LMFDB - Embedded newform 1350.3.b.b.1349.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1350,3,Mod(1349,1350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1350.1349"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1350.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,8,0,0,0,0,0,0,0,0,0,0,0,16,0,0,-116] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$36.7848356886$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1349.4
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1350.1349
Dual form			1350.3.b.b.1349.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +5.00000i q^{7} +2.82843 q^{8} +8.48528i q^{11} +1.00000i q^{13} +7.07107i q^{14} +4.00000 q^{16} -25.4558 q^{17} -29.0000 q^{19} +12.0000i q^{22} -8.48528 q^{23} +1.41421i q^{26} +10.0000i q^{28} +16.9706i q^{29} -10.0000 q^{31} +5.65685 q^{32} -36.0000 q^{34} -25.0000i q^{37} -41.0122 q^{38} +16.9706i q^{41} -14.0000i q^{43} +16.9706i q^{44} -12.0000 q^{46} -8.48528 q^{47} +24.0000 q^{49} +2.00000i q^{52} -50.9117 q^{53} +14.1421i q^{56} +24.0000i q^{58} -93.3381i q^{59} +23.0000 q^{61} -14.1421 q^{62} +8.00000 q^{64} -19.0000i q^{67} -50.9117 q^{68} +101.823i q^{71} +97.0000i q^{73} -35.3553i q^{74} -58.0000 q^{76} -42.4264 q^{77} -77.0000 q^{79} +24.0000i q^{82} -118.794 q^{83} -19.7990i q^{86} +24.0000i q^{88} +76.3675i q^{89} -5.00000 q^{91} -16.9706 q^{92} -12.0000 q^{94} -49.0000i q^{97} +33.9411 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} + 16 q^{16} - 116 q^{19} - 40 q^{31} - 144 q^{34} - 48 q^{46} + 96 q^{49} + 92 q^{61} + 32 q^{64} - 232 q^{76} - 308 q^{79} - 20 q^{91} - 48 q^{94}+O(q^{100}) $ 4 * q + 8 * q^4 + 16 * q^16 - 116 * q^19 - 40 * q^31 - 144 * q^34 - 48 * q^46 + 96 * q^49 + 92 * q^61 + 32 * q^64 - 232 * q^76 - 308 * q^79 - 20 * q^91 - 48 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.00000i	0.714286i	0.934050	+	0.357143i	$0.116249\pi$
$7$	5.00000i	0.714286i	−0.934050	+	0.357143i	$0.883751\pi$
$8$	2.82843	0.353553
$9$	0	0
$10$	0	0
$11$	8.48528i	0.771389i	0.922627	+	0.385695i	$0.126038\pi$
$11$	8.48528i	0.771389i	−0.922627	+	0.385695i	$0.873962\pi$
$12$	0	0
$13$	1.00000i	0.0769231i	0.999260	+	0.0384615i	$0.0122457\pi$
$13$	1.00000i	0.0769231i	−0.999260	+	0.0384615i	$0.987754\pi$
$14$	7.07107i	0.505076i
$15$	0	0
$16$	4.00000	0.250000
$17$	−25.4558	−1.49740	−0.748701	−	0.662908i	$-0.769322\pi$
$17$	−25.4558	−1.49740	−0.748701	+	0.662908i	$0.769322\pi$
$18$	0	0
$19$	−29.0000	−1.52632	−0.763158	−	0.646212i	$-0.776352\pi$
$19$	−29.0000	−1.52632	−0.763158	+	0.646212i	$0.776352\pi$
$20$	0	0
$21$	0	0
$22$	12.0000i	0.545455i
$23$	−8.48528	−0.368925	−0.184463	−	0.982840i	$-0.559054\pi$
$23$	−8.48528	−0.368925	−0.184463	+	0.982840i	$0.559054\pi$
$24$	0	0
$25$	0	0
$26$	1.41421i	0.0543928i
$27$	0	0
$28$	10.0000i	0.357143i
$29$	16.9706i	0.585192i	0.956236	+	0.292596i	$0.0945191\pi$
$29$	16.9706i	0.585192i	−0.956236	+	0.292596i	$0.905481\pi$
$30$	0	0
$31$	−10.0000	−0.322581	−0.161290	−	0.986907i	$-0.551566\pi$
$31$	−10.0000	−0.322581	−0.161290	+	0.986907i	$0.551566\pi$
$32$	5.65685	0.176777
$33$	0	0
$34$	−36.0000	−1.05882
$35$	0	0
$36$	0	0
$37$	− 25.0000i	− 0.675676i	−0.941204	−	0.337838i	$-0.890304\pi$
$37$	− 25.0000i	− 0.675676i	0.941204	−	0.337838i	$-0.109696\pi$
$38$	−41.0122	−1.07927
$39$	0	0
$40$	0	0
$41$	16.9706i	0.413916i	0.978350	+	0.206958i	$0.0663564\pi$
$41$	16.9706i	0.413916i	−0.978350	+	0.206958i	$0.933644\pi$
$42$	0	0
$43$	− 14.0000i	− 0.325581i	−0.986661	−	0.162791i	$-0.947950\pi$
$43$	− 14.0000i	− 0.325581i	0.986661	−	0.162791i	$-0.0520495\pi$
$44$	16.9706i	0.385695i
$45$	0	0
$46$	−12.0000	−0.260870
$47$	−8.48528	−0.180538	−0.0902690	−	0.995917i	$-0.528773\pi$
$47$	−8.48528	−0.180538	−0.0902690	+	0.995917i	$0.528773\pi$
$48$	0	0
$49$	24.0000	0.489796
$50$	0	0
$51$	0	0
$52$	2.00000i	0.0384615i
$53$	−50.9117	−0.960598	−0.480299	−	0.877105i	$-0.659472\pi$
$53$	−50.9117	−0.960598	−0.480299	+	0.877105i	$0.659472\pi$
$54$	0	0
$55$	0	0
$56$	14.1421i	0.252538i
$57$	0	0
$58$	24.0000i	0.413793i
$59$	− 93.3381i	− 1.58200i	−0.611815	−	0.791001i	$-0.709560\pi$
$59$	− 93.3381i	− 1.58200i	0.611815	−	0.791001i	$-0.290440\pi$
$60$	0	0
$61$	23.0000	0.377049	0.188525	−	0.982068i	$-0.439629\pi$
$61$	23.0000	0.377049	0.188525	+	0.982068i	$0.439629\pi$
$62$	−14.1421	−0.228099
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	− 19.0000i	− 0.283582i	−0.989897	−	0.141791i	$-0.954714\pi$
$67$	− 19.0000i	− 0.283582i	0.989897	−	0.141791i	$-0.0452861\pi$
$68$	−50.9117	−0.748701
$69$	0	0
$70$	0	0
$71$	101.823i	1.43413i	0.697005	+	0.717066i	$0.254515\pi$
$71$	101.823i	1.43413i	−0.697005	+	0.717066i	$0.745485\pi$
$72$	0	0
$73$	97.0000i	1.32877i	0.747392	+	0.664384i	$0.231306\pi$
$73$	97.0000i	1.32877i	−0.747392	+	0.664384i	$0.768694\pi$
$74$	− 35.3553i	− 0.477775i
$75$	0	0
$76$	−58.0000	−0.763158
$77$	−42.4264	−0.550992
$78$	0	0
$79$	−77.0000	−0.974684	−0.487342	−	0.873211i	$-0.662033\pi$
