LMFDB - Embedded newform 4608.2.d.n.2305.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(2305,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4608.2305");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.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:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1536)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2305.3
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4608.2305
Dual form			4608.2.d.n.2305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{5} -4.24264 q^{7} +O(q^{10})$	$q+1.41421i q^{5} -4.24264 q^{7} +6.00000i q^{11} +5.65685i q^{13} +6.00000 q^{17} +4.00000i q^{19} -2.82843 q^{23} +3.00000 q^{25} -1.41421i q^{29} +1.41421 q^{31} -6.00000i q^{35} +8.48528i q^{37} -2.00000 q^{41} +2.82843 q^{47} +11.0000 q^{49} -9.89949i q^{53} -8.48528 q^{55} +4.00000i q^{59} -8.48528i q^{61} -8.00000 q^{65} +8.00000i q^{67} +2.82843 q^{71} -8.00000 q^{73} -25.4558i q^{77} -12.7279 q^{79} -2.00000i q^{83} +8.48528i q^{85} +2.00000 q^{89} -24.0000i q^{91} -5.65685 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 24 q^{17} + 12 q^{25} - 8 q^{41} + 44 q^{49} - 32 q^{65} - 32 q^{73} + 8 q^{89} + 8 q^{97}+O(q^{100}) $ 4 * q + 24 * q^17 + 12 * q^25 - 8 * q^41 + 44 * q^49 - 32 * q^65 - 32 * q^73 + 8 * q^89 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	1.41421i	0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$	1.41421i	0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$	0	0
$7$	−4.24264	−1.60357	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−4.24264	−1.60357	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.00000i	1.80907i	0.426401	+	0.904534i	$0.359781\pi$
$11$	6.00000i	1.80907i	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	5.65685i	1.56893i	0.620174	+	0.784465i	$0.287062\pi$
$13$	5.65685i	1.56893i	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 1.41421i	− 0.262613i	−0.991342	−	0.131306i	$-0.958083\pi$
$29$	− 1.41421i	− 0.262613i	0.991342	−	0.131306i	$-0.0419172\pi$
$30$	0	0
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 6.00000i	− 1.01419i
$36$	0	0
$37$	8.48528i	1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$	8.48528i	1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.89949i	− 1.35980i	−0.733305	−	0.679900i	$-0.762023\pi$
$53$	− 9.89949i	− 1.35980i	0.733305	−	0.679900i	$-0.237977\pi$
$54$	0	0
$55$	−8.48528	−1.14416
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	− 8.48528i	− 1.08643i	−0.839594	−	0.543214i	$-0.817207\pi$
$61$	− 8.48528i	− 1.08643i	0.839594	−	0.543214i	$-0.182793\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.00000	−0.992278
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.82843	0.335673	0.167836	−	0.985815i	$-0.446322\pi$
$71$	2.82843	0.335673	0.167836	+	0.985815i	$0.446322\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 25.4558i	− 2.90096i
$78$	0	0
$79$	−12.7279	−1.43200	−0.716002	−	0.698099i	$-0.754030\pi$
$79$	−12.7279	−1.43200	−0.716002	+	0.698099i	$0.754030\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 2.00000i	− 0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$	− 2.00000i	− 0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$	0	0
$85$	8.48528i	0.920358i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	− 24.0000i	− 2.51588i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.65685	−0.580381
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.41421i	− 0.140720i	−0.997522	−	0.0703598i	$-0.977585\pi$
$101$	− 1.41421i	− 0.140720i	0.997522	−	0.0703598i	$-0.0224147\pi$
$102$	0	0
$103$	7.07107	0.696733	0.348367	−	0.937358i	$-0.386736\pi$
