555	−0.226277	−	0.974063i	$-0.572656\pi$
$137$	−62.0000	−0.452555	−0.226277	+	0.974063i	$0.572656\pi$
$138$	0	0
$139$	173.205i	1.24608i	0.782190	+	0.623040i	$0.214103\pi$
$139$	173.205i	1.24608i	−0.782190	+	0.623040i	$0.785897\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	173.205i	1.21122i
$144$	0	0
$145$	70.0000	0.482759
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−115.000	−0.771812	−0.385906	−	0.922538i	$-0.626111\pi$
$149$	−115.000	−0.771812	−0.385906	+	0.922538i	$0.626111\pi$
$150$	0	0
$151$	43.3013i	0.286763i	0.989667	+	0.143382i	$0.0457977\pi$
$151$	43.3013i	0.286763i	−0.989667	+	0.143382i	$0.954202\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	375.855i	2.42487i
$156$	0	0
$157$	−20.0000	−0.127389	−0.0636943	−	0.997969i	$-0.520288\pi$
$157$	−20.0000	−0.127389	−0.0636943	+	0.997969i	$0.520288\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−30.0000	−0.186335
$162$	0	0
$163$	− 103.923i	− 0.637565i	−0.947828	−	0.318782i	$-0.896726\pi$
$163$	− 103.923i	− 0.637565i	0.947828	−	0.318782i	$-0.103274\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	245.951i	1.47276i	0.676567	+	0.736381i	$0.263467\pi$
$167$	245.951i	1.47276i	−0.676567	+	0.736381i	$0.736533\pi$
$168$	0	0
$169$	231.000	1.36686
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−127.000	−0.734104	−0.367052	−	0.930200i	$-0.619633\pi$
$173$	−127.000	−0.734104	−0.367052	+	0.930200i	$0.619633\pi$
$174$	0	0
$175$	− 207.846i	− 1.18769i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 233.827i	− 1.30630i	−0.757231	−	0.653148i	$-0.773448\pi$
$179$	− 233.827i	− 1.30630i	0.757231	−	0.653148i	$-0.226552\pi$
$180$	0	0
$181$	−56.0000	−0.309392	−0.154696	−	0.987962i	$-0.549440\pi$
$181$	−56.0000	−0.309392	−0.154696	+	0.987962i	$0.549440\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−70.0000	−0.378378
$186$	0	0
$187$	69.2820i	0.370492i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	34.6410i	0.181367i	0.995880	+	0.0906833i	$0.0289051\pi$
$191$	34.6410i	0.181367i	−0.995880	+	0.0906833i	$0.971095\pi$
$192$	0	0
$193$	65.0000	0.336788	0.168394	−	0.985720i	$-0.446142\pi$
$193$	65.0000	0.336788	0.168394	+	0.985720i	$0.446142\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−253.000	−1.28426	−0.642132	−	0.766594i	$-0.721950\pi$
$197$	−253.000	−1.28426	−0.642132	+	0.766594i	$0.721950\pi$
$198$	0	0
$199$	− 129.904i	− 0.652783i	−0.945235	−	0.326391i	$-0.894167\pi$
$199$	− 129.904i	− 0.652783i	0.945235	−	0.326391i	$-0.105833\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	86.6025i	0.426613i
$204$	0	0
$205$	350.000	1.70732
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−90.0000	−0.430622
$210$	0	0
$211$	− 148.956i	− 0.705954i	−0.935632	−	0.352977i	$-0.885169\pi$
$211$	− 148.956i	− 0.705954i	0.935632	−	0.352977i	$-0.114831\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	121.244i	0.563924i
$216$	0	0
$217$	−465.000	−2.14286
$218$	0	0
$219$	0	0
$220$	0	0
$221$	160.000	0.723982
$222$	0	0
$223$	− 34.6410i	− 0.155341i	−0.996979	−	0.0776704i	$-0.975252\pi$
$223$	− 34.6410i	− 0.155341i	0.996979	−	0.0776704i	$-0.0247482\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	90.0666i	0.396769i	0.980124	+	0.198385i	$0.0635695\pi$
$227$	90.0666i	0.396769i	−0.980124	+	0.198385i	$0.936430\pi$
$228$	0	0
$229$	−146.000	−0.637555	−0.318777	−	0.947830i	$-0.603272\pi$
$229$	−146.000	−0.637555	−0.318777	+	0.947830i	$0.603272\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	334.000	1.43348	0.716738	−	0.697342i	$-0.245634\pi$
$233$	334.000	1.43348	0.716738	+	0.697342i	$0.245634\pi$
$234$	0	0
$235$	− 606.218i	− 2.57965i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.3205i	0.0724707i	0.999343	+	0.0362354i	$0.0115366\pi$
$239$	17.3205i	0.0724707i	−0.999343	+	0.0362354i	$0.988463\pi$
$240$	0	0