$79$	−77.0000	−0.974684	−0.487342	+	0.873211i	$0.662033\pi$
$80$	0	0
$81$	0	0
$82$	24.0000i	0.292683i
$83$	−118.794	−1.43125	−0.715626	−	0.698484i	$-0.753859\pi$
$83$	−118.794	−1.43125	−0.715626	+	0.698484i	$0.753859\pi$
$84$	0	0
$85$	0	0
$86$	− 19.7990i	− 0.230221i
$87$	0	0
$88$	24.0000i	0.272727i
$89$	76.3675i	0.858062i	0.903290	+	0.429031i	$0.141145\pi$
$89$	76.3675i	0.858062i	−0.903290	+	0.429031i	$0.858855\pi$
$90$	0	0
$91$	−5.00000	−0.0549451
$92$	−16.9706	−0.184463
$93$	0	0
$94$	−12.0000	−0.127660
$95$	0	0
$96$	0	0
$97$	− 49.0000i	− 0.505155i	−0.967577	−	0.252577i	$-0.918722\pi$
$97$	− 49.0000i	− 0.505155i	0.967577	−	0.252577i	$-0.0812782\pi$
$98$	33.9411	0.346338
$99$	0	0
$100$	0	0
$101$	135.765i	1.34420i	0.740459	+	0.672101i	$0.234608\pi$
$101$	135.765i	1.34420i	−0.740459	+	0.672101i	$0.765392\pi$
$102$	0	0
$103$	163.000i	1.58252i	0.611477	+	0.791262i	$0.290576\pi$
$103$	163.000i	1.58252i	−0.611477	+	0.791262i	$0.709424\pi$
$104$	2.82843i	0.0271964i
$105$	0	0
$106$	−72.0000	−0.679245
$107$	−25.4558	−0.237905	−0.118953	−	0.992900i	$-0.537954\pi$
$107$	−25.4558	−0.237905	−0.118953	+	0.992900i	$0.537954\pi$
$108$	0	0
$109$	−2.00000	−0.0183486	−0.00917431	−	0.999958i	$-0.502920\pi$
$109$	−2.00000	−0.0183486	−0.00917431	+	0.999958i	$0.502920\pi$
$110$	0	0
$111$	0	0
$112$	20.0000i	0.178571i
$113$	−110.309	−0.976183	−0.488091	−	0.872793i	$-0.662307\pi$
$113$	−110.309	−0.976183	−0.488091	+	0.872793i	$0.662307\pi$
$114$	0	0
$115$	0	0
$116$	33.9411i	0.292596i
$117$	0	0
$118$	− 132.000i	− 1.11864i
$119$	− 127.279i	− 1.06957i
$120$	0	0
$121$	49.0000	0.404959
$122$	32.5269	0.266614
$123$	0	0
$124$	−20.0000	−0.161290
$125$	0	0
$126$	0	0
$127$	− 178.000i	− 1.40157i	−0.713370	−	0.700787i	$-0.752832\pi$
$127$	− 178.000i	− 1.40157i	0.713370	−	0.700787i	$-0.247168\pi$
$128$	11.3137	0.0883883
$129$	0	0
$130$	0	0
$131$	118.794i	0.906824i	0.891301	+	0.453412i	$0.149793\pi$
$131$	118.794i	0.906824i	−0.891301	+	0.453412i	$0.850207\pi$
$132$	0	0
$133$	− 145.000i	− 1.09023i
$134$	− 26.8701i	− 0.200523i
$135$	0	0
$136$	−72.0000	−0.529412
$137$	93.3381	0.681300	0.340650	−	0.940190i	$-0.389353\pi$
$137$	93.3381	0.681300	0.340650	+	0.940190i	$0.389353\pi$
$138$	0	0
$139$	163.000	1.17266	0.586331	−	0.810072i	$-0.300572\pi$
$139$	163.000	1.17266	0.586331	+	0.810072i	$0.300572\pi$
$140$	0	0
$141$	0	0
$142$	144.000i	1.01408i
$143$	−8.48528	−0.0593376
$144$	0	0
$145$	0	0
$146$	137.179i	0.939580i
$147$	0	0
$148$	− 50.0000i	− 0.337838i
$149$	33.9411i	0.227793i	0.993493	+	0.113896i	$0.0363332\pi$
$149$	33.9411i	0.227793i	−0.993493	+	0.113896i	$0.963667\pi$
$150$	0	0
$151$	149.000	0.986755	0.493377	−	0.869815i	$-0.335762\pi$
$151$	149.000	0.986755	0.493377	+	0.869815i	$0.335762\pi$
$152$	−82.0244	−0.539634
$153$	0	0
$154$	−60.0000	−0.389610
$155$	0	0
$156$	0	0
$157$	242.000i	1.54140i	0.637197	+	0.770701i	$0.280094\pi$
$157$	242.000i	1.54140i	−0.637197	+	0.770701i	$0.719906\pi$
$158$	−108.894	−0.689205
$159$	0	0
$160$	0	0
$161$	− 42.4264i	− 0.263518i
$162$	0	0
$163$	− 173.000i	− 1.06135i	−0.847575	−	0.530675i	$-0.821939\pi$
$163$	− 173.000i	− 1.06135i	0.847575	−	0.530675i	$-0.178061\pi$
$164$	33.9411i	0.206958i
$165$	0	0
$166$	−168.000	−1.01205
$167$	−195.161	−1.16863	−0.584316	−	0.811526i	$-0.698637\pi$
$167$	−195.161	−1.16863	−0.584316	+	0.811526i	$0.698637\pi$
$168$	0	0
$169$	168.000	0.994083
$170$	0	0
$171$	0	0
$172$	− 28.0000i	− 0.162791i
$173$	−322.441	−1.86382	−0.931910	−	0.362691i	$-0.881858\pi$
$173$	−322.441	−1.86382	−0.931910	+	0.362691i	$0.881858\pi$
$174$	0	0
$175$	0	0
$176$	33.9411i	0.192847i
$177$	0	0
$178$	108.000i	0.606742i
$179$	305.470i	1.70654i	0.521472	+	0.853269i	$0.325383\pi$
$179$	305.470i	1.70654i	−0.521472	+	0.853269i	$0.674617\pi$
$180$	0	0
$181$	263.000	1.45304	0.726519	−	0.687146i	$-0.241137\pi$
$181$	263.000	1.45304	0.726519	+	0.687146i	$0.241137\pi$
$182$	−7.07107	−0.0388520
$183$	0	0
$184$	−24.0000	−0.130435
$185$	0	0
$186$	0	0
$187$	− 216.000i	− 1.15508i
$188$	−16.9706	−0.0902690
$189$	0	0
$190$	0	0
$191$	− 144.250i	− 0.755234i	−0.925962	−	0.377617i	$-0.876744\pi$
$191$	− 144.250i	− 0.755234i	0.925962	−	0.377617i	$-0.123256\pi$
$192$	0	0
$193$	− 71.0000i	− 0.367876i	−0.982938	−	0.183938i	$-0.941115\pi$
$193$	− 71.0000i	− 0.367876i	0.982938	−	0.183938i	$-0.0588845\pi$
$194$	− 69.2965i	− 0.357198i
$195$	0	0
$196$	48.0000	0.244898
$197$	76.3675	0.387652	0.193826	−	0.981036i	$-0.437910\pi$
$197$	76.3675	0.387652	0.193826	+	0.981036i	$0.437910\pi$
$198$	0	0
$199$	−173.000	−0.869347	−0.434673	−	0.900588i	$-0.643136\pi$
$199$	−173.000	−0.869347	−0.434673	+	0.900588i	$0.643136\pi$
$200$	0	0
$201$	0	0
$202$	192.000i	0.950495i
$203$	−84.8528	−0.417994
$204$	0	0
$205$	0	0
$206$	230.517i	1.11901i
$207$	0	0
$208$	4.00000i	0.0192308i
$209$	− 246.073i	− 1.17738i
$210$	0	0
$211$	341.000	1.61611	0.808057	−	0.589104i	$-0.200519\pi$
$211$	341.000	1.61611	0.808057	+	0.589104i	$0.200519\pi$
$212$	−101.823	−0.480299