$103$	7.07107	0.696733	0.348367	+	0.937358i	$0.386736\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	16.9706i	1.62549i	0.582623	+	0.812743i	$0.302026\pi$
$109$	16.9706i	1.62549i	−0.582623	+	0.812743i	$0.697974\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	− 4.00000i	− 0.373002i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−25.4558	−2.33353
$120$	0	0
$121$	−25.0000	−2.27273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.3137i	1.01193i
$126$	0	0
$127$	−12.7279	−1.12942	−0.564710	−	0.825289i	$-0.691012\pi$
$127$	−12.7279	−1.12942	−0.564710	+	0.825289i	$0.691012\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.0000i	1.04844i	0.851581	+	0.524222i	$0.175644\pi$
$131$	12.0000i	1.04844i	−0.851581	+	0.524222i	$0.824356\pi$
$132$	0	0
$133$	− 16.9706i	− 1.47153i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−33.9411	−2.83830
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 4.24264i	− 0.347571i	−0.984784	−	0.173785i	$-0.944400\pi$
$149$	− 4.24264i	− 0.347571i	0.984784	−	0.173785i	$-0.0555999\pi$
$150$	0	0
$151$	−12.7279	−1.03578	−0.517892	−	0.855446i	$-0.673283\pi$
$151$	−12.7279	−1.03578	−0.517892	+	0.855446i	$0.673283\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.00000i	0.160644i
$156$	0	0
$157$	2.82843i	0.225733i	0.993610	+	0.112867i	$0.0360032\pi$
$157$	2.82843i	0.225733i	−0.993610	+	0.112867i	$0.963997\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	8.00000i	0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$	8.00000i	0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.6274	1.75096	0.875481	−	0.483252i	$-0.160545\pi$
$167$	22.6274	1.75096	0.875481	+	0.483252i	$0.160545\pi$
$168$	0	0
$169$	−19.0000	−1.46154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.07107i	0.537603i	0.963196	+	0.268802i	$0.0866276\pi$
$173$	7.07107i	0.537603i	−0.963196	+	0.268802i	$0.913372\pi$
$174$	0	0
$175$	−12.7279	−0.962140
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 20.0000i	− 1.49487i	−0.664335	−	0.747435i	$-0.731285\pi$
$179$	− 20.0000i	− 1.49487i	0.664335	−	0.747435i	$-0.268715\pi$
$180$	0	0
$181$	16.9706i	1.26141i	0.776022	+	0.630706i	$0.217235\pi$
$181$	16.9706i	1.26141i	−0.776022	+	0.630706i	$0.782765\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	36.0000i	2.63258i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4.24264i	− 0.302276i	−0.988513	−	0.151138i	$-0.951706\pi$
$197$	− 4.24264i	− 0.302276i	0.988513	−	0.151138i	$-0.0482937\pi$
$198$	0	0
$199$	−4.24264	−0.300753	−0.150376	−	0.988629i	$-0.548049\pi$
$199$	−4.24264	−0.300753	−0.150376	+	0.988629i	$0.548049\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	− 2.82843i	− 0.197546i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−24.0000	−1.66011
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	0	0
$220$	0	0
$221$	33.9411i	2.28313i
$222$	0	0
$223$	−15.5563	−1.04173	−0.520865	−	0.853639i	$-0.674391\pi$
$223$	−15.5563	−1.04173	−0.520865	+	0.853639i	$0.674391\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 14.0000i	− 0.929213i	−0.885517	−	0.464606i	$-0.846196\pi$
$227$	− 14.0000i	− 0.929213i	0.885517	−	0.464606i	$-0.153804\pi$
$228$	0	0
$229$	− 22.6274i	− 1.49526i	−0.664114	−	0.747631i	$-0.731191\pi$
$229$	− 22.6274i	− 1.49526i	0.664114	−	0.747631i	$-0.268809\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	4.00000i	0.260931i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.3137	0.731823	0.365911	−	0.930650i	$-0.380757\pi$
$239$	11.3137	0.731823	0.365911	+	0.930650i	$0.380757\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	15.5563i	0.993859i
$246$	0	0
$247$	−22.6274	−1.43975
