LMFDB - Embedded newform 3600.3.e.q.3151.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3600,3,Mod(3151,3600)] 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("3600.3151"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3600.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,36,0,0,0,0,0,0,0,0,0,0,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$98.0928951697$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3151.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3600.3151
Dual form			3600.3.e.q.3151.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.46410i q^{7} -3.46410i q^{11} +6.00000 q^{13} +18.0000 q^{17} -13.8564i q^{19} -6.92820i q^{23} +28.0000 q^{29} +48.4974i q^{31} -30.0000 q^{37} -2.00000 q^{41} -62.3538i q^{43} +55.4256i q^{47} +37.0000 q^{49} +102.000 q^{53} -100.459i q^{59} -74.0000 q^{61} +96.9948i q^{67} +6.92820i q^{71} -132.000 q^{73} -12.0000 q^{77} -103.923i q^{79} +117.779i q^{83} -14.0000 q^{89} -20.7846i q^{91} -24.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{13} + 36 q^{17} + 56 q^{29} - 60 q^{37} - 4 q^{41} + 74 q^{49} + 204 q^{53} - 148 q^{61} - 264 q^{73} - 24 q^{77} - 28 q^{89} - 48 q^{97}+O(q^{100}) $ 2 * q + 12 * q^13 + 36 * q^17 + 56 * q^29 - 60 * q^37 - 4 * q^41 + 74 * q^49 + 204 * q^53 - 148 * q^61 - 264 * q^73 - 24 * q^77 - 28 * q^89 - 48 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.46410i	− 0.494872i	−0.968904	−	0.247436i	$-0.920412\pi$
$7$	− 3.46410i	− 0.494872i	0.968904	−	0.247436i	$-0.0795879\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.46410i	− 0.314918i	−0.987525	−	0.157459i	$-0.949670\pi$
$11$	− 3.46410i	− 0.314918i	0.987525	−	0.157459i	$-0.0503303\pi$
$12$	0	0
$13$	6.00000	0.461538	0.230769	−	0.973009i	$-0.425876\pi$
$13$	6.00000	0.461538	0.230769	+	0.973009i	$0.425876\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.0000	1.05882	0.529412	−	0.848365i	$-0.322413\pi$
$17$	18.0000	1.05882	0.529412	+	0.848365i	$0.322413\pi$
$18$	0	0
$19$	− 13.8564i	− 0.729285i	−0.931148	−	0.364642i	$-0.881191\pi$
$19$	− 13.8564i	− 0.729285i	0.931148	−	0.364642i	$-0.118809\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 6.92820i	− 0.301226i	−0.988593	−	0.150613i	$-0.951875\pi$
$23$	− 6.92820i	− 0.301226i	0.988593	−	0.150613i	$-0.0481248\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	28.0000	0.965517	0.482759	−	0.875753i	$-0.339635\pi$
$29$	28.0000	0.965517	0.482759	+	0.875753i	$0.339635\pi$
$30$	0	0
$31$	48.4974i	1.56443i	0.623007	+	0.782216i	$0.285911\pi$
$31$	48.4974i	1.56443i	−0.623007	+	0.782216i	$0.714089\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−30.0000	−0.810811	−0.405405	−	0.914137i	$-0.632870\pi$
$37$	−30.0000	−0.810811	−0.405405	+	0.914137i	$0.632870\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.0487805	−0.0243902	−	0.999703i	$-0.507764\pi$
$41$	−2.00000	−0.0487805	−0.0243902	+	0.999703i	$0.507764\pi$
$42$	0	0
$43$	− 62.3538i	− 1.45009i	−0.688702	−	0.725045i	$-0.741819\pi$
$43$	− 62.3538i	− 1.45009i	0.688702	−	0.725045i	$-0.258181\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	55.4256i	1.17927i	0.807670	+	0.589634i	$0.200728\pi$
$47$	55.4256i	1.17927i	−0.807670	+	0.589634i	$0.799272\pi$
$48$	0	0
$49$	37.0000	0.755102
$50$	0	0
$51$	0	0
$52$	0	0
$53$	102.000	1.92453	0.962264	−	0.272117i	$-0.0877238\pi$
$53$	102.000	1.92453	0.962264	+	0.272117i	$0.0877238\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 100.459i	− 1.70269i	−0.524603	−	0.851347i	$-0.675786\pi$