$213$	0	0
$214$	−36.0000	−0.168224
$215$	0	0
$216$	0	0
$217$	− 50.0000i	− 0.230415i
$218$	−2.82843	−0.0129744
$219$	0	0
$220$	0	0
$221$	− 25.4558i	− 0.115185i
$222$	0	0
$223$	58.0000i	0.260090i	0.991508	+	0.130045i	$0.0415121\pi$
$223$	58.0000i	0.260090i	−0.991508	+	0.130045i	$0.958488\pi$
$224$	28.2843i	0.126269i
$225$	0	0
$226$	−156.000	−0.690265
$227$	−135.765	−0.598082	−0.299041	−	0.954240i	$-0.596667\pi$
$227$	−135.765	−0.598082	−0.299041	+	0.954240i	$0.596667\pi$
$228$	0	0
$229$	262.000	1.14410	0.572052	−	0.820217i	$-0.306147\pi$
$229$	262.000	1.14410	0.572052	+	0.820217i	$0.306147\pi$
$230$	0	0
$231$	0	0
$232$	48.0000i	0.206897i
$233$	305.470	1.31103	0.655515	−	0.755182i	$-0.272451\pi$
$233$	305.470	1.31103	0.655515	+	0.755182i	$0.272451\pi$
$234$	0	0
$235$	0	0
$236$	− 186.676i	− 0.791001i
$237$	0	0
$238$	− 180.000i	− 0.756303i
$239$	− 169.706i	− 0.710065i	−0.934854	−	0.355033i	$-0.884470\pi$
$239$	− 169.706i	− 0.710065i	0.934854	−	0.355033i	$-0.115530\pi$
$240$	0	0
$241$	−121.000	−0.502075	−0.251037	−	0.967977i	$-0.580772\pi$
$241$	−121.000	−0.502075	−0.251037	+	0.967977i	$0.580772\pi$
$242$	69.2965	0.286349
$243$	0	0
$244$	46.0000	0.188525
$245$	0	0
$246$	0	0
$247$	− 29.0000i	− 0.117409i
$248$	−28.2843	−0.114049
$249$	0	0
$250$	0	0
$251$	356.382i	1.41985i	0.704278	+	0.709924i	$0.251271\pi$
$251$	356.382i	1.41985i	−0.704278	+	0.709924i	$0.748729\pi$
$252$	0	0
$253$	− 72.0000i	− 0.284585i
$254$	− 251.730i	− 0.991063i
$255$	0	0
$256$	16.0000	0.0625000
$257$	84.8528	0.330167	0.165083	−	0.986280i	$-0.447211\pi$
$257$	84.8528	0.330167	0.165083	+	0.986280i	$0.447211\pi$
$258$	0	0
$259$	125.000	0.482625
$260$	0	0
$261$	0	0
$262$	168.000i	0.641221i
$263$	−169.706	−0.645269	−0.322634	−	0.946524i	$-0.604568\pi$
$263$	−169.706	−0.645269	−0.322634	+	0.946524i	$0.604568\pi$
$264$	0	0
$265$	0	0
$266$	− 205.061i	− 0.770906i
$267$	0	0
$268$	− 38.0000i	− 0.141791i
$269$	280.014i	1.04095i	0.853878	+	0.520473i	$0.174244\pi$
$269$	280.014i	1.04095i	−0.853878	+	0.520473i	$0.825756\pi$
$270$	0	0
$271$	29.0000	0.107011	0.0535055	−	0.998568i	$-0.482961\pi$
$271$	29.0000	0.107011	0.0535055	+	0.998568i	$0.482961\pi$
$272$	−101.823	−0.374351
$273$	0	0
$274$	132.000	0.481752
$275$	0	0
$276$	0	0
$277$	− 382.000i	− 1.37906i	−0.724256	−	0.689531i	$-0.757817\pi$
$277$	− 382.000i	− 1.37906i	0.724256	−	0.689531i	$-0.242183\pi$
$278$	230.517	0.829197
$279$	0	0
$280$	0	0
$281$	− 220.617i	− 0.785115i	−0.919727	−	0.392558i	$-0.871590\pi$
$281$	− 220.617i	− 0.785115i	0.919727	−	0.392558i	$-0.128410\pi$
$282$	0	0
$283$	− 62.0000i	− 0.219081i	−0.993982	−	0.109541i	$-0.965062\pi$
$283$	− 62.0000i	− 0.219081i	0.993982	−	0.109541i	$-0.0349380\pi$
$284$	203.647i	0.717066i
$285$	0	0
$286$	−12.0000	−0.0419580
$287$	−84.8528	−0.295654
$288$	0	0
$289$	359.000	1.24221
$290$	0	0
$291$	0	0
$292$	194.000i	0.664384i
$293$	−313.955	−1.07152	−0.535760	−	0.844370i	$-0.679975\pi$
$293$	−313.955	−1.07152	−0.535760	+	0.844370i	$0.679975\pi$
$294$	0	0
$295$	0	0
$296$	− 70.7107i	− 0.238887i
$297$	0	0
$298$	48.0000i	0.161074i
$299$	− 8.48528i	− 0.0283789i
$300$	0	0
$301$	70.0000	0.232558
$302$	210.718	0.697741
$303$	0	0
$304$	−116.000	−0.381579
$305$	0	0
$306$	0	0
$307$	− 34.0000i	− 0.110749i	−0.998466	−	0.0553746i	$-0.982365\pi$
$307$	− 34.0000i	− 0.110749i	0.998466	−	0.0553746i	$-0.0176353\pi$
$308$	−84.8528	−0.275496
$309$	0	0
$310$	0	0
$311$	− 59.3970i	− 0.190987i	−0.995430	−	0.0954935i	$-0.969557\pi$
$311$	− 59.3970i	− 0.190987i	0.995430	−	0.0954935i	$-0.0304429\pi$
$312$	0	0
$313$	− 239.000i	− 0.763578i	−0.924249	−	0.381789i	$-0.875308\pi$
$313$	− 239.000i	− 0.763578i	0.924249	−	0.381789i	$-0.124692\pi$
$314$	342.240i	1.08994i
$315$	0	0
$316$	−154.000	−0.487342
$317$	16.9706	0.0535349	0.0267674	−	0.999642i	$-0.491479\pi$
$317$	16.9706	0.0535349	0.0267674	+	0.999642i	$0.491479\pi$
$318$	0	0
$319$	−144.000	−0.451411
$320$	0	0
$321$	0	0
$322$	− 60.0000i	− 0.186335i
$323$	738.219	2.28551
$324$	0	0
$325$	0	0
$326$	− 244.659i	− 0.750488i
$327$	0	0
$328$	48.0000i	0.146341i
$329$	− 42.4264i	− 0.128956i
$330$	0	0
$331$	−499.000	−1.50755	−0.753776	−	0.657131i	$-0.771770\pi$
$331$	−499.000	−1.50755	−0.753776	+	0.657131i	$0.771770\pi$
$332$	−237.588	−0.715626
$333$	0	0
$334$	−276.000	−0.826347
$335$	0	0
$336$	0	0
$337$	− 145.000i	− 0.430267i	−0.976585	−	0.215134i	$-0.930981\pi$
$337$	− 145.000i	− 0.430267i	0.976585	−	0.215134i	$-0.0690187\pi$
$338$	237.588	0.702923
$339$	0	0
$340$	0	0
$341$	− 84.8528i	− 0.248835i
$342$	0	0
$343$	365.000i	1.06414i
$344$	− 39.5980i	− 0.115110i
$345$	0	0
$346$	−456.000	−1.31792
$347$	441.235	1.27157	0.635785	−	0.771866i	$-0.280677\pi$
$347$	441.235	1.27157	0.635785	+	0.771866i	$0.280677\pi$
$348$	0	0
$349$	121.000	0.346705	0.173352	−	0.984860i	$-0.444540\pi$
$349$	121.000	0.346705	0.173352	+	0.984860i	$0.444540\pi$
$350$	0	0
$351$	0	0
$352$	48.0000i	0.136364i
$353$	441.235	1.24996	0.624978	−	0.780642i	$-0.285108\pi$