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 18.0000i	− 1.13615i	−0.822977	−	0.568075i	$-0.807688\pi$
$251$	− 18.0000i	− 1.13615i	0.822977	−	0.568075i	$-0.192312\pi$
$252$	0	0
$253$	− 16.9706i	− 1.06693i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	− 36.0000i	− 2.23693i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.9706	−1.04645	−0.523225	−	0.852195i	$-0.675271\pi$
$263$	−16.9706	−1.04645	−0.523225	+	0.852195i	$0.675271\pi$
$264$	0	0
$265$	14.0000	0.860013
$266$	0	0
$267$	0	0
$268$	0	0
$269$	12.7279i	0.776035i	0.921652	+	0.388018i	$0.126840\pi$
$269$	12.7279i	0.776035i	−0.921652	+	0.388018i	$0.873160\pi$
$270$	0	0
$271$	−4.24264	−0.257722	−0.128861	−	0.991663i	$-0.541132\pi$
$271$	−4.24264	−0.257722	−0.128861	+	0.991663i	$0.541132\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.0000i	1.08544i
$276$	0	0
$277$	− 16.9706i	− 1.01966i	−0.860274	−	0.509831i	$-0.829708\pi$
$277$	− 16.9706i	− 1.01966i	0.860274	−	0.509831i	$-0.170292\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.48528	0.500870
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	24.0416i	1.40453i	0.711917	+	0.702264i	$0.247827\pi$
$293$	24.0416i	1.40453i	−0.711917	+	0.702264i	$0.752173\pi$
$294$	0	0
$295$	−5.65685	−0.329355
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 16.0000i	− 0.925304i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12.0000	0.687118
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−22.6274	−1.28308	−0.641542	−	0.767088i	$-0.721705\pi$
$311$	−22.6274	−1.28308	−0.641542	+	0.767088i	$0.721705\pi$
$312$	0	0
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 21.2132i	− 1.19145i	−0.803188	−	0.595726i	$-0.796864\pi$
$317$	− 21.2132i	− 1.19145i	0.803188	−	0.595726i	$-0.203136\pi$
$318$	0	0
$319$	8.48528	0.475085
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.0000i	1.33540i
$324$	0	0
$325$	16.9706i	0.941357i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	− 16.0000i	− 0.879440i	−0.898135	−	0.439720i	$-0.855078\pi$
$331$	− 16.0000i	− 0.879440i	0.898135	−	0.439720i	$-0.144922\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.48528i	0.459504i
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.00000i	0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$	2.00000i	0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$	0	0
$349$	− 14.1421i	− 0.757011i	−0.925599	−	0.378506i	$-0.876438\pi$
$349$	− 14.1421i	− 0.757011i	0.925599	−	0.378506i	$-0.123562\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	4.00000i	0.212298i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.1127	1.64207	0.821033	−	0.570881i	$-0.193398\pi$
$359$	31.1127	1.64207	0.821033	+	0.570881i	$0.193398\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 11.3137i	− 0.592187i
$366$	0	0
$367$	12.7279	0.664392	0.332196	−	0.943210i	$-0.392210\pi$
$367$	12.7279	0.664392	0.332196	+	0.943210i	$0.392210\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	42.0000i	2.18053i
$372$	0	0
$373$	− 19.7990i	− 1.02515i	−0.858642	−	0.512576i	$-0.828691\pi$
$373$	− 19.7990i	− 1.02515i	0.858642	−	0.512576i	$-0.171309\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	− 4.00000i	− 0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$	− 4.00000i	− 0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.65685	0.289052	0.144526	−	0.989501i	$-0.453834\pi$
$383$	5.65685	0.289052	0.144526	+	0.989501i	$0.453834\pi$
$384$	0	0
$385$	36.0000	1.83473
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1.41421i	− 0.0717035i	−0.999357	−	0.0358517i	$-0.988586\pi$
$389$	− 1.41421i	− 0.0717035i	0.999357	−	0.0358517i	$-0.0114144\pi$