$241$	134.000	0.556017	0.278008	−	0.960579i	$-0.410326\pi$
$241$	134.000	0.556017	0.278008	+	0.960579i	$0.410326\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	182.000	0.742857
$246$	0	0
$247$	207.846i	0.841482i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	207.846i	0.828072i	0.910260	+	0.414036i	$0.135881\pi$
$251$	207.846i	0.828072i	−0.910260	+	0.414036i	$0.864119\pi$
$252$	0	0
$253$	−30.0000	−0.118577
$254$	0	0
$255$	0	0
$256$	0	0
$257$	268.000	1.04280	0.521401	−	0.853312i	$-0.325410\pi$
$257$	268.000	1.04280	0.521401	+	0.853312i	$0.325410\pi$
$258$	0	0
$259$	− 86.6025i	− 0.334373i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 433.013i	− 1.64644i	−0.567725	−	0.823218i	$-0.692176\pi$
$263$	− 433.013i	− 1.64644i	0.567725	−	0.823218i	$-0.307824\pi$
$264$	0	0
$265$	−329.000	−1.24151
$266$	0	0
$267$	0	0
$268$	0	0
$269$	350.000	1.30112	0.650558	−	0.759457i	$-0.274535\pi$
$269$	350.000	1.30112	0.650558	+	0.759457i	$0.274535\pi$
$270$	0	0
$271$	− 36.3731i	− 0.134218i	−0.997746	−	0.0671090i	$-0.978622\pi$
$271$	− 36.3731i	− 0.134218i	0.997746	−	0.0671090i	$-0.0213775\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 207.846i	− 0.755804i
$276$	0	0
$277$	520.000	1.87726	0.938628	−	0.344931i	$-0.112098\pi$
$277$	520.000	1.87726	0.938628	+	0.344931i	$0.112098\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−440.000	−1.56584	−0.782918	−	0.622125i	$-0.786270\pi$
$281$	−440.000	−1.56584	−0.782918	+	0.622125i	$0.786270\pi$
$282$	0	0
$283$	329.090i	1.16286i	0.813596	+	0.581430i	$0.197507\pi$
$283$	329.090i	1.16286i	−0.813596	+	0.581430i	$0.802493\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	433.013i	1.50876i
$288$	0	0
$289$	−225.000	−0.778547
$290$	0	0
$291$	0	0
$292$	0	0
$293$	218.000	0.744027	0.372014	−	0.928227i	$-0.378667\pi$
$293$	218.000	0.744027	0.372014	+	0.928227i	$0.378667\pi$
$294$	0	0
$295$	− 242.487i	− 0.821990i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	69.2820i	0.231712i
$300$	0	0
$301$	−150.000	−0.498339
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−448.000	−1.46885
$306$	0	0
$307$	207.846i	0.677023i	0.940962	+	0.338512i	$0.109923\pi$
$307$	207.846i	0.677023i	−0.940962	+	0.338512i	$0.890077\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 294.449i	− 0.946780i	−0.880853	−	0.473390i	$-0.843030\pi$
$311$	− 294.449i	− 0.946780i	0.880853	−	0.473390i	$-0.156970\pi$
$312$	0	0
$313$	485.000	1.54952	0.774760	−	0.632255i	$-0.217870\pi$
$313$	485.000	1.54952	0.774760	+	0.632255i	$0.217870\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−217.000	−0.684543	−0.342271	−	0.939601i	$-0.611196\pi$
$317$	−217.000	−0.684543	−0.342271	+	0.939601i	$0.611196\pi$
$318$	0	0
$319$	86.6025i	0.271481i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	83.1384i	0.257395i
$324$	0	0
$325$	−480.000	−1.47692
$326$	0	0
$327$	0	0
$328$	0	0
$329$	750.000	2.27964
$330$	0	0
$331$	− 433.013i	− 1.30820i	−0.756410	−	0.654098i	$-0.773048\pi$
$331$	− 433.013i	− 1.30820i	0.756410	−	0.654098i	$-0.226952\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	606.218i	1.80961i
$336$	0	0
$337$	−310.000	−0.919881	−0.459941	−	0.887950i	$-0.652129\pi$
$337$	−310.000	−0.919881	−0.459941	+	0.887950i	$0.652129\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−465.000	−1.36364
$342$	0	0
$343$	− 199.186i	− 0.580717i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	216.506i	0.623938i	0.950092	+	0.311969i	$0.100988\pi$
$347$	216.506i	0.623938i	−0.950092	+	0.311969i	$0.899012\pi$
$348$	0	0
$349$	−74.0000	−0.212034	−0.106017	−	0.994364i	$-0.533810\pi$
$349$	−74.0000	−0.212034	−0.106017	+	0.994364i	$0.533810\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	394.000	1.11615	0.558074	−	0.829791i	$-0.311541\pi$
$353$	394.000	1.11615	0.558074	+	0.829791i	$0.311541\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 571.577i	− 1.59214i	−0.605207	−	0.796068i	$-0.706910\pi$