$353$	441.235	1.24996	0.624978	+	0.780642i	$0.285108\pi$
$354$	0	0
$355$	0	0
$356$	152.735i	0.429031i
$357$	0	0
$358$	432.000i	1.20670i
$359$	432.749i	1.20543i	0.797957	+	0.602715i	$0.205914\pi$
$359$	432.749i	1.20543i	−0.797957	+	0.602715i	$0.794086\pi$
$360$	0	0
$361$	480.000	1.32964
$362$	371.938	1.02745
$363$	0	0
$364$	−10.0000	−0.0274725
$365$	0	0
$366$	0	0
$367$	149.000i	0.405995i	0.979179	+	0.202997i	$0.0650683\pi$
$367$	149.000i	0.405995i	−0.979179	+	0.202997i	$0.934932\pi$
$368$	−33.9411	−0.0922313
$369$	0	0
$370$	0	0
$371$	− 254.558i	− 0.686141i
$372$	0	0
$373$	361.000i	0.967828i	0.875115	+	0.483914i	$0.160785\pi$
$373$	361.000i	0.967828i	−0.875115	+	0.483914i	$0.839215\pi$
$374$	− 305.470i	− 0.816765i
$375$	0	0
$376$	−24.0000	−0.0638298
$377$	−16.9706	−0.0450148
$378$	0	0
$379$	−173.000	−0.456464	−0.228232	−	0.973607i	$-0.573295\pi$
$379$	−173.000	−0.456464	−0.228232	+	0.973607i	$0.573295\pi$
$380$	0	0
$381$	0	0
$382$	− 204.000i	− 0.534031i
$383$	678.823	1.77238	0.886191	−	0.463320i	$-0.153342\pi$
$383$	678.823	1.77238	0.886191	+	0.463320i	$0.153342\pi$
$384$	0	0
$385$	0	0
$386$	− 100.409i	− 0.260127i
$387$	0	0
$388$	− 98.0000i	− 0.252577i
$389$	− 415.779i	− 1.06884i	−0.845219	−	0.534420i	$-0.820530\pi$
$389$	− 415.779i	− 1.06884i	0.845219	−	0.534420i	$-0.179470\pi$
$390$	0	0
$391$	216.000	0.552430
$392$	67.8823	0.173169
$393$	0	0
$394$	108.000	0.274112
$395$	0	0
$396$	0	0
$397$	− 286.000i	− 0.720403i	−0.932875	−	0.360202i	$-0.882708\pi$
$397$	− 286.000i	− 0.720403i	0.932875	−	0.360202i	$-0.117292\pi$
$398$	−244.659	−0.614721
$399$	0	0
$400$	0	0
$401$	− 339.411i	− 0.846412i	−0.906033	−	0.423206i	$-0.860905\pi$
$401$	− 339.411i	− 0.846412i	0.906033	−	0.423206i	$-0.139095\pi$
$402$	0	0
$403$	− 10.0000i	− 0.0248139i
$404$	271.529i	0.672101i
$405$	0	0
$406$	−120.000	−0.295567
$407$	212.132	0.521209
$408$	0	0
$409$	−215.000	−0.525672	−0.262836	−	0.964840i	$-0.584658\pi$
$409$	−215.000	−0.525672	−0.262836	+	0.964840i	$0.584658\pi$
$410$	0	0
$411$	0	0
$412$	326.000i	0.791262i
$413$	466.690	1.13000
$414$	0	0
$415$	0	0
$416$	5.65685i	0.0135982i
$417$	0	0
$418$	− 348.000i	− 0.832536i
$419$	− 602.455i	− 1.43784i	−0.695093	−	0.718920i	$-0.744637\pi$
$419$	− 602.455i	− 1.43784i	0.695093	−	0.718920i	$-0.255363\pi$
$420$	0	0
$421$	455.000	1.08076	0.540380	−	0.841421i	$-0.318280\pi$
$421$	455.000	1.08076	0.540380	+	0.841421i	$0.318280\pi$
$422$	482.247	1.14276
$423$	0	0
$424$	−144.000	−0.339623
$425$	0	0
$426$	0	0
$427$	115.000i	0.269321i
$428$	−50.9117	−0.118953
$429$	0	0
$430$	0	0
$431$	− 330.926i	− 0.767810i	−0.923373	−	0.383905i	$-0.874579\pi$
$431$	− 330.926i	− 0.767810i	0.923373	−	0.383905i	$-0.125421\pi$
$432$	0	0
$433$	− 218.000i	− 0.503464i	−0.967797	−	0.251732i	$-0.919000\pi$
$433$	− 218.000i	− 0.503464i	0.967797	−	0.251732i	$-0.0810002\pi$
$434$	− 70.7107i	− 0.162928i
$435$	0	0
$436$	−4.00000	−0.00917431
$437$	246.073	0.563096
$438$	0	0
$439$	−374.000	−0.851936	−0.425968	−	0.904738i	$-0.640066\pi$
$439$	−374.000	−0.851936	−0.425968	+	0.904738i	$0.640066\pi$
$440$	0	0
$441$	0	0
$442$	− 36.0000i	− 0.0814480i
$443$	−16.9706	−0.0383083	−0.0191541	−	0.999817i	$-0.506097\pi$
$443$	−16.9706	−0.0383083	−0.0191541	+	0.999817i	$0.506097\pi$
$444$	0	0
$445$	0	0
$446$	82.0244i	0.183911i
$447$	0	0
$448$	40.0000i	0.0892857i
$449$	483.661i	1.07720i	0.842563	+	0.538598i	$0.181046\pi$
$449$	483.661i	1.07720i	−0.842563	+	0.538598i	$0.818954\pi$
$450$	0	0
$451$	−144.000	−0.319290
$452$	−220.617	−0.488091
$453$	0	0
$454$	−192.000	−0.422907
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 0.0481400i	−0.999710	−	0.0240700i	$-0.992338\pi$
$457$	− 22.0000i	− 0.0481400i	0.999710	−	0.0240700i	$-0.00766247\pi$
$458$	370.524	0.809004
$459$	0	0
$460$	0	0
$461$	− 398.808i	− 0.865094i	−0.901611	−	0.432547i	$-0.857615\pi$
$461$	− 398.808i	− 0.865094i	0.901611	−	0.432547i	$-0.142385\pi$
$462$	0	0
$463$	883.000i	1.90713i	0.301191	+	0.953564i	$0.402616\pi$
$463$	883.000i	1.90713i	−0.301191	+	0.953564i	$0.597384\pi$
$464$	67.8823i	0.146298i
$465$	0	0
$466$	432.000	0.927039
$467$	−127.279	−0.272547	−0.136273	−	0.990671i	$-0.543513\pi$
$467$	−127.279	−0.272547	−0.136273	+	0.990671i	$0.543513\pi$
$468$	0	0
$469$	95.0000	0.202559
$470$	0	0
$471$	0	0
$472$	− 264.000i	− 0.559322i
$473$	118.794	0.251150
$474$	0	0
$475$	0	0
$476$	− 254.558i	− 0.534787i
$477$	0	0
$478$	− 240.000i	− 0.502092i
$479$	− 390.323i	− 0.814870i	−0.913234	−	0.407435i	$-0.866423\pi$
$479$	− 390.323i	− 0.814870i	0.913234	−	0.407435i	$-0.133577\pi$
$480$	0	0
$481$	25.0000	0.0519751
$482$	−171.120	−0.355020
$483$	0	0
$484$	98.0000	0.202479
$485$	0	0
$486$	0	0
$487$	317.000i	0.650924i	0.945555	+	0.325462i	$0.105520\pi$
$487$	317.000i	0.650924i	−0.945555	+	0.325462i	$0.894480\pi$
$488$	65.0538	0.133307
$489$	0	0
$490$	0	0
$491$	− 364.867i	− 0.743110i	−0.928411	−	0.371555i	$-0.878825\pi$
$491$	− 364.867i	− 0.743110i	0.928411	−	0.371555i	$-0.121175\pi$
$492$	0	0