$390$	0	0
$391$	−16.9706	−0.858238
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 18.0000i	− 0.905678i
$396$	0	0
$397$	− 31.1127i	− 1.56150i	−0.624843	−	0.780751i	$-0.714837\pi$
$397$	− 31.1127i	− 1.56150i	0.624843	−	0.780751i	$-0.285163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−50.9117	−2.52360
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 16.9706i	− 0.835067i
$414$	0	0
$415$	2.82843	0.138842
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 22.0000i	− 1.07477i	−0.843337	−	0.537385i	$-0.819412\pi$
$419$	− 22.0000i	− 1.07477i	0.843337	−	0.537385i	$-0.180588\pi$
$420$	0	0
$421$	11.3137i	0.551396i	0.961244	+	0.275698i	$0.0889090\pi$
$421$	11.3137i	0.551396i	−0.961244	+	0.275698i	$0.911091\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	18.0000	0.873128
$426$	0	0
$427$	36.0000i	1.74216i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.7990	−0.953684	−0.476842	−	0.878989i	$-0.658219\pi$
$431$	−19.7990	−0.953684	−0.476842	+	0.878989i	$0.658219\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 11.3137i	− 0.541208i
$438$	0	0
$439$	−9.89949	−0.472477	−0.236239	−	0.971695i	$-0.575915\pi$
$439$	−9.89949	−0.472477	−0.236239	+	0.971695i	$0.575915\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 22.0000i	− 1.04525i	−0.852562	−	0.522626i	$-0.824953\pi$
$443$	− 22.0000i	− 1.04525i	0.852562	−	0.522626i	$-0.175047\pi$
$444$	0	0
$445$	2.82843i	0.134080i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	− 12.0000i	− 0.565058i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	33.9411	1.59118
$456$	0	0
$457$	−42.0000	−1.96468	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−42.0000	−1.96468	−0.982339	+	0.187112i	$0.940087\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.3553i	1.64666i	0.567561	+	0.823331i	$0.307887\pi$
$461$	35.3553i	1.64666i	−0.567561	+	0.823331i	$0.692113\pi$
$462$	0	0
$463$	−29.6985	−1.38021	−0.690103	−	0.723711i	$-0.742435\pi$
$463$	−29.6985	−1.38021	−0.690103	+	0.723711i	$0.742435\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.0000i	0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$	18.0000i	0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$	0	0
$469$	− 33.9411i	− 1.56726i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	12.0000i	0.550598i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.48528	−0.387702	−0.193851	−	0.981031i	$-0.562098\pi$
$479$	−8.48528	−0.387702	−0.193851	+	0.981031i	$0.562098\pi$
$480$	0	0
$481$	−48.0000	−2.18861
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.82843i	0.128432i
$486$	0	0
$487$	35.3553	1.60210	0.801052	−	0.598595i	$-0.204274\pi$
$487$	35.3553	1.60210	0.801052	+	0.598595i	$0.204274\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 12.0000i	− 0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$491$	− 12.0000i	− 0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$492$	0	0
$493$	− 8.48528i	− 0.382158i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	28.0000i	1.25345i	0.779240	+	0.626726i	$0.215605\pi$
$499$	28.0000i	1.25345i	−0.779240	+	0.626726i	$0.784395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14.1421	−0.630567	−0.315283	−	0.948998i	$-0.602100\pi$
$503$	−14.1421	−0.630567	−0.315283	+	0.948998i	$0.602100\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.89949i	0.438787i	0.975636	+	0.219394i	$0.0704079\pi$
$509$	9.89949i	0.438787i	−0.975636	+	0.219394i	$0.929592\pi$
$510$	0	0
$511$	33.9411	1.50147
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.0000i	0.440653i
$516$	0	0
$517$	16.9706i	0.746364i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	0	0