$359$	− 571.577i	− 1.59214i	0.605207	−	0.796068i	$-0.293090\pi$
$360$	0	0
$361$	253.000	0.700831
$362$	0	0
$363$	0	0
$364$	0	0
$365$	385.000	1.05479
$366$	0	0
$367$	562.917i	1.53383i	0.641747	+	0.766916i	$0.278210\pi$
$367$	562.917i	1.53383i	−0.641747	+	0.766916i	$0.721790\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 407.032i	− 1.09712i
$372$	0	0
$373$	40.0000	0.107239	0.0536193	−	0.998561i	$-0.482924\pi$
$373$	40.0000	0.107239	0.0536193	+	0.998561i	$0.482924\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	200.000	0.530504
$378$	0	0
$379$	685.892i	1.80974i	0.425686	+	0.904871i	$0.360033\pi$
$379$	685.892i	1.80974i	−0.425686	+	0.904871i	$0.639967\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	360.267i	0.940644i	0.882495	+	0.470322i	$0.155862\pi$
$383$	360.267i	0.940644i	−0.882495	+	0.470322i	$0.844138\pi$
$384$	0	0
$385$	525.000	1.36364
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−475.000	−1.22108	−0.610540	−	0.791986i	$-0.709047\pi$
$389$	−475.000	−1.22108	−0.610540	+	0.791986i	$0.709047\pi$
$390$	0	0
$391$	27.7128i	0.0708768i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 48.4974i	− 0.122778i
$396$	0	0
$397$	−260.000	−0.654912	−0.327456	−	0.944866i	$-0.606191\pi$
$397$	−260.000	−0.654912	−0.327456	+	0.944866i	$0.606191\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−740.000	−1.84539	−0.922693	−	0.385535i	$-0.874017\pi$
$401$	−740.000	−1.84539	−0.922693	+	0.385535i	$0.874017\pi$
$402$	0	0
$403$	1073.87i	2.66469i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 86.6025i	− 0.212783i
$408$	0	0
$409$	659.000	1.61125	0.805623	−	0.592428i	$-0.201830\pi$
$409$	659.000	1.61125	0.805623	+	0.592428i	$0.201830\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	300.000	0.726392
$414$	0	0
$415$	206.114i	0.496660i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 588.897i	− 1.40548i	−0.711445	−	0.702741i	$-0.751959\pi$
$419$	− 588.897i	− 1.40548i	0.711445	−	0.702741i	$-0.248041\pi$
$420$	0	0
$421$	496.000	1.17815	0.589074	−	0.808079i	$-0.299493\pi$
$421$	496.000	1.17815	0.589074	+	0.808079i	$0.299493\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−192.000	−0.451765
$426$	0	0
$427$	− 554.256i	− 1.29802i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 571.577i	− 1.32616i	−0.748547	−	0.663082i	$-0.769248\pi$
$431$	− 571.577i	− 1.32616i	0.748547	−	0.663082i	$-0.230752\pi$
$432$	0	0
$433$	−235.000	−0.542725	−0.271363	−	0.962477i	$-0.587474\pi$
$433$	−235.000	−0.542725	−0.271363	+	0.962477i	$0.587474\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−36.0000	−0.0823799
$438$	0	0
$439$	− 413.960i	− 0.942962i	−0.881876	−	0.471481i	$-0.843720\pi$
$439$	− 413.960i	− 0.942962i	0.881876	−	0.471481i	$-0.156280\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 575.041i	− 1.29806i	−0.760763	−	0.649030i	$-0.775175\pi$
$443$	− 575.041i	− 1.29806i	0.760763	−	0.649030i	$-0.224825\pi$
$444$	0	0
$445$	−70.0000	−0.157303
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−470.000	−1.04677	−0.523385	−	0.852096i	$-0.675331\pi$
$449$	−470.000	−1.04677	−0.523385	+	0.852096i	$0.675331\pi$
$450$	0	0
$451$	433.013i	0.960117i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1212.44i	− 2.66469i
$456$	0	0
$457$	−325.000	−0.711160	−0.355580	−	0.934646i	$-0.615717\pi$
$457$	−325.000	−0.711160	−0.355580	+	0.934646i	$0.615717\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−655.000	−1.42082	−0.710412	−	0.703786i	$-0.751491\pi$
$461$	−655.000	−1.42082	−0.710412	+	0.703786i	$0.751491\pi$
$462$	0	0
$463$	164.545i	0.355388i	0.984086	+	0.177694i	$0.0568638\pi$
$463$	164.545i	0.355388i	−0.984086	+	0.177694i	$0.943136\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 57.1577i	− 0.122393i	−0.998126	−	0.0611967i	$-0.980508\pi$