$493$	− 432.000i	− 0.876268i
$494$	− 41.0122i	− 0.0830206i
$495$	0	0
$496$	−40.0000	−0.0806452
$497$	−509.117	−1.02438
$498$	0	0
$499$	−710.000	−1.42285	−0.711423	−	0.702764i	$-0.751949\pi$
$499$	−710.000	−1.42285	−0.711423	+	0.702764i	$0.751949\pi$
$500$	0	0
$501$	0	0
$502$	504.000i	1.00398i
$503$	280.014	0.556688	0.278344	−	0.960481i	$-0.410214\pi$
$503$	280.014	0.556688	0.278344	+	0.960481i	$0.410214\pi$
$504$	0	0
$505$	0	0
$506$	− 101.823i	− 0.201232i
$507$	0	0
$508$	− 356.000i	− 0.700787i
$509$	263.044i	0.516785i	0.966040	+	0.258393i	$0.0831928\pi$
$509$	263.044i	0.516785i	−0.966040	+	0.258393i	$0.916807\pi$
$510$	0	0
$511$	−485.000	−0.949119
$512$	22.6274	0.0441942
$513$	0	0
$514$	120.000	0.233463
$515$	0	0
$516$	0	0
$517$	− 72.0000i	− 0.139265i
$518$	176.777	0.341268
$519$	0	0
$520$	0	0
$521$	687.308i	1.31921i	0.751613	+	0.659604i	$0.229276\pi$
$521$	687.308i	1.31921i	−0.751613	+	0.659604i	$0.770724\pi$
$522$	0	0
$523$	763.000i	1.45889i	0.684039	+	0.729446i	$0.260222\pi$
$523$	763.000i	1.45889i	−0.684039	+	0.729446i	$0.739778\pi$
$524$	237.588i	0.453412i
$525$	0	0
$526$	−240.000	−0.456274
$527$	254.558	0.483033
$528$	0	0
$529$	−457.000	−0.863894
$530$	0	0
$531$	0	0
$532$	− 290.000i	− 0.545113i
$533$	−16.9706	−0.0318397
$534$	0	0
$535$	0	0
$536$	− 53.7401i	− 0.100261i
$537$	0	0
$538$	396.000i	0.736059i
$539$	203.647i	0.377823i
$540$	0	0
$541$	−313.000	−0.578558	−0.289279	−	0.957245i	$-0.593416\pi$
$541$	−313.000	−0.578558	−0.289279	+	0.957245i	$0.593416\pi$
$542$	41.0122	0.0756683
$543$	0	0
$544$	−144.000	−0.264706
$545$	0	0
$546$	0	0
$547$	− 139.000i	− 0.254113i	−0.991895	−	0.127057i	$-0.959447\pi$
$547$	− 139.000i	− 0.254113i	0.991895	−	0.127057i	$-0.0405530\pi$
$548$	186.676	0.340650
$549$	0	0
$550$	0	0
$551$	− 492.146i	− 0.893188i
$552$	0	0
$553$	− 385.000i	− 0.696203i
$554$	− 540.230i	− 0.975144i
$555$	0	0
$556$	326.000	0.586331
$557$	280.014	0.502719	0.251359	−	0.967894i	$-0.419122\pi$
$557$	280.014	0.502719	0.251359	+	0.967894i	$0.419122\pi$
$558$	0	0
$559$	14.0000	0.0250447
$560$	0	0
$561$	0	0
$562$	− 312.000i	− 0.555160i
$563$	−593.970	−1.05501	−0.527504	−	0.849552i	$-0.676872\pi$
$563$	−593.970	−1.05501	−0.527504	+	0.849552i	$0.676872\pi$
$564$	0	0
$565$	0	0
$566$	− 87.6812i	− 0.154914i
$567$	0	0
$568$	288.000i	0.507042i
$569$	− 619.426i	− 1.08862i	−0.838884	−	0.544311i	$-0.816791\pi$
$569$	− 619.426i	− 1.08862i	0.838884	−	0.544311i	$-0.183209\pi$
$570$	0	0
$571$	−163.000	−0.285464	−0.142732	−	0.989761i	$-0.545589\pi$
$571$	−163.000	−0.285464	−0.142732	+	0.989761i	$0.545589\pi$
$572$	−16.9706	−0.0296688
$573$	0	0
$574$	−120.000	−0.209059
$575$	0	0
$576$	0	0
$577$	1127.00i	1.95321i	0.215050	+	0.976603i	$0.431009\pi$
$577$	1127.00i	1.95321i	−0.215050	+	0.976603i	$0.568991\pi$
$578$	507.703	0.878378
$579$	0	0
$580$	0	0
$581$	− 593.970i	− 1.02232i
$582$	0	0
$583$	− 432.000i	− 0.740995i
$584$	274.357i	0.469790i
$585$	0	0
$586$	−444.000	−0.757679
$587$	1009.75	1.72018	0.860092	−	0.510138i	$-0.170406\pi$
$587$	1009.75	1.72018	0.860092	+	0.510138i	$0.170406\pi$
$588$	0	0
$589$	290.000	0.492360
$590$	0	0
$591$	0	0
$592$	− 100.000i	− 0.168919i
$593$	356.382	0.600981	0.300491	−	0.953785i	$-0.402850\pi$
$593$	356.382	0.600981	0.300491	+	0.953785i	$0.402850\pi$
$594$	0	0
$595$	0	0
$596$	67.8823i	0.113896i
$597$	0	0
$598$	− 12.0000i	− 0.0200669i
$599$	899.440i	1.50157i	0.660547	+	0.750784i	$0.270324\pi$
$599$	899.440i	1.50157i	−0.660547	+	0.750784i	$0.729676\pi$
$600$	0	0
$601$	−166.000	−0.276206	−0.138103	−	0.990418i	$-0.544101\pi$
$601$	−166.000	−0.276206	−0.138103	+	0.990418i	$0.544101\pi$
$602$	98.9949	0.164443
$603$	0	0
$604$	298.000	0.493377
$605$	0	0
$606$	0	0
$607$	− 523.000i	− 0.861614i	−0.902444	−	0.430807i	$-0.858229\pi$
$607$	− 523.000i	− 0.861614i	0.902444	−	0.430807i	$-0.141771\pi$
$608$	−164.049	−0.269817
$609$	0	0
$610$	0	0
$611$	− 8.48528i	− 0.0138875i
$612$	0	0
$613$	− 335.000i	− 0.546493i	−0.961944	−	0.273246i	$-0.911903\pi$
$613$	− 335.000i	− 0.546493i	0.961944	−	0.273246i	$-0.0880974\pi$
$614$	− 48.0833i	− 0.0783115i
$615$	0	0
$616$	−120.000	−0.194805
$617$	−500.632	−0.811396	−0.405698	−	0.914007i	$-0.632972\pi$
$617$	−500.632	−0.811396	−0.405698	+	0.914007i	$0.632972\pi$
$618$	0	0
$619$	−5.00000	−0.00807754	−0.00403877	−	0.999992i	$-0.501286\pi$
$619$	−5.00000	−0.00807754	−0.00403877	+	0.999992i	$0.501286\pi$
$620$	0	0
$621$	0	0
$622$	− 84.0000i	− 0.135048i
$623$	−381.838	−0.612902
$624$	0	0
$625$	0	0
$626$	− 337.997i	− 0.539931i
$627$	0	0
$628$	484.000i	0.770701i
$629$	636.396i	1.01176i
$630$	0	0
$631$	245.000	0.388273	0.194136	−	0.980975i	$-0.437810\pi$
$631$	245.000	0.388273	0.194136	+	0.980975i	$0.437810\pi$
$632$	−217.789	−0.344603
$633$	0	0
$634$	24.0000	0.0378549
$635$	0	0
$636$	0	0
$637$	24.0000i	0.0376766i
$638$	−203.647	−0.319196
$639$	0	0
$640$	0	0
$641$	644.881i	1.00606i	0.864270	+	0.503028i	$0.167781\pi$
$641$	644.881i	1.00606i	−0.864270	+	0.503028i	$0.832219\pi$