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.48528	0.369625
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 11.3137i	− 0.490051i
$534$	0	0
$535$	5.65685	0.244567
$536$	0	0
$537$	0	0
$538$	0	0
$539$	66.0000i	2.84282i
$540$	0	0
$541$	22.6274i	0.972829i	0.873728	+	0.486414i	$0.161695\pi$
$541$	22.6274i	0.972829i	−0.873728	+	0.486414i	$0.838305\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−24.0000	−1.02805
$546$	0	0
$547$	40.0000i	1.71028i	0.518400	+	0.855138i	$0.326528\pi$
$547$	40.0000i	1.71028i	−0.518400	+	0.855138i	$0.673472\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.65685	0.240990
$552$	0	0
$553$	54.0000	2.29631
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.07107i	− 0.299611i	−0.988716	−	0.149805i	$-0.952135\pi$
$557$	− 7.07107i	− 0.299611i	0.988716	−	0.149805i	$-0.0478647\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 46.0000i	− 1.93867i	−0.245745	−	0.969334i	$-0.579033\pi$
$563$	− 46.0000i	− 1.93867i	0.245745	−	0.969334i	$-0.420967\pi$
$564$	0	0
$565$	14.1421i	0.594964i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	28.0000i	1.17176i	0.810397	+	0.585882i	$0.199252\pi$
$571$	28.0000i	1.17176i	−0.810397	+	0.585882i	$0.800748\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.48528	−0.353861
$576$	0	0
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.48528i	0.352029i
$582$	0	0
$583$	59.3970	2.45997
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44.0000i	1.81607i	0.418890	+	0.908037i	$0.362419\pi$
$587$	44.0000i	1.81607i	−0.418890	+	0.908037i	$0.637581\pi$
$588$	0	0
$589$	5.65685i	0.233087i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	− 36.0000i	− 1.47586i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.1127	1.27123	0.635615	−	0.772006i	$-0.280747\pi$
$599$	31.1127	1.27123	0.635615	+	0.772006i	$0.280747\pi$
$600$	0	0
$601$	4.00000	0.163163	0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000	0.163163	0.0815817	+	0.996667i	$0.474003\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 35.3553i	− 1.43740i
$606$	0	0
$607$	9.89949	0.401808	0.200904	−	0.979611i	$-0.435612\pi$
$607$	9.89949	0.401808	0.200904	+	0.979611i	$0.435612\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16.0000i	0.647291i
$612$	0	0
$613$	− 2.82843i	− 0.114239i	−0.998367	−	0.0571195i	$-0.981808\pi$
$613$	− 2.82843i	− 0.114239i	0.998367	−	0.0571195i	$-0.0181916\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.48528	−0.339956
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	50.9117i	2.02998i
$630$	0	0
$631$	1.41421	0.0562990	0.0281495	−	0.999604i	$-0.491039\pi$
$631$	1.41421	0.0562990	0.0281495	+	0.999604i	$0.491039\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 18.0000i	− 0.714308i
$636$	0	0
$637$	62.2254i	2.46546i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−46.0000	−1.81689	−0.908445	−	0.418004i	$-0.862730\pi$
$641$	−46.0000	−1.81689	−0.908445	+	0.418004i	$0.862730\pi$
$642$	0	0
$643$	− 32.0000i	− 1.26196i	−0.775800	−	0.630978i	$-0.782654\pi$
$643$	− 32.0000i	− 1.26196i	0.775800	−	0.630978i	$-0.217346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.1421	0.555985	0.277992	−	0.960583i	$-0.410331\pi$
$647$	14.1421	0.555985	0.277992	+	0.960583i	$0.410331\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.6985i	1.16219i	0.813835	+	0.581096i	$0.197376\pi$
$653$	29.6985i	1.16219i	−0.813835	+	0.581096i	$0.802624\pi$
$654$	0	0
$655$	−16.9706	−0.663095
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 36.0000i	− 1.40236i	−0.712984	−	0.701180i	$-0.752657\pi$
$659$	− 36.0000i	− 1.40236i	0.712984	−	0.701180i	$-0.247343\pi$
$660$	0	0