$467$	− 57.1577i	− 0.122393i	0.998126	−	0.0611967i	$-0.0194917\pi$
$468$	0	0
$469$	−750.000	−1.59915
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−150.000	−0.317125
$474$	0	0
$475$	− 249.415i	− 0.525085i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 329.090i	− 0.687035i	−0.939146	−	0.343517i	$-0.888382\pi$
$479$	− 329.090i	− 0.687035i	0.939146	−	0.343517i	$-0.111618\pi$
$480$	0	0
$481$	−200.000	−0.415800
$482$	0	0
$483$	0	0
$484$	0	0
$485$	175.000	0.360825
$486$	0	0
$487$	519.615i	1.06697i	0.845809	+	0.533486i	$0.179118\pi$
$487$	519.615i	1.06697i	−0.845809	+	0.533486i	$0.820882\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	216.506i	0.440950i	0.975393	+	0.220475i	$0.0707607\pi$
$491$	216.506i	0.440950i	−0.975393	+	0.220475i	$0.929239\pi$
$492$	0	0
$493$	80.0000	0.162272
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 45.0333i	− 0.0902471i	−0.998981	−	0.0451236i	$-0.985632\pi$
$499$	− 45.0333i	− 0.0902471i	0.998981	−	0.0451236i	$-0.0143682\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 384.515i	− 0.764444i	−0.924071	−	0.382222i	$-0.875159\pi$
$503$	− 384.515i	− 0.764444i	0.924071	−	0.382222i	$-0.124841\pi$
$504$	0	0
$505$	−1085.00	−2.14851
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−265.000	−0.520629	−0.260314	−	0.965524i	$-0.583826\pi$
$509$	−265.000	−0.520629	−0.260314	+	0.965524i	$0.583826\pi$
$510$	0	0
$511$	476.314i	0.932121i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 969.948i	− 1.88340i
$516$	0	0
$517$	750.000	1.45068
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−380.000	−0.729367	−0.364683	−	0.931132i	$-0.618823\pi$
$521$	−380.000	−0.729367	−0.364683	+	0.931132i	$0.618823\pi$
$522$	0	0
$523$	− 623.538i	− 1.19223i	−0.802898	−	0.596117i	$-0.796709\pi$
$523$	− 623.538i	− 1.19223i	0.802898	−	0.596117i	$-0.203291\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	429.549i	0.815083i
$528$	0	0
$529$	517.000	0.977316
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1000.00	1.87617
$534$	0	0
$535$	− 909.327i	− 1.69968i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	225.167i	0.417749i
$540$	0	0
$541$	532.000	0.983364	0.491682	−	0.870775i	$-0.336382\pi$
$541$	532.000	0.983364	0.491682	+	0.870775i	$0.336382\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	938.000	1.72110
$546$	0	0
$547$	− 900.666i	− 1.64656i	−0.567638	−	0.823278i	$-0.692143\pi$
$547$	− 900.666i	− 1.64656i	0.567638	−	0.823278i	$-0.307857\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	103.923i	0.188608i
$552$	0	0
$553$	60.0000	0.108499
$554$	0	0
$555$	0	0
$556$	0	0
$557$	89.0000	0.159785	0.0798923	−	0.996804i	$-0.474542\pi$
$557$	89.0000	0.159785	0.0798923	+	0.996804i	$0.474542\pi$
$558$	0	0
$559$	346.410i	0.619696i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 303.109i	− 0.538382i	−0.963087	−	0.269191i	$-0.913244\pi$
$563$	− 303.109i	− 0.538382i	0.963087	−	0.269191i	$-0.0867562\pi$
$564$	0	0
$565$	518.000	0.916814
$566$	0	0
$567$	0	0
$568$	0	0
$569$	100.000	0.175747	0.0878735	−	0.996132i	$-0.471993\pi$
$569$	100.000	0.175747	0.0878735	+	0.996132i	$0.471993\pi$
$570$	0	0
$571$	339.482i	0.594539i	0.954794	+	0.297270i	$0.0960760\pi$
$571$	339.482i	0.594539i	−0.954794	+	0.297270i	$0.903924\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 83.1384i	− 0.144589i
$576$	0	0
$577$	−730.000	−1.26516	−0.632582	−	0.774493i	$-0.718005\pi$
$577$	−730.000	−1.26516	−0.632582	+	0.774493i	$0.718005\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−255.000	−0.438898
$582$	0	0
$583$	− 407.032i	− 0.698168i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	801.940i	1.36617i	0.730341	+	0.683083i	$0.239361\pi$
$587$	801.940i	1.36617i	−0.730341	+	0.683083i	$0.760639\pi$
$588$	0	0
$589$	−558.000	−0.947368
$590$	0	0
$591$	0	0
$592$	0	0