$59$	− 100.459i	− 1.70269i	0.524603	−	0.851347i	$-0.324214\pi$
$60$	0	0
$61$	−74.0000	−1.21311	−0.606557	−	0.795040i	$-0.707450\pi$
$61$	−74.0000	−1.21311	−0.606557	+	0.795040i	$0.707450\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	96.9948i	1.44768i	0.689966	+	0.723842i	$0.257626\pi$
$67$	96.9948i	1.44768i	−0.689966	+	0.723842i	$0.742374\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.92820i	0.0975803i	0.998809	+	0.0487902i	$0.0155366\pi$
$71$	6.92820i	0.0975803i	−0.998809	+	0.0487902i	$0.984463\pi$
$72$	0	0
$73$	−132.000	−1.80822	−0.904110	−	0.427301i	$-0.859465\pi$
$73$	−132.000	−1.80822	−0.904110	+	0.427301i	$0.859465\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.0000	−0.155844
$78$	0	0
$79$	− 103.923i	− 1.31548i	−0.753244	−	0.657741i	$-0.771512\pi$
$79$	− 103.923i	− 1.31548i	0.753244	−	0.657741i	$-0.228488\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	117.779i	1.41903i	0.704691	+	0.709515i	$0.251086\pi$
$83$	117.779i	1.41903i	−0.704691	+	0.709515i	$0.748914\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.0000	−0.157303	−0.0786517	−	0.996902i	$-0.525061\pi$
$89$	−14.0000	−0.157303	−0.0786517	+	0.996902i	$0.525061\pi$
$90$	0	0
$91$	− 20.7846i	− 0.228402i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−24.0000	−0.247423	−0.123711	−	0.992318i	$-0.539480\pi$
$97$	−24.0000	−0.247423	−0.123711	+	0.992318i	$0.539480\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	20.0000	0.198020	0.0990099	−	0.995086i	$-0.468432\pi$
$101$	20.0000	0.198020	0.0990099	+	0.995086i	$0.468432\pi$
$102$	0	0
$103$	− 114.315i	− 1.10986i	−0.831898	−	0.554929i	$-0.812745\pi$
$103$	− 114.315i	− 1.10986i	0.831898	−	0.554929i	$-0.187255\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 34.6410i	− 0.323748i	−0.986811	−	0.161874i	$-0.948246\pi$
$107$	− 34.6410i	− 0.323748i	0.986811	−	0.161874i	$-0.0517538\pi$
$108$	0	0
$109$	86.0000	0.788991	0.394495	−	0.918898i	$-0.370919\pi$
$109$	86.0000	0.788991	0.394495	+	0.918898i	$0.370919\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	126.000	1.11504	0.557522	−	0.830162i	$-0.311752\pi$
$113$	126.000	1.11504	0.557522	+	0.830162i	$0.311752\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 62.3538i	− 0.523982i
$120$	0	0
$121$	109.000	0.900826
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 183.597i	− 1.44565i	−0.691032	−	0.722824i	$-0.742844\pi$
$127$	− 183.597i	− 1.44565i	0.691032	−	0.722824i	$-0.257156\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	45.0333i	0.343766i	0.985117	+	0.171883i	$0.0549851\pi$
$131$	45.0333i	0.343766i	−0.985117	+	0.171883i	$0.945015\pi$
$132$	0	0
$133$	−48.0000	−0.360902
$134$	0	0
$135$	0	0
$136$	0	0
$137$	102.000	0.744526	0.372263	−	0.928127i	$-0.378582\pi$
$137$	102.000	0.744526	0.372263	+	0.928127i	$0.378582\pi$
$138$	0	0
$139$	− 90.0666i	− 0.647961i	−0.946064	−	0.323981i	$-0.894979\pi$
$139$	− 90.0666i	− 0.647961i	0.946064	−	0.323981i	$-0.105021\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 20.7846i	− 0.145347i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	248.000	1.66443	0.832215	−	0.554453i	$-0.187073\pi$
$149$	248.000	1.66443	0.832215	+	0.554453i	$0.187073\pi$
$150$	0	0
$151$	− 76.2102i	− 0.504704i	−0.967636	−	0.252352i	$-0.918796\pi$
$151$	− 76.2102i	− 0.504704i	0.967636	−	0.252352i	$-0.0812040\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−162.000	−1.03185	−0.515924	−	0.856635i	$-0.672551\pi$