$661$	8.48528i	0.330039i	0.986290	+	0.165020i	$0.0527687\pi$
$661$	8.48528i	0.330039i	−0.986290	+	0.165020i	$0.947231\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	24.0000	0.930680
$666$	0	0
$667$	4.00000i	0.154881i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	50.9117	1.96542
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 43.8406i	− 1.68493i	−0.538750	−	0.842466i	$-0.681103\pi$
$677$	− 43.8406i	− 1.68493i	0.538750	−	0.842466i	$-0.318897\pi$
$678$	0	0
$679$	−8.48528	−0.325635
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30.0000i	1.14792i	0.818884	+	0.573959i	$0.194593\pi$
$683$	30.0000i	1.14792i	−0.818884	+	0.573959i	$0.805407\pi$
$684$	0	0
$685$	− 25.4558i	− 0.972618i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	56.0000	2.13343
$690$	0	0
$691$	32.0000i	1.21734i	0.793424	+	0.608669i	$0.208296\pi$
$691$	32.0000i	1.21734i	−0.793424	+	0.608669i	$0.791704\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 38.1838i	− 1.44218i	−0.692841	−	0.721090i	$-0.743641\pi$
$701$	− 38.1838i	− 1.44218i	0.692841	−	0.721090i	$-0.256359\pi$
$702$	0	0
$703$	−33.9411	−1.28011
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000i	0.225653i
$708$	0	0
$709$	16.9706i	0.637343i	0.947865	+	0.318671i	$0.103237\pi$
$709$	16.9706i	0.637343i	−0.947865	+	0.318671i	$0.896763\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.00000	−0.149801
$714$	0	0
$715$	− 48.0000i	− 1.79510i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	31.1127	1.16031	0.580154	−	0.814507i	$-0.302992\pi$
$719$	31.1127	1.16031	0.580154	+	0.814507i	$0.302992\pi$
$720$	0	0
$721$	−30.0000	−1.11726
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 4.24264i	− 0.157568i
$726$	0	0
$727$	26.8701	0.996555	0.498278	−	0.867018i	$-0.333966\pi$
$727$	26.8701	0.996555	0.498278	+	0.867018i	$0.333966\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	28.2843i	1.04470i	0.852730	+	0.522352i	$0.174945\pi$
$733$	28.2843i	1.04470i	−0.852730	+	0.522352i	$0.825055\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−48.0000	−1.76810
$738$	0	0
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	28.2843	1.03765	0.518825	−	0.854881i	$-0.326370\pi$
$743$	28.2843	1.03765	0.518825	+	0.854881i	$0.326370\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	16.9706i	0.620091i
$750$	0	0
$751$	−7.07107	−0.258027	−0.129013	−	0.991643i	$-0.541181\pi$
$751$	−7.07107	−0.258027	−0.129013	+	0.991643i	$0.541181\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 18.0000i	− 0.655087i
$756$	0	0
$757$	45.2548i	1.64481i	0.568899	+	0.822407i	$0.307370\pi$
$757$	45.2548i	1.64481i	−0.568899	+	0.822407i	$0.692630\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−46.0000	−1.66750	−0.833749	−	0.552143i	$-0.813810\pi$
$761$	−46.0000	−1.66750	−0.833749	+	0.552143i	$0.813810\pi$
$762$	0	0
$763$	− 72.0000i	− 2.60658i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−22.6274	−0.817029
$768$	0	0
$769$	−28.0000	−1.00971	−0.504853	−	0.863205i	$-0.668453\pi$
$769$	−28.0000	−1.00971	−0.504853	+	0.863205i	$0.668453\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 18.3848i	− 0.661254i	−0.943761	−	0.330627i	$-0.892740\pi$
$773$	− 18.3848i	− 0.661254i	0.943761	−	0.330627i	$-0.107260\pi$
$774$	0	0
$775$	4.24264	0.152400
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 8.00000i	− 0.286630i
$780$	0	0
$781$	16.9706i	0.607254i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.00000	−0.142766
$786$	0	0
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−42.4264	−1.50851
$792$	0	0
$793$	48.0000	1.70453
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 24.0416i	− 0.851598i	−0.904818	−	0.425799i	$-0.859993\pi$