$593$	982.000	1.65599	0.827993	−	0.560738i	$-0.189483\pi$
$593$	982.000	1.65599	0.827993	+	0.560738i	$0.189483\pi$
$594$	0	0
$595$	− 484.974i	− 0.815083i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	225.167i	0.375904i	0.982178	+	0.187952i	$0.0601850\pi$
$599$	225.167i	0.375904i	−0.982178	+	0.187952i	$0.939815\pi$
$600$	0	0
$601$	251.000	0.417637	0.208819	−	0.977954i	$-0.433038\pi$
$601$	251.000	0.417637	0.208819	+	0.977954i	$0.433038\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−322.000	−0.532231
$606$	0	0
$607$	− 381.051i	− 0.627761i	−0.949462	−	0.313881i	$-0.898371\pi$
$607$	− 381.051i	− 0.627761i	0.949462	−	0.313881i	$-0.101629\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1732.05i	− 2.83478i
$612$	0	0
$613$	−650.000	−1.06036	−0.530179	−	0.847885i	$-0.677875\pi$
$613$	−650.000	−1.06036	−0.530179	+	0.847885i	$0.677875\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−758.000	−1.22853	−0.614263	−	0.789102i	$-0.710546\pi$
$617$	−758.000	−1.22853	−0.614263	+	0.789102i	$0.710546\pi$
$618$	0	0
$619$	− 173.205i	− 0.279814i	−0.990165	−	0.139907i	$-0.955320\pi$
$619$	− 173.205i	− 0.279814i	0.990165	−	0.139907i	$-0.0446804\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 86.6025i	− 0.139009i
$624$	0	0
$625$	−649.000	−1.03840
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−80.0000	−0.127186
$630$	0	0
$631$	− 119.512i	− 0.189400i	−0.995506	−	0.0947001i	$-0.969811\pi$
$631$	− 119.512i	− 0.189400i	0.995506	−	0.0947001i	$-0.0301892\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	181.865i	0.286402i
$636$	0	0
$637$	520.000	0.816327
$638$	0	0
$639$	0	0
$640$	0	0
$641$	910.000	1.41966	0.709828	−	0.704375i	$-0.248772\pi$
$641$	910.000	1.41966	0.709828	+	0.704375i	$0.248772\pi$
$642$	0	0
$643$	− 34.6410i	− 0.0538741i	−0.999637	−	0.0269370i	$-0.991425\pi$
$643$	− 34.6410i	− 0.0538741i	0.999637	−	0.0269370i	$-0.00857536\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 914.523i	− 1.41348i	−0.707472	−	0.706741i	$-0.750165\pi$
$647$	− 914.523i	− 1.41348i	0.707472	−	0.706741i	$-0.249835\pi$
$648$	0	0
$649$	300.000	0.462250
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−103.000	−0.157734	−0.0788668	−	0.996885i	$-0.525130\pi$
$653$	−103.000	−0.157734	−0.0788668	+	0.996885i	$0.525130\pi$
$654$	0	0
$655$	− 1151.81i	− 1.75849i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 60.6218i	− 0.0919906i	−0.998942	−	0.0459953i	$-0.985354\pi$
$659$	− 60.6218i	− 0.0919906i	0.998942	−	0.0459953i	$-0.0146459\pi$
$660$	0	0
$661$	−578.000	−0.874433	−0.437216	−	0.899356i	$-0.644036\pi$
$661$	−578.000	−0.874433	−0.437216	+	0.899356i	$0.644036\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	630.000	0.947368
$666$	0	0
$667$	34.6410i	0.0519356i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 554.256i	− 0.826015i
$672$	0	0
$673$	845.000	1.25557	0.627786	−	0.778386i	$-0.283961\pi$
$673$	845.000	1.25557	0.627786	+	0.778386i	$0.283961\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1154.00	1.70458	0.852290	−	0.523070i	$-0.175214\pi$
$677$	1154.00	1.70458	0.852290	+	0.523070i	$0.175214\pi$
$678$	0	0
$679$	216.506i	0.318861i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	187.061i	0.273882i	0.990579	+	0.136941i	$0.0437271\pi$
$683$	187.061i	0.273882i	−0.990579	+	0.136941i	$0.956273\pi$
$684$	0	0
$685$	434.000	0.633577
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−940.000	−1.36430
$690$	0	0
$691$	491.902i	0.711870i	0.934511	+	0.355935i	$0.115838\pi$
$691$	491.902i	0.711870i	−0.934511	+	0.355935i	$0.884162\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1212.44i	− 1.74451i
$696$	0	0
$697$	400.000	0.573888
$698$	0	0
$699$	0	0
$700$	0	0
$701$	215.000	0.306705	0.153352	−	0.988172i	$-0.450993\pi$
$701$	215.000	0.306705	0.153352	+	0.988172i	$0.450993\pi$
$702$	0	0