$157$	−162.000	−1.03185	−0.515924	+	0.856635i	$0.672551\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−24.0000	−0.149068
$162$	0	0
$163$	41.5692i	0.255026i	0.991837	+	0.127513i	$0.0406994\pi$
$163$	41.5692i	0.255026i	−0.991837	+	0.127513i	$0.959301\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	117.779i	0.705266i	0.935762	+	0.352633i	$0.114714\pi$
$167$	117.779i	0.705266i	−0.935762	+	0.352633i	$0.885286\pi$
$168$	0	0
$169$	−133.000	−0.786982
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−30.0000	−0.173410	−0.0867052	−	0.996234i	$-0.527634\pi$
$173$	−30.0000	−0.173410	−0.0867052	+	0.996234i	$0.527634\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 142.028i	− 0.793453i	−0.917937	−	0.396727i	$-0.870146\pi$
$179$	− 142.028i	− 0.793453i	0.917937	−	0.396727i	$-0.129854\pi$
$180$	0	0
$181$	86.0000	0.475138	0.237569	−	0.971371i	$-0.423649\pi$
$181$	86.0000	0.475138	0.237569	+	0.971371i	$0.423649\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 62.3538i	− 0.333443i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 284.056i	− 1.48721i	−0.668621	−	0.743603i	$-0.733115\pi$
$191$	− 284.056i	− 1.48721i	0.668621	−	0.743603i	$-0.266885\pi$
$192$	0	0
$193$	−84.0000	−0.435233	−0.217617	−	0.976034i	$-0.569828\pi$
$193$	−84.0000	−0.435233	−0.217617	+	0.976034i	$0.569828\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000	0.0304569	0.0152284	−	0.999884i	$-0.495152\pi$
$197$	6.00000	0.0304569	0.0152284	+	0.999884i	$0.495152\pi$
$198$	0	0
$199$	62.3538i	0.313336i	0.987651	+	0.156668i	$0.0500752\pi$
$199$	62.3538i	0.313336i	−0.987651	+	0.156668i	$0.949925\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 96.9948i	− 0.477807i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−48.0000	−0.229665
$210$	0	0
$211$	− 200.918i	− 0.952218i	−0.879386	−	0.476109i	$-0.842047\pi$
$211$	− 200.918i	− 0.952218i	0.879386	−	0.476109i	$-0.157953\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	168.000	0.774194
$218$	0	0
$219$	0	0
$220$	0	0
$221$	108.000	0.488688
$222$	0	0
$223$	79.6743i	0.357284i	0.983914	+	0.178642i	$0.0571704\pi$
$223$	79.6743i	0.357284i	−0.983914	+	0.178642i	$0.942830\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 311.769i	− 1.37343i	−0.726926	−	0.686716i	$-0.759052\pi$
$227$	− 311.769i	− 1.37343i	0.726926	−	0.686716i	$-0.240948\pi$
$228$	0	0
$229$	130.000	0.567686	0.283843	−	0.958871i	$-0.408391\pi$
$229$	130.000	0.567686	0.283843	+	0.958871i	$0.408391\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−126.000	−0.540773	−0.270386	−	0.962752i	$-0.587151\pi$
$233$	−126.000	−0.540773	−0.270386	+	0.962752i	$0.587151\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 436.477i	− 1.82626i	−0.407665	−	0.913131i	$-0.633657\pi$
$239$	− 436.477i	− 1.82626i	0.407665	−	0.913131i	$-0.366343\pi$
$240$	0	0
$241$	118.000	0.489627	0.244813	−	0.969570i	$-0.421273\pi$
$241$	118.000	0.489627	0.244813	+	0.969570i	$0.421273\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 83.1384i	− 0.336593i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 384.515i	− 1.53193i	−0.642880	−	0.765967i	$-0.722261\pi$
$251$	− 384.515i	− 1.53193i	0.642880	−	0.765967i	$-0.277739\pi$
$252$	0	0
$253$	−24.0000	−0.0948617
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−138.000	−0.536965	−0.268482	−	0.963285i	$-0.586522\pi$
$257$	−138.000	−0.536965	−0.268482	+	0.963285i	$0.586522\pi$