$797$	− 24.0416i	− 0.851598i	0.904818	−	0.425799i	$-0.140007\pi$
$798$	0	0
$799$	16.9706	0.600375
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 48.0000i	− 1.69388i
$804$	0	0
$805$	16.9706i	0.598134i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	20.0000i	0.702295i	0.936320	+	0.351147i	$0.114208\pi$
$811$	20.0000i	0.702295i	−0.936320	+	0.351147i	$0.885792\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.3137	−0.396302
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.41421i	− 0.0493564i	−0.999695	−	0.0246782i	$-0.992144\pi$
$821$	− 1.41421i	− 0.0493564i	0.999695	−	0.0246782i	$-0.00785611\pi$
$822$	0	0
$823$	41.0122	1.42960	0.714798	−	0.699331i	$-0.246519\pi$
$823$	41.0122	1.42960	0.714798	+	0.699331i	$0.246519\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	0	0
$829$	− 28.2843i	− 0.982353i	−0.871060	−	0.491177i	$-0.836567\pi$
$829$	− 28.2843i	− 0.982353i	0.871060	−	0.491177i	$-0.163433\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	66.0000	2.28676
$834$	0	0
$835$	32.0000i	1.10741i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	19.7990	0.683537	0.341769	−	0.939784i	$-0.388974\pi$
$839$	19.7990	0.683537	0.341769	+	0.939784i	$0.388974\pi$
$840$	0	0
$841$	27.0000	0.931034
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 26.8701i	− 0.924358i
$846$	0	0
$847$	106.066	3.64447
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 24.0000i	− 0.822709i
$852$	0	0
$853$	42.4264i	1.45265i	0.687350	+	0.726326i	$0.258774\pi$
$853$	42.4264i	1.45265i	−0.687350	+	0.726326i	$0.741226\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	0	0
$859$	− 32.0000i	− 1.09183i	−0.837842	−	0.545913i	$-0.816183\pi$
$859$	− 32.0000i	− 1.09183i	0.837842	−	0.545913i	$-0.183817\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.3137	−0.385123	−0.192562	−	0.981285i	$-0.561680\pi$
$863$	−11.3137	−0.385123	−0.192562	+	0.981285i	$0.561680\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 76.3675i	− 2.59059i
$870$	0	0
$871$	−45.2548	−1.53340
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 48.0000i	− 1.62270i
$876$	0	0
$877$	− 36.7696i	− 1.24162i	−0.783961	−	0.620810i	$-0.786804\pi$
$877$	− 36.7696i	− 1.24162i	0.783961	−	0.620810i	$-0.213196\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	0	0
$883$	24.0000i	0.807664i	0.914833	+	0.403832i	$0.132322\pi$
$883$	24.0000i	0.807664i	−0.914833	+	0.403832i	$0.867678\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.5980	1.32957	0.664785	−	0.747035i	$-0.268523\pi$
$887$	39.5980	1.32957	0.664785	+	0.747035i	$0.268523\pi$
$888$	0	0
$889$	54.0000	1.81110
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11.3137i	0.378599i
$894$	0	0
$895$	28.2843	0.945439
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2.00000i	− 0.0667037i
$900$	0	0
$901$	− 59.3970i	− 1.97880i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.0000	−0.797787
$906$	0	0
$907$	32.0000i	1.06254i	0.847202	+	0.531271i	$0.178286\pi$
$907$	32.0000i	1.06254i	−0.847202	+	0.531271i	$0.821714\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−28.2843	−0.937100	−0.468550	−	0.883437i	$-0.655223\pi$
$911$	−28.2843	−0.937100	−0.468550	+	0.883437i	$0.655223\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 50.9117i	− 1.68125i
$918$	0	0
$919$	−26.8701	−0.886361	−0.443181	−	0.896432i	$-0.646150\pi$
$919$	−26.8701	−0.886361	−0.443181	+	0.896432i	$0.646150\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	16.0000i	0.526646i
$924$	0	0
$925$	25.4558i	0.836983i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	44.0000i	1.44204i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−50.9117	−1.66499