$703$	− 103.923i	− 0.147828i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1342.34i	− 1.89864i
$708$	0	0
$709$	532.000	0.750353	0.375176	−	0.926953i	$-0.377582\pi$
$709$	532.000	0.750353	0.375176	+	0.926953i	$0.377582\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−186.000	−0.260870
$714$	0	0
$715$	− 1212.44i	− 1.69571i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1143.15i	1.58992i	0.606661	+	0.794961i	$0.292509\pi$
$719$	1143.15i	1.58992i	−0.606661	+	0.794961i	$0.707491\pi$
$720$	0	0
$721$	1200.00	1.66436
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−240.000	−0.331034
$726$	0	0
$727$	1238.42i	1.70346i	0.523980	+	0.851731i	$0.324447\pi$
$727$	1238.42i	1.70346i	−0.523980	+	0.851731i	$0.675553\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	138.564i	0.189554i
$732$	0	0
$733$	−950.000	−1.29604	−0.648022	−	0.761622i	$-0.724403\pi$
$733$	−950.000	−1.29604	−0.648022	+	0.761622i	$0.724403\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−750.000	−1.01764
$738$	0	0
$739$	581.969i	0.787509i	0.919216	+	0.393754i	$0.128824\pi$
$739$	581.969i	0.787509i	−0.919216	+	0.393754i	$0.871176\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 866.025i	− 1.16558i	−0.812623	−	0.582790i	$-0.801961\pi$
$743$	− 866.025i	− 1.16558i	0.812623	−	0.582790i	$-0.198039\pi$
$744$	0	0
$745$	805.000	1.08054
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1125.00	1.50200
$750$	0	0
$751$	174.937i	0.232939i	0.993194	+	0.116469i	$0.0371577\pi$
$751$	174.937i	0.232939i	−0.993194	+	0.116469i	$0.962842\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 303.109i	− 0.401469i
$756$	0	0
$757$	−830.000	−1.09643	−0.548217	−	0.836336i	$-0.684693\pi$
$757$	−830.000	−1.09643	−0.548217	+	0.836336i	$0.684693\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−560.000	−0.735874	−0.367937	−	0.929851i	$-0.619936\pi$
$761$	−560.000	−0.735874	−0.367937	+	0.929851i	$0.619936\pi$
$762$	0	0
$763$	1160.47i	1.52094i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 692.820i	− 0.903286i
$768$	0	0
$769$	−331.000	−0.430429	−0.215215	−	0.976567i	$-0.569045\pi$
$769$	−331.000	−0.430429	−0.215215	+	0.976567i	$0.569045\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−298.000	−0.385511	−0.192755	−	0.981247i	$-0.561742\pi$
$773$	−298.000	−0.385511	−0.192755	+	0.981247i	$0.561742\pi$
$774$	0	0
$775$	− 1288.65i	− 1.66277i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	519.615i	0.667029i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	140.000	0.178344
$786$	0	0
$787$	1108.51i	1.40853i	0.709938	+	0.704265i	$0.248723\pi$
$787$	1108.51i	1.40853i	−0.709938	+	0.704265i	$0.751277\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	640.859i	0.810188i
$792$	0	0
$793$	−1280.00	−1.61412
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1303.00	−1.63488	−0.817440	−	0.576013i	$-0.804608\pi$
$797$	−1303.00	−1.63488	−0.817440	+	0.576013i	$0.804608\pi$
$798$	0	0
$799$	− 692.820i	− 0.867109i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	476.314i	0.593168i
$804$	0	0
$805$	210.000	0.260870
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−170.000	−0.210136	−0.105068	−	0.994465i	$-0.533506\pi$
$809$	−170.000	−0.210136	−0.105068	+	0.994465i	$0.533506\pi$
$810$	0	0
$811$	779.423i	0.961064i	0.876977	+	0.480532i	$0.159556\pi$
$811$	779.423i	0.961064i	−0.876977	+	0.480532i	$0.840444\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	727.461i	0.892591i
$816$	0	0
$817$	−180.000	−0.220318
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1090.00	−1.32765	−0.663825	−	0.747888i	$-0.731068\pi$
$821$	−1090.00	−1.32765	−0.663825	+	0.747888i	$0.731068\pi$
$822$	0	0
$823$	943.968i	1.14698i	0.819211	+	0.573492i	$0.194412\pi$
$823$	943.968i	1.14698i	−0.819211	+	0.573492i	$0.805588\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	83.1384i	0.100530i	0.998736	+	0.0502651i	$0.0160066\pi$