$258$	0	0
$259$	103.923i	0.401247i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	394.908i	1.50155i	0.660558	+	0.750775i	$0.270320\pi$
$263$	394.908i	1.50155i	−0.660558	+	0.750775i	$0.729680\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	212.000	0.788104	0.394052	−	0.919088i	$-0.371073\pi$
$269$	212.000	0.788104	0.394052	+	0.919088i	$0.371073\pi$
$270$	0	0
$271$	− 228.631i	− 0.843656i	−0.906676	−	0.421828i	$-0.861389\pi$
$271$	− 228.631i	− 0.843656i	0.906676	−	0.421828i	$-0.138611\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	186.000	0.671480	0.335740	−	0.941955i	$-0.391014\pi$
$277$	186.000	0.671480	0.335740	+	0.941955i	$0.391014\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	394.000	1.40214	0.701068	−	0.713095i	$-0.252707\pi$
$281$	394.000	1.40214	0.701068	+	0.713095i	$0.252707\pi$
$282$	0	0
$283$	471.118i	1.66473i	0.554230	+	0.832364i	$0.313013\pi$
$283$	471.118i	1.66473i	−0.554230	+	0.832364i	$0.686987\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.92820i	0.0241401i
$288$	0	0
$289$	35.0000	0.121107
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.00000	0.0204778	0.0102389	−	0.999948i	$-0.496741\pi$
$293$	6.00000	0.0204778	0.0102389	+	0.999948i	$0.496741\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 41.5692i	− 0.139027i
$300$	0	0
$301$	−216.000	−0.717608
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	339.482i	1.10580i	0.833246	+	0.552902i	$0.186480\pi$
$307$	339.482i	1.10580i	−0.833246	+	0.552902i	$0.813520\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	242.487i	0.779701i	0.920878	+	0.389851i	$0.127473\pi$
$311$	242.487i	0.779701i	−0.920878	+	0.389851i	$0.872527\pi$
$312$	0	0
$313$	108.000	0.345048	0.172524	−	0.985005i	$-0.444808\pi$
$313$	108.000	0.345048	0.172524	+	0.985005i	$0.444808\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−102.000	−0.321767	−0.160883	−	0.986973i	$-0.551434\pi$
$317$	−102.000	−0.321767	−0.160883	+	0.986973i	$0.551434\pi$
$318$	0	0
$319$	− 96.9948i	− 0.304059i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 249.415i	− 0.772184i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	192.000	0.583587
$330$	0	0
$331$	235.559i	0.711658i	0.934551	+	0.355829i	$0.115802\pi$
$331$	235.559i	0.711658i	−0.934551	+	0.355829i	$0.884198\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	372.000	1.10386	0.551929	−	0.833891i	$-0.313892\pi$
$337$	372.000	1.10386	0.551929	+	0.833891i	$0.313892\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	168.000	0.492669
$342$	0	0
$343$	− 297.913i	− 0.868550i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	145.492i	0.419286i	0.977778	+	0.209643i	$0.0672302\pi$
$347$	145.492i	0.419286i	−0.977778	+	0.209643i	$0.932770\pi$
$348$	0	0
$349$	334.000	0.957020	0.478510	−	0.878082i	$-0.341177\pi$
$349$	334.000	0.957020	0.478510	+	0.878082i	$0.341177\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−486.000	−1.37677	−0.688385	−	0.725345i	$-0.741680\pi$
$353$	−486.000	−1.37677	−0.688385	+	0.725345i	$0.741680\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 512.687i	− 1.42810i	−0.700096	−	0.714049i	$-0.746859\pi$
$359$	− 512.687i	− 1.42810i	0.700096	−	0.714049i	$-0.253141\pi$
$360$	0	0
$361$	169.000	0.468144
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 239.023i	− 0.651289i	−0.945492	−	0.325644i	$-0.894419\pi$
$367$	− 239.023i	− 0.651289i	0.945492	−	0.325644i	$-0.105581\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 353.338i	− 0.952395i