$936$	0	0
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.41421i	0.0461020i	0.999734	+	0.0230510i	$0.00733802\pi$
$941$	1.41421i	0.0461020i	−0.999734	+	0.0230510i	$0.992662\pi$
$942$	0	0
$943$	5.65685	0.184213
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	0	0
$949$	− 45.2548i	− 1.46903i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	16.0000i	0.517748i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	76.3675	2.46604
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	0	0
$964$	0	0
$965$	16.9706i	0.546302i
$966$	0	0
$967$	21.2132	0.682171	0.341085	−	0.940032i	$-0.389205\pi$
$967$	21.2132	0.682171	0.341085	+	0.940032i	$0.389205\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14.0000i	0.449281i	0.974442	+	0.224641i	$0.0721208\pi$
$971$	14.0000i	0.449281i	−0.974442	+	0.224641i	$0.927879\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	12.0000i	0.383522i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.2843	0.902128	0.451064	−	0.892492i	$-0.351045\pi$
$983$	28.2843	0.902128	0.451064	+	0.892492i	$0.351045\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	4.24264	0.134772	0.0673860	−	0.997727i	$-0.478534\pi$
$991$	4.24264	0.134772	0.0673860	+	0.997727i	$0.478534\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 6.00000i	− 0.190213i
$996$	0	0
$997$	31.1127i	0.985349i	0.870214	+	0.492675i	$0.163981\pi$
$997$	31.1127i	0.985349i	−0.870214	+	0.492675i	$0.836019\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.d.n.2305.3		4
3.2	odd	2		1536.2.d.c.769.1		4
4.3	odd	2	inner	4608.2.d.n.2305.4		4
8.3	odd	2	inner	4608.2.d.n.2305.2		4
8.5	even	2	inner	4608.2.d.n.2305.1		4
12.11	even	2		1536.2.d.c.769.3		4
16.3	odd	4		4608.2.a.g.1.2		2
16.5	even	4		4608.2.a.g.1.1		2
16.11	odd	4		4608.2.a.l.1.1		2
16.13	even	4		4608.2.a.l.1.2		2
24.5	odd	2		1536.2.d.c.769.4		4
24.11	even	2		1536.2.d.c.769.2		4
48.5	odd	4		1536.2.a.d.1.2	yes	2
48.11	even	4		1536.2.a.i.1.2	yes	2
48.29	odd	4		1536.2.a.i.1.1	yes	2
48.35	even	4		1536.2.a.d.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.a.d.1.1	✓	2	48.35	even	4
1536.2.a.d.1.2	yes	2	48.5	odd	4
1536.2.a.i.1.1	yes	2	48.29	odd	4
1536.2.a.i.1.2	yes	2	48.11	even	4
1536.2.d.c.769.1		4	3.2	odd	2
1536.2.d.c.769.2		4	24.11	even	2
1536.2.d.c.769.3		4	12.11	even	2
1536.2.d.c.769.4		4	24.5	odd	2
4608.2.a.g.1.1		2	16.5	even	4
4608.2.a.g.1.2		2	16.3	odd	4
4608.2.a.l.1.1		2	16.11	odd	4
4608.2.a.l.1.2		2	16.13	even	4
4608.2.d.n.2305.1		4	8.5	even	2	inner
4608.2.d.n.2305.2		4	8.3	odd	2	inner
4608.2.d.n.2305.3		4	1.1	even	1	trivial
4608.2.d.n.2305.4		4	4.3	odd	2	inner

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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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{5} -4.24264 q^{7} +O(q^{10})\)	\(q+1.41421i q^{5} -4.24264 q^{7} +6.00000i q^{11} +5.65685i q^{13} +6.00000 q^{17} +4.00000i q^{19} -2.82843 q^{23} +3.00000 q^{25} -1.41421i q^{29} +1.41421 q^{31} -6.00000i q^{35} +8.48528i q^{37} -2.00000 q^{41} +2.82843 q^{47} +11.0000 q^{49} -9.89949i q^{53} -8.48528 q^{55} +4.00000i q^{59} -8.48528i q^{61} -8.00000 q^{65} +8.00000i q^{67} +2.82843 q^{71} -8.00000 q^{73} -25.4558i q^{77} -12.7279 q^{79} -2.00000i q^{83} +8.48528i q^{85} +2.00000 q^{89} -24.0000i q^{91} -5.65685 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 24 q^{17} + 12 q^{25} - 8 q^{41} + 44 q^{49} - 32 q^{65} - 32 q^{73} + 8 q^{89} + 8 q^{97}+O(q^{100}) \) 4 * q + 24 * q^17 + 12 * q^25 - 8 * q^41 + 44 * q^49 - 32 * q^65 - 32 * q^73 + 8 * q^89 + 8 * q^97

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