$827$	83.1384i	0.100530i	−0.998736	+	0.0502651i	$0.983993\pi$
$828$	0	0
$829$	−542.000	−0.653800	−0.326900	−	0.945059i	$-0.606004\pi$
$829$	−542.000	−0.653800	−0.326900	+	0.945059i	$0.606004\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	208.000	0.249700
$834$	0	0
$835$	− 1721.66i	− 2.06187i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 692.820i	− 0.825769i	−0.910783	−	0.412885i	$-0.864521\pi$
$839$	− 692.820i	− 0.825769i	0.910783	−	0.412885i	$-0.135479\pi$
$840$	0	0
$841$	−741.000	−0.881094
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1617.00	−1.91361
$846$	0	0
$847$	− 398.372i	− 0.470333i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 34.6410i	− 0.0407062i
$852$	0	0
$853$	−290.000	−0.339977	−0.169988	−	0.985446i	$-0.554373\pi$
$853$	−290.000	−0.339977	−0.169988	+	0.985446i	$0.554373\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−368.000	−0.429405	−0.214702	−	0.976680i	$-0.568878\pi$
$857$	−368.000	−0.429405	−0.214702	+	0.976680i	$0.568878\pi$
$858$	0	0
$859$	1066.94i	1.24208i	0.783780	+	0.621038i	$0.213289\pi$
$859$	1066.94i	1.24208i	−0.783780	+	0.621038i	$0.786711\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1018.45i	1.18012i	0.807358	+	0.590061i	$0.200896\pi$
$863$	1018.45i	1.18012i	−0.807358	+	0.590061i	$0.799104\pi$
$864$	0	0
$865$	889.000	1.02775
$866$	0	0
$867$	0	0
$868$	0	0
$869$	60.0000	0.0690449
$870$	0	0
$871$	1732.05i	1.98858i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 60.6218i	− 0.0692820i
$876$	0	0
$877$	1330.00	1.51653	0.758267	−	0.651944i	$-0.226046\pi$
$877$	1330.00	1.51653	0.758267	+	0.651944i	$0.226046\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1060.00	1.20318	0.601589	−	0.798806i	$-0.294534\pi$
$881$	1060.00	1.20318	0.601589	+	0.798806i	$0.294534\pi$
$882$	0	0
$883$	− 1506.88i	− 1.70655i	−0.521461	−	0.853275i	$-0.674613\pi$
$883$	− 1506.88i	− 1.70655i	0.521461	−	0.853275i	$-0.325387\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	713.605i	0.804515i	0.915527	+	0.402258i	$0.131774\pi$
$887$	713.605i	0.804515i	−0.915527	+	0.402258i	$0.868226\pi$
$888$	0	0
$889$	−225.000	−0.253093
$890$	0	0
$891$	0	0
$892$	0	0
$893$	900.000	1.00784
$894$	0	0
$895$	1636.79i	1.82881i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	536.936i	0.597259i
$900$	0	0
$901$	−376.000	−0.417314
$902$	0	0
$903$	0	0
$904$	0	0
$905$	392.000	0.433149
$906$	0	0
$907$	1333.68i	1.47043i	0.677835	+	0.735215i	$0.262919\pi$
$907$	1333.68i	1.47043i	−0.677835	+	0.735215i	$0.737081\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1541.53i	1.69212i	0.533084	+	0.846062i	$0.321033\pi$
$911$	1541.53i	1.69212i	−0.533084	+	0.846062i	$0.678967\pi$
$912$	0	0
$913$	−255.000	−0.279299
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1425.00	1.55398
$918$	0	0
$919$	909.327i	0.989474i	0.869043	+	0.494737i	$0.164736\pi$
$919$	909.327i	0.989474i	−0.869043	+	0.494737i	$0.835264\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	240.000	0.259459
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1340.00	−1.44241	−0.721206	−	0.692721i	$-0.756412\pi$
$929$	−1340.00	−1.44241	−0.721206	+	0.692721i	$0.756412\pi$
$930$	0	0
$931$	270.200i	0.290225i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 484.974i	− 0.518689i
$936$	0	0
$937$	−1225.00	−1.30736	−0.653682	−	0.756769i	$-0.726777\pi$
$937$	−1225.00	−1.30736	−0.653682	+	0.756769i	$0.726777\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−115.000	−0.122210	−0.0611052	−	0.998131i	$-0.519463\pi$
$941$	−115.000	−0.122210	−0.0611052	+	0.998131i	$0.519463\pi$
$942$	0	0
$943$	173.205i	0.183675i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	687.624i	0.726108i	0.931768	+	0.363054i	$0.118266\pi$
$947$	687.624i	0.726108i	−0.931768	+	0.363054i	$0.881734\pi$
$948$	0	0
$949$	1100.00	1.15911