$642$	0	0
$643$	82.0000i	0.127527i	0.997965	+	0.0637636i	$0.0203104\pi$
$643$	82.0000i	0.127527i	−0.997965	+	0.0637636i	$0.979690\pi$
$644$	− 84.8528i	− 0.131759i
$645$	0	0
$646$	1044.00	1.61610
$647$	−458.205	−0.708200	−0.354100	−	0.935208i	$-0.615213\pi$
$647$	−458.205	−0.708200	−0.354100	+	0.935208i	$0.615213\pi$
$648$	0	0
$649$	792.000	1.22034
$650$	0	0
$651$	0	0
$652$	− 346.000i	− 0.530675i
$653$	322.441	0.493784	0.246892	−	0.969043i	$-0.420591\pi$
$653$	322.441	0.493784	0.246892	+	0.969043i	$0.420591\pi$
$654$	0	0
$655$	0	0
$656$	67.8823i	0.103479i
$657$	0	0
$658$	− 60.0000i	− 0.0911854i
$659$	1035.20i	1.57087i	0.618943	+	0.785436i	$0.287561\pi$
$659$	1035.20i	1.57087i	−0.618943	+	0.785436i	$0.712439\pi$
$660$	0	0
$661$	143.000	0.216339	0.108169	−	0.994132i	$-0.465501\pi$
$661$	143.000	0.216339	0.108169	+	0.994132i	$0.465501\pi$
$662$	−705.693	−1.06600
$663$	0	0
$664$	−336.000	−0.506024
$665$	0	0
$666$	0	0
$667$	− 144.000i	− 0.215892i
$668$	−390.323	−0.584316
$669$	0	0
$670$	0	0
$671$	195.161i	0.290852i
$672$	0	0
$673$	− 671.000i	− 0.997028i	−0.866882	−	0.498514i	$-0.833879\pi$
$673$	− 671.000i	− 0.997028i	0.866882	−	0.498514i	$-0.166121\pi$
$674$	− 205.061i	− 0.304245i
$675$	0	0
$676$	336.000	0.497041
$677$	−8.48528	−0.0125337	−0.00626683	−	0.999980i	$-0.501995\pi$
$677$	−8.48528	−0.0125337	−0.00626683	+	0.999980i	$0.501995\pi$
$678$	0	0
$679$	245.000	0.360825
$680$	0	0
$681$	0	0
$682$	− 120.000i	− 0.175953i
$683$	101.823	0.149083	0.0745413	−	0.997218i	$-0.476251\pi$
$683$	101.823	0.149083	0.0745413	+	0.997218i	$0.476251\pi$
$684$	0	0
$685$	0	0
$686$	516.188i	0.752461i
$687$	0	0
$688$	− 56.0000i	− 0.0813953i
$689$	− 50.9117i	− 0.0738921i
$690$	0	0
$691$	−850.000	−1.23010	−0.615051	−	0.788488i	$-0.710864\pi$
$691$	−850.000	−1.23010	−0.615051	+	0.788488i	$0.710864\pi$
$692$	−644.881	−0.931910
$693$	0	0
$694$	624.000	0.899135
$695$	0	0
$696$	0	0
$697$	− 432.000i	− 0.619799i
$698$	171.120	0.245157
$699$	0	0
$700$	0	0
$701$	− 280.014i	− 0.399450i	−0.979852	−	0.199725i	$-0.935995\pi$
$701$	− 280.014i	− 0.399450i	0.979852	−	0.199725i	$-0.0640048\pi$
$702$	0	0
$703$	725.000i	1.03129i
$704$	67.8823i	0.0964237i
$705$	0	0
$706$	624.000	0.883853
$707$	−678.823	−0.960145
$708$	0	0
$709$	1201.00	1.69394	0.846968	−	0.531645i	$-0.178426\pi$
$709$	1201.00	1.69394	0.846968	+	0.531645i	$0.178426\pi$
$710$	0	0
$711$	0	0
$712$	216.000i	0.303371i
$713$	84.8528	0.119008
$714$	0	0
$715$	0	0
$716$	610.940i	0.853269i
$717$	0	0
$718$	612.000i	0.852368i
$719$	− 967.322i	− 1.34537i	−0.739928	−	0.672686i	$-0.765141\pi$
$719$	− 967.322i	− 1.34537i	0.739928	−	0.672686i	$-0.234859\pi$
$720$	0	0
$721$	−815.000	−1.13037
$722$	678.823	0.940197
$723$	0	0
$724$	526.000	0.726519
$725$	0	0
$726$	0	0
$727$	950.000i	1.30674i	0.757039	+	0.653370i	$0.226645\pi$
$727$	950.000i	1.30674i	−0.757039	+	0.653370i	$0.773355\pi$
$728$	−14.1421	−0.0194260
$729$	0	0
$730$	0	0
$731$	356.382i	0.487526i
$732$	0	0
$733$	− 98.0000i	− 0.133697i	−0.997763	−	0.0668486i	$-0.978706\pi$
$733$	− 98.0000i	− 0.133697i	0.997763	−	0.0668486i	$-0.0212944\pi$
$734$	210.718i	0.287081i
$735$	0	0
$736$	−48.0000	−0.0652174
$737$	161.220	0.218752
$738$	0	0
$739$	−1262.00	−1.70771	−0.853857	−	0.520508i	$-0.825742\pi$
$739$	−1262.00	−1.70771	−0.853857	+	0.520508i	$0.825742\pi$
$740$	0	0
$741$	0	0
$742$	− 360.000i	− 0.485175i
$743$	−772.161	−1.03925	−0.519624	−	0.854395i	$-0.673928\pi$
$743$	−772.161	−1.03925	−0.519624	+	0.854395i	$0.673928\pi$
$744$	0	0
$745$	0	0
$746$	510.531i	0.684358i
$747$	0	0
$748$	− 432.000i	− 0.577540i
$749$	− 127.279i	− 0.169932i
$750$	0	0
$751$	197.000	0.262317	0.131158	−	0.991361i	$-0.458130\pi$
$751$	197.000	0.262317	0.131158	+	0.991361i	$0.458130\pi$
$752$	−33.9411	−0.0451345
$753$	0	0
$754$	−24.0000	−0.0318302
$755$	0	0
$756$	0	0
$757$	− 241.000i	− 0.318362i	−0.987249	−	0.159181i	$-0.949115\pi$
$757$	− 241.000i	− 0.318362i	0.987249	−	0.159181i	$-0.0508853\pi$
$758$	−244.659	−0.322769
$759$	0	0
$760$	0	0
$761$	500.632i	0.657860i	0.944354	+	0.328930i	$0.106688\pi$
$761$	500.632i	0.657860i	−0.944354	+	0.328930i	$0.893312\pi$
$762$	0	0
$763$	− 10.0000i	− 0.0131062i
$764$	− 288.500i	− 0.377617i
$765$	0	0
$766$	960.000	1.25326
$767$	93.3381	0.121692
$768$	0	0
$769$	−431.000	−0.560468	−0.280234	−	0.959932i	$-0.590412\pi$
$769$	−431.000	−0.560468	−0.280234	+	0.959932i	$0.590412\pi$
$770$	0	0
$771$	0	0
$772$	− 142.000i	− 0.183938i
$773$	−254.558	−0.329312	−0.164656	−	0.986351i	$-0.552651\pi$
$773$	−254.558	−0.329312	−0.164656	+	0.986351i	$0.552651\pi$
$774$	0	0
$775$	0	0
$776$	− 138.593i	− 0.178599i
$777$	0	0
$778$	− 588.000i	− 0.755784i
$779$	− 492.146i	− 0.631767i
$780$	0	0
$781$	−864.000	−1.10627
$782$	305.470	0.390627
$783$	0	0
$784$	96.0000	0.122449
$785$	0	0
$786$	0	0
$787$	125.000i	0.158831i	0.996842	+	0.0794155i	$0.0253054\pi$
$787$	125.000i	0.158831i	−0.996842	+	0.0794155i	$0.974695\pi$
$788$	152.735	0.193826
$789$	0	0