$372$	0	0
$373$	726.000	1.94638	0.973190	−	0.230001i	$-0.0738730\pi$
$373$	726.000	1.94638	0.973190	+	0.230001i	$0.0738730\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	168.000	0.445623
$378$	0	0
$379$	− 644.323i	− 1.70006i	−0.526734	−	0.850030i	$-0.676584\pi$
$379$	− 644.323i	− 1.70006i	0.526734	−	0.850030i	$-0.323416\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 581.969i	− 1.51950i	−0.650214	−	0.759751i	$-0.725321\pi$
$383$	− 581.969i	− 1.51950i	0.650214	−	0.759751i	$-0.274679\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	200.000	0.514139	0.257069	−	0.966393i	$-0.417243\pi$
$389$	200.000	0.514139	0.257069	+	0.966393i	$0.417243\pi$
$390$	0	0
$391$	− 124.708i	− 0.318945i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−342.000	−0.861461	−0.430730	−	0.902481i	$-0.641744\pi$
$397$	−342.000	−0.861461	−0.430730	+	0.902481i	$0.641744\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−622.000	−1.55112	−0.775561	−	0.631273i	$-0.782533\pi$
$401$	−622.000	−1.55112	−0.775561	+	0.631273i	$0.782533\pi$
$402$	0	0
$403$	290.985i	0.722046i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	103.923i	0.255339i
$408$	0	0
$409$	−370.000	−0.904645	−0.452323	−	0.891854i	$-0.649404\pi$
$409$	−370.000	−0.904645	−0.452323	+	0.891854i	$0.649404\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−348.000	−0.842615
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 800.207i	− 1.90980i	−0.296923	−	0.954902i	$-0.595960\pi$
$419$	− 800.207i	− 1.90980i	0.296923	−	0.954902i	$-0.404040\pi$
$420$	0	0
$421$	178.000	0.422803	0.211401	−	0.977399i	$-0.432197\pi$
$421$	178.000	0.422803	0.211401	+	0.977399i	$0.432197\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	256.344i	0.600336i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 152.420i	− 0.353644i	−0.984243	−	0.176822i	$-0.943418\pi$
$431$	− 152.420i	− 0.353644i	0.984243	−	0.176822i	$-0.0565817\pi$
$432$	0	0
$433$	468.000	1.08083	0.540416	−	0.841398i	$-0.318267\pi$
$433$	468.000	1.08083	0.540416	+	0.841398i	$0.318267\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−96.0000	−0.219680
$438$	0	0
$439$	− 408.764i	− 0.931125i	−0.885015	−	0.465563i	$-0.845852\pi$
$439$	− 408.764i	− 0.931125i	0.885015	−	0.465563i	$-0.154148\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 367.195i	− 0.828882i	−0.910076	−	0.414441i	$-0.863977\pi$
$443$	− 367.195i	− 0.828882i	0.910076	−	0.414441i	$-0.136023\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	314.000	0.699332	0.349666	−	0.936874i	$-0.386295\pi$
$449$	314.000	0.699332	0.349666	+	0.936874i	$0.386295\pi$
$450$	0	0
$451$	6.92820i	0.0153619i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−96.0000	−0.210066	−0.105033	−	0.994469i	$-0.533495\pi$
$457$	−96.0000	−0.210066	−0.105033	+	0.994469i	$0.533495\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	152.000	0.329718	0.164859	−	0.986317i	$-0.447283\pi$
$461$	152.000	0.329718	0.164859	+	0.986317i	$0.447283\pi$
$462$	0	0
$463$	197.454i	0.426466i	0.977001	+	0.213233i	$0.0683994\pi$
$463$	197.454i	0.426466i	−0.977001	+	0.213233i	$0.931601\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 436.477i	− 0.934640i	−0.884088	−	0.467320i	$-0.845220\pi$
$467$	− 436.477i	− 0.934640i	0.884088	−	0.467320i	$-0.154780\pi$
$468$	0	0
$469$	336.000	0.716418
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−216.000	−0.456660