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−44.0000	−0.0461700	−0.0230850	−	0.999734i	$-0.507349\pi$
$953$	−44.0000	−0.0461700	−0.0230850	+	0.999734i	$0.507349\pi$
$954$	0	0
$955$	− 242.487i	− 0.253913i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	536.936i	0.559891i
$960$	0	0
$961$	−1922.00	−2.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−455.000	−0.471503
$966$	0	0
$967$	− 251.147i	− 0.259718i	−0.991532	−	0.129859i	$-0.958548\pi$
$967$	− 251.147i	− 0.259718i	0.991532	−	0.129859i	$-0.0414525\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	337.750i	0.347837i	0.984760	+	0.173919i	$0.0556430\pi$
$971$	337.750i	0.347837i	−0.984760	+	0.173919i	$0.944357\pi$
$972$	0	0
$973$	1500.00	1.54162
$974$	0	0
$975$	0	0
$976$	0	0
$977$	346.000	0.354145	0.177073	−	0.984198i	$-0.443337\pi$
$977$	346.000	0.354145	0.177073	+	0.984198i	$0.443337\pi$
$978$	0	0
$979$	− 86.6025i	− 0.0884602i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	793.279i	0.806998i	0.914980	+	0.403499i	$0.132206\pi$
$983$	793.279i	0.806998i	−0.914980	+	0.403499i	$0.867794\pi$
$984$	0	0
$985$	1771.00	1.79797
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−60.0000	−0.0606673
$990$	0	0
$991$	− 1054.82i	− 1.06440i	−0.846619	−	0.532199i	$-0.821366\pi$
$991$	− 1054.82i	− 1.06440i	0.846619	−	0.532199i	$-0.178634\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	909.327i	0.913896i
$996$	0	0
$997$	−260.000	−0.260782	−0.130391	−	0.991463i	$-0.541623\pi$
$997$	−260.000	−0.260782	−0.130391	+	0.991463i	$0.541623\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.g.a.703.1		2
3.2	odd	2		1728.3.g.f.703.1		2
4.3	odd	2	inner	1728.3.g.a.703.2		2
8.3	odd	2		108.3.d.a.55.1	✓	2
8.5	even	2		108.3.d.a.55.2	yes	2
12.11	even	2		1728.3.g.f.703.2		2
24.5	odd	2		108.3.d.b.55.1	yes	2
24.11	even	2		108.3.d.b.55.2	yes	2
72.5	odd	6		324.3.f.h.55.1		2
72.11	even	6		324.3.f.h.271.1		2
72.13	even	6		324.3.f.c.55.1		2
72.29	odd	6		324.3.f.b.271.1		2
72.43	odd	6		324.3.f.c.271.1		2
72.59	even	6		324.3.f.b.55.1		2
72.61	even	6		324.3.f.i.271.1		2
72.67	odd	6		324.3.f.i.55.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.3.d.a.55.1	✓	2	8.3	odd	2
108.3.d.a.55.2	yes	2	8.5	even	2
108.3.d.b.55.1	yes	2	24.5	odd	2
108.3.d.b.55.2	yes	2	24.11	even	2
324.3.f.b.55.1		2	72.59	even	6
324.3.f.b.271.1		2	72.29	odd	6
324.3.f.c.55.1		2	72.13	even	6
324.3.f.c.271.1		2	72.43	odd	6
324.3.f.h.55.1		2	72.5	odd	6
324.3.f.h.271.1		2	72.11	even	6
324.3.f.i.55.1		2	72.67	odd	6
324.3.f.i.271.1		2	72.61	even	6
1728.3.g.a.703.1		2	1.1	even	1	trivial
1728.3.g.a.703.2		2	4.3	odd	2	inner
1728.3.g.f.703.1		2	3.2	odd	2
1728.3.g.f.703.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.00000 q^{5} -8.66025i q^{7} +O(q^{10})\)	\(q-7.00000 q^{5} -8.66025i q^{7} -8.66025i q^{11} -20.0000 q^{13} -8.00000 q^{17} -10.3923i q^{19} -3.46410i q^{23} +24.0000 q^{25} -10.0000 q^{29} -53.6936i q^{31} +60.6218i q^{35} +10.0000 q^{37} -50.0000 q^{41} -17.3205i q^{43} +86.6025i q^{47} -26.0000 q^{49} +47.0000 q^{53} +60.6218i q^{55} +34.6410i q^{59} +64.0000 q^{61} +140.000 q^{65} -86.6025i q^{67} -55.0000 q^{73} -75.0000 q^{77} +6.92820i q^{79} -29.4449i q^{83} +56.0000 q^{85} +10.0000 q^{89} +173.205i q^{91} +72.7461i q^{95} -25.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{5}+O(q^{10}) \) 2 * q - 14 * q^5	\( 2 q - 14 q^{5} - 40 q^{13} - 16 q^{17} + 48 q^{25} - 20 q^{29} + 20 q^{37} - 100 q^{41} - 52 q^{49} + 94 q^{53} + 128 q^{61} + 280 q^{65} - 110 q^{73} - 150 q^{77} + 112 q^{85} + 20 q^{89} - 50 q^{97}+O(q^{100}) \) 2 * q - 14 * q^5 - 40 * q^13 - 16 * q^17 + 48 * q^25 - 20 * q^29 + 20 * q^37 - 100 * q^41 - 52 * q^49 + 94 * q^53 + 128 * q^61 + 280 * q^65 - 110 * q^73 - 150 * q^77 + 112 * q^85 + 20 * q^89 - 50 * q^97

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