$790$	0	0
$791$	− 551.543i	− 0.697273i
$792$	0	0
$793$	23.0000i	0.0290038i
$794$	− 404.465i	− 0.509402i
$795$	0	0
$796$	−346.000	−0.434673
$797$	848.528	1.06465	0.532326	−	0.846539i	$-0.321318\pi$
$797$	848.528	1.06465	0.532326	+	0.846539i	$0.321318\pi$
$798$	0	0
$799$	216.000	0.270338
$800$	0	0
$801$	0	0
$802$	− 480.000i	− 0.598504i
$803$	−823.072	−1.02500
$804$	0	0
$805$	0	0
$806$	− 14.1421i	− 0.0175461i
$807$	0	0
$808$	384.000i	0.475248i
$809$	− 1120.06i	− 1.38450i	−0.721660	−	0.692248i	$-0.756620\pi$
$809$	− 1120.06i	− 1.38450i	0.721660	−	0.692248i	$-0.243380\pi$
$810$	0	0
$811$	−322.000	−0.397041	−0.198520	−	0.980097i	$-0.563614\pi$
$811$	−322.000	−0.397041	−0.198520	+	0.980097i	$0.563614\pi$
$812$	−169.706	−0.208997
$813$	0	0
$814$	300.000	0.368550
$815$	0	0
$816$	0	0
$817$	406.000i	0.496940i
$818$	−304.056	−0.371706
$819$	0	0
$820$	0	0
$821$	873.984i	1.06454i	0.846576	+	0.532268i	$0.178660\pi$
$821$	873.984i	1.06454i	−0.846576	+	0.532268i	$0.821340\pi$
$822$	0	0
$823$	− 269.000i	− 0.326853i	−0.986556	−	0.163426i	$-0.947745\pi$
$823$	− 269.000i	− 0.326853i	0.986556	−	0.163426i	$-0.0522547\pi$
$824$	461.034i	0.559507i
$825$	0	0
$826$	660.000	0.799031
$827$	−25.4558	−0.0307809	−0.0153905	−	0.999882i	$-0.504899\pi$
$827$	−25.4558	−0.0307809	−0.0153905	+	0.999882i	$0.504899\pi$
$828$	0	0
$829$	1105.00	1.33293	0.666466	−	0.745536i	$-0.267806\pi$
$829$	1105.00	1.33293	0.666466	+	0.745536i	$0.267806\pi$
$830$	0	0
$831$	0	0
$832$	8.00000i	0.00961538i
$833$	−610.940	−0.733422
$834$	0	0
$835$	0	0
$836$	− 492.146i	− 0.588692i
$837$	0	0
$838$	− 852.000i	− 1.01671i
$839$	118.794i	0.141590i	0.997491	+	0.0707950i	$0.0225536\pi$
$839$	118.794i	0.141590i	−0.997491	+	0.0707950i	$0.977446\pi$
$840$	0	0
$841$	553.000	0.657551
$842$	643.467	0.764213
$843$	0	0
$844$	682.000	0.808057
$845$	0	0
$846$	0	0
$847$	245.000i	0.289256i
$848$	−203.647	−0.240149
$849$	0	0
$850$	0	0
$851$	212.132i	0.249274i
$852$	0	0
$853$	− 1391.00i	− 1.63072i	−0.578958	−	0.815358i	$-0.696540\pi$
$853$	− 1391.00i	− 1.63072i	0.578958	−	0.815358i	$-0.303460\pi$
$854$	162.635i	0.190439i
$855$	0	0
$856$	−72.0000	−0.0841121
$857$	−543.058	−0.633673	−0.316837	−	0.948480i	$-0.602621\pi$
$857$	−543.058	−0.633673	−0.316837	+	0.948480i	$0.602621\pi$
$858$	0	0
$859$	−845.000	−0.983702	−0.491851	−	0.870679i	$-0.663680\pi$
$859$	−845.000	−0.983702	−0.491851	+	0.870679i	$0.663680\pi$
$860$	0	0
$861$	0	0
$862$	− 468.000i	− 0.542923i
$863$	−1247.34	−1.44535	−0.722675	−	0.691188i	$-0.757087\pi$
$863$	−1247.34	−1.44535	−0.722675	+	0.691188i	$0.757087\pi$
$864$	0	0
$865$	0	0
$866$	− 308.299i	− 0.356003i
$867$	0	0
$868$	− 100.000i	− 0.115207i
$869$	− 653.367i	− 0.751860i
$870$	0	0
$871$	19.0000	0.0218140
$872$	−5.65685	−0.00648722
$873$	0	0
$874$	348.000	0.398169
$875$	0	0
$876$	0	0
$877$	1151.00i	1.31243i	0.754575	+	0.656214i	$0.227843\pi$
$877$	1151.00i	1.31243i	−0.754575	+	0.656214i	$0.772157\pi$
$878$	−528.916	−0.602410
$879$	0	0
$880$	0	0
$881$	1298.25i	1.47361i	0.676107	+	0.736804i	$0.263666\pi$
$881$	1298.25i	1.47361i	−0.676107	+	0.736804i	$0.736334\pi$
$882$	0	0
$883$	− 677.000i	− 0.766704i	−0.923602	−	0.383352i	$-0.874770\pi$
$883$	− 677.000i	− 0.766704i	0.923602	−	0.383352i	$-0.125230\pi$
$884$	− 50.9117i	− 0.0575924i
$885$	0	0
$886$	−24.0000	−0.0270880
$887$	−1035.20	−1.16708	−0.583542	−	0.812083i	$-0.698334\pi$
$887$	−1035.20	−1.16708	−0.583542	+	0.812083i	$0.698334\pi$
$888$	0	0
$889$	890.000	1.00112
$890$	0	0
$891$	0	0
$892$	116.000i	0.130045i
$893$	246.073	0.275558
$894$	0	0
$895$	0	0
$896$	56.5685i	0.0631345i
$897$	0	0
$898$	684.000i	0.761693i
$899$	− 169.706i	− 0.188772i
$900$	0	0
$901$	1296.00	1.43840
$902$	−203.647	−0.225772
$903$	0	0
$904$	−312.000	−0.345133
$905$	0	0
$906$	0	0
$907$	5.00000i	0.00551268i	0.999996	+	0.00275634i	$0.000877371\pi$
$907$	5.00000i	0.00551268i	−0.999996	+	0.00275634i	$0.999123\pi$
$908$	−271.529	−0.299041
$909$	0	0
$910$	0	0
$911$	1001.26i	1.09908i	0.835467	+	0.549541i	$0.185197\pi$
$911$	1001.26i	1.09908i	−0.835467	+	0.549541i	$0.814803\pi$
$912$	0	0
$913$	− 1008.00i	− 1.10405i
$914$	− 31.1127i	− 0.0340402i
$915$	0	0
$916$	524.000	0.572052
$917$	−593.970	−0.647731
$918$	0	0
$919$	−1190.00	−1.29489	−0.647443	−	0.762114i	$-0.724162\pi$
$919$	−1190.00	−1.29489	−0.647443	+	0.762114i	$0.724162\pi$
$920$	0	0
$921$	0	0
$922$	− 564.000i	− 0.611714i
$923$	−101.823	−0.110318
$924$	0	0
$925$	0	0
$926$	1248.75i	1.34854i
$927$	0	0
$928$	96.0000i	0.103448i
$929$	475.176i	0.511492i	0.966744	+	0.255746i	$0.0823210\pi$
$929$	475.176i	0.511492i	−0.966744	+	0.255746i	$0.917679\pi$
$930$	0	0
$931$	−696.000	−0.747583
$932$	610.940	0.655515
$933$	0	0
$934$	−180.000	−0.192719
$935$	0	0
$936$	0	0
$937$	1775.00i	1.89434i	0.320727	+	0.947172i	$0.396073\pi$
$937$	1775.00i	1.89434i	−0.320727	+	0.947172i	$0.603927\pi$
$938$	134.350	0.143231
$939$	0	0
$940$	0	0
$941$	− 161.220i	− 0.171329i	−0.996324	−	0.0856644i	$-0.972699\pi$