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	297.913i	0.621947i	0.950419	+	0.310974i	$0.100655\pi$
$479$	297.913i	0.621947i	−0.950419	+	0.310974i	$0.899345\pi$
$480$	0	0
$481$	−180.000	−0.374220
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.1769i	0.0640183i	0.999488	+	0.0320092i	$0.0101906\pi$
$487$	31.1769i	0.0640183i	−0.999488	+	0.0320092i	$0.989809\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	751.710i	1.53098i	0.643449	+	0.765489i	$0.277503\pi$
$491$	751.710i	1.53098i	−0.643449	+	0.765489i	$0.722497\pi$
$492$	0	0
$493$	504.000	1.02231
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0000	0.0482897
$498$	0	0
$499$	− 304.841i	− 0.610904i	−0.952208	−	0.305452i	$-0.901192\pi$
$499$	− 304.841i	− 0.610904i	0.952208	−	0.305452i	$-0.0988075\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	443.405i	0.881521i	0.897625	+	0.440760i	$0.145291\pi$
$503$	443.405i	0.881521i	−0.897625	+	0.440760i	$0.854709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	580.000	1.13949	0.569745	−	0.821822i	$-0.307042\pi$
$509$	580.000	1.13949	0.569745	+	0.821822i	$0.307042\pi$
$510$	0	0
$511$	457.261i	0.894836i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	192.000	0.371373
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−262.000	−0.502879	−0.251440	−	0.967873i	$-0.580904\pi$
$521$	−262.000	−0.502879	−0.251440	+	0.967873i	$0.580904\pi$
$522$	0	0
$523$	− 429.549i	− 0.821317i	−0.911789	−	0.410658i	$-0.865299\pi$
$523$	− 429.549i	− 0.821317i	0.911789	−	0.410658i	$-0.134701\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	872.954i	1.65646i
$528$	0	0
$529$	481.000	0.909263
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.0000	−0.0225141
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 128.172i	− 0.237795i
$540$	0	0
$541$	−58.0000	−0.107209	−0.0536044	−	0.998562i	$-0.517071\pi$
$541$	−58.0000	−0.107209	−0.0536044	+	0.998562i	$0.517071\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1004.59i	1.83654i	0.395950	+	0.918272i	$0.370415\pi$
$547$	1004.59i	1.83654i	−0.395950	+	0.918272i	$0.629585\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 387.979i	− 0.704137i
$552$	0	0
$553$	−360.000	−0.650995
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−498.000	−0.894075	−0.447038	−	0.894515i	$-0.647521\pi$
$557$	−498.000	−0.894075	−0.447038	+	0.894515i	$0.647521\pi$
$558$	0	0
$559$	− 374.123i	− 0.669272i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 533.472i	− 0.947552i	−0.880645	−	0.473776i	$-0.842891\pi$
$563$	− 533.472i	− 0.947552i	0.880645	−	0.473776i	$-0.157109\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−502.000	−0.882250	−0.441125	−	0.897446i	$-0.645420\pi$
$569$	−502.000	−0.882250	−0.441125	+	0.897446i	$0.645420\pi$
$570$	0	0
$571$	13.8564i	0.0242669i	0.999926	+	0.0121335i	$0.00386229\pi$
$571$	13.8564i	0.0242669i	−0.999926	+	0.0121335i	$0.996138\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−504.000	−0.873484	−0.436742	−	0.899587i	$-0.643868\pi$
$577$	−504.000	−0.873484	−0.436742	+	0.899587i	$0.643868\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	408.000	0.702238
$582$	0	0
$583$	− 353.338i	− 0.606069i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1032.30i	− 1.75861i	−0.476262	−	0.879303i	$-0.658009\pi$
$587$	− 1032.30i	− 1.75861i	0.476262	−	0.879303i	$-0.341991\pi$
$588$	0	0
$589$	672.000	1.14092
$590$	0	0
$591$	0	0
$592$	0	0