$941$	− 161.220i	− 0.171329i	0.996324	−	0.0856644i	$-0.0273013\pi$
$942$	0	0
$943$	− 144.000i	− 0.152704i
$944$	− 373.352i	− 0.395500i
$945$	0	0
$946$	168.000	0.177590
$947$	−568.514	−0.600331	−0.300166	−	0.953887i	$-0.597042\pi$
$947$	−568.514	−0.600331	−0.300166	+	0.953887i	$0.597042\pi$
$948$	0	0
$949$	−97.0000	−0.102213
$950$	0	0
$951$	0	0
$952$	− 360.000i	− 0.378151i
$953$	−1807.36	−1.89650	−0.948250	−	0.317524i	$-0.897149\pi$
$953$	−1807.36	−1.89650	−0.948250	+	0.317524i	$0.897149\pi$
$954$	0	0
$955$	0	0
$956$	− 339.411i	− 0.355033i
$957$	0	0
$958$	− 552.000i	− 0.576200i
$959$	466.690i	0.486643i
$960$	0	0
$961$	−861.000	−0.895942
$962$	35.3553	0.0367519
$963$	0	0
$964$	−242.000	−0.251037
$965$	0	0
$966$	0	0
$967$	701.000i	0.724922i	0.931999	+	0.362461i	$0.118063\pi$
$967$	701.000i	0.724922i	−0.931999	+	0.362461i	$0.881937\pi$
$968$	138.593	0.143175
$969$	0	0
$970$	0	0
$971$	381.838i	0.393242i	0.980480	+	0.196621i	$0.0629968\pi$
$971$	381.838i	0.393242i	−0.980480	+	0.196621i	$0.937003\pi$
$972$	0	0
$973$	815.000i	0.837616i
$974$	448.306i	0.460273i
$975$	0	0
$976$	92.0000	0.0942623
$977$	−322.441	−0.330031	−0.165016	−	0.986291i	$-0.552767\pi$
$977$	−322.441	−0.330031	−0.165016	+	0.986291i	$0.552767\pi$
$978$	0	0
$979$	−648.000	−0.661900
$980$	0	0
$981$	0	0
$982$	− 516.000i	− 0.525458i
$983$	−42.4264	−0.0431601	−0.0215801	−	0.999767i	$-0.506870\pi$
$983$	−42.4264	−0.0431601	−0.0215801	+	0.999767i	$0.506870\pi$
$984$	0	0
$985$	0	0
$986$	− 610.940i	− 0.619615i
$987$	0	0
$988$	− 58.0000i	− 0.0587045i
$989$	118.794i	0.120115i
$990$	0	0
$991$	−475.000	−0.479314	−0.239657	−	0.970858i	$-0.577035\pi$
$991$	−475.000	−0.479314	−0.239657	+	0.970858i	$0.577035\pi$
$992$	−56.5685	−0.0570247
$993$	0	0
$994$	−720.000	−0.724346
$995$	0	0
$996$	0	0
$997$	− 1534.00i	− 1.53862i	−0.638878	−	0.769308i	$-0.720601\pi$
$997$	− 1534.00i	− 1.53862i	0.638878	−	0.769308i	$-0.279399\pi$
$998$	−1004.09	−1.00610
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.3.b.b.1349.4		4
3.2	odd	2	inner	1350.3.b.b.1349.2		4
5.2	odd	4		1350.3.d.d.701.2		2
5.3	odd	4		54.3.b.a.53.1	✓	2
5.4	even	2	inner	1350.3.b.b.1349.1		4
15.2	even	4		1350.3.d.d.701.1		2
15.8	even	4		54.3.b.a.53.2	yes	2
15.14	odd	2	inner	1350.3.b.b.1349.3		4
20.3	even	4		432.3.e.d.161.1		2
40.3	even	4		1728.3.e.f.1025.2		2
40.13	odd	4		1728.3.e.l.1025.2		2
45.13	odd	12		162.3.d.a.107.2		4
45.23	even	12		162.3.d.a.107.1		4
45.38	even	12		162.3.d.a.53.2		4
45.43	odd	12		162.3.d.a.53.1		4
60.23	odd	4		432.3.e.d.161.2		2
120.53	even	4		1728.3.e.l.1025.1		2
120.83	odd	4		1728.3.e.f.1025.1		2
180.23	odd	12		1296.3.q.i.593.2		4
180.43	even	12		1296.3.q.i.1025.2		4
180.83	odd	12		1296.3.q.i.1025.1		4
180.103	even	12		1296.3.q.i.593.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.b.a.53.1	✓	2	5.3	odd	4
54.3.b.a.53.2	yes	2	15.8	even	4
162.3.d.a.53.1		4	45.43	odd	12
162.3.d.a.53.2		4	45.38	even	12
162.3.d.a.107.1		4	45.23	even	12
162.3.d.a.107.2		4	45.13	odd	12
432.3.e.d.161.1		2	20.3	even	4
432.3.e.d.161.2		2	60.23	odd	4
1296.3.q.i.593.1		4	180.103	even	12
1296.3.q.i.593.2		4	180.23	odd	12
1296.3.q.i.1025.1		4	180.83	odd	12
1296.3.q.i.1025.2		4	180.43	even	12
1350.3.b.b.1349.1		4	5.4	even	2	inner
1350.3.b.b.1349.2		4	3.2	odd	2	inner
1350.3.b.b.1349.3		4	15.14	odd	2	inner
1350.3.b.b.1349.4		4	1.1	even	1	trivial
1350.3.d.d.701.1		2	15.2	even	4
1350.3.d.d.701.2		2	5.2	odd	4
1728.3.e.f.1025.1		2	120.83	odd	4
1728.3.e.f.1025.2		2	40.3	even	4
1728.3.e.l.1025.1		2	120.53	even	4
1728.3.e.l.1025.2		2	40.13	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1350.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +5.00000i q^{7} +2.82843 q^{8} +8.48528i q^{11} +1.00000i q^{13} +7.07107i q^{14} +4.00000 q^{16} -25.4558 q^{17} -29.0000 q^{19} +12.0000i q^{22} -8.48528 q^{23} +1.41421i q^{26} +10.0000i q^{28} +16.9706i q^{29} -10.0000 q^{31} +5.65685 q^{32} -36.0000 q^{34} -25.0000i q^{37} -41.0122 q^{38} +16.9706i q^{41} -14.0000i q^{43} +16.9706i q^{44} -12.0000 q^{46} -8.48528 q^{47} +24.0000 q^{49} +2.00000i q^{52} -50.9117 q^{53} +14.1421i q^{56} +24.0000i q^{58} -93.3381i q^{59} +23.0000 q^{61} -14.1421 q^{62} +8.00000 q^{64} -19.0000i q^{67} -50.9117 q^{68} +101.823i q^{71} +97.0000i q^{73} -35.3553i q^{74} -58.0000 q^{76} -42.4264 q^{77} -77.0000 q^{79} +24.0000i q^{82} -118.794 q^{83} -19.7990i q^{86} +24.0000i q^{88} +76.3675i q^{89} -5.00000 q^{91} -16.9706 q^{92} -12.0000 q^{94} -49.0000i q^{97} +33.9411 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} + 16 q^{16} - 116 q^{19} - 40 q^{31} - 144 q^{34} - 48 q^{46} + 96 q^{49} + 92 q^{61} + 32 q^{64} - 232 q^{76} - 308 q^{79} - 20 q^{91} - 48 q^{94}+O(q^{100}) \) 4 * q + 8 * q^4 + 16 * q^16 - 116 * q^19 - 40 * q^31 - 144 * q^34 - 48 * q^46 + 96 * q^49 + 92 * q^61 + 32 * q^64 - 232 * q^76 - 308 * q^79 - 20 * q^91 - 48 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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