$593$	462.000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$593$	462.000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 734.390i	− 1.22603i	−0.790073	−	0.613013i	$-0.789957\pi$
$599$	− 734.390i	− 1.22603i	0.790073	−	0.613013i	$-0.210043\pi$
$600$	0	0
$601$	−902.000	−1.50083	−0.750416	−	0.660966i	$-0.770147\pi$
$601$	−902.000	−1.50083	−0.750416	+	0.660966i	$0.770147\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 329.090i	− 0.542158i	−0.962557	−	0.271079i	$-0.912620\pi$
$607$	− 329.090i	− 0.542158i	0.962557	−	0.271079i	$-0.0873804\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	332.554i	0.544278i
$612$	0	0
$613$	−966.000	−1.57586	−0.787928	−	0.615767i	$-0.788846\pi$
$613$	−966.000	−1.57586	−0.787928	+	0.615767i	$0.788846\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−846.000	−1.37115	−0.685575	−	0.728002i	$-0.740449\pi$
$617$	−846.000	−1.37115	−0.685575	+	0.728002i	$0.740449\pi$
$618$	0	0
$619$	478.046i	0.772288i	0.922439	+	0.386144i	$0.126193\pi$
$619$	478.046i	0.772288i	−0.922439	+	0.386144i	$0.873807\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	48.4974i	0.0778450i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−540.000	−0.858506
$630$	0	0
$631$	− 242.487i	− 0.384290i	−0.981367	−	0.192145i	$-0.938456\pi$
$631$	− 242.487i	− 0.384290i	0.981367	−	0.192145i	$-0.0615444\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	222.000	0.348509
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−26.0000	−0.0405616	−0.0202808	−	0.999794i	$-0.506456\pi$
$641$	−26.0000	−0.0405616	−0.0202808	+	0.999794i	$0.506456\pi$
$642$	0	0
$643$	− 387.979i	− 0.603389i	−0.953405	−	0.301695i	$-0.902448\pi$
$643$	− 387.979i	− 0.603389i	0.953405	−	0.301695i	$-0.0975523\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	678.964i	1.04940i	0.851286	+	0.524702i	$0.175823\pi$
$647$	678.964i	1.04940i	−0.851286	+	0.524702i	$0.824177\pi$
$648$	0	0
$649$	−348.000	−0.536210
$650$	0	0
$651$	0	0
$652$	0	0
$653$	858.000	1.31394	0.656968	−	0.753919i	$-0.271839\pi$
$653$	858.000	1.31394	0.656968	+	0.753919i	$0.271839\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	155.885i	0.236547i	0.992981	+	0.118274i	$0.0377360\pi$
$659$	155.885i	0.236547i	−0.992981	+	0.118274i	$0.962264\pi$
$660$	0	0
$661$	1102.00	1.66717	0.833585	−	0.552390i	$-0.186284\pi$
$661$	1102.00	1.66717	0.833585	+	0.552390i	$0.186284\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 193.990i	− 0.290839i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	256.344i	0.382032i
$672$	0	0
$673$	−180.000	−0.267459	−0.133730	−	0.991018i	$-0.542695\pi$
$673$	−180.000	−0.267459	−0.133730	+	0.991018i	$0.542695\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	810.000	1.19645	0.598227	−	0.801326i	$-0.295872\pi$
$677$	810.000	1.19645	0.598227	+	0.801326i	$0.295872\pi$
$678$	0	0
$679$	83.1384i	0.122442i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	893.738i	1.30855i	0.756258	+	0.654274i	$0.227026\pi$
$683$	893.738i	1.30855i	−0.756258	+	0.654274i	$0.772974\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	612.000	0.888244
$690$	0	0
$691$	166.277i	0.240632i	0.992736	+	0.120316i	$0.0383908\pi$
$691$	166.277i	0.240632i	−0.992736	+	0.120316i	$0.961609\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−36.0000	−0.0516499
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1012.00	1.44365	0.721826	−	0.692075i	$-0.243303\pi$
$701$	1012.00	1.44365	0.721826	+	0.692075i	$0.243303\pi$
$702$	0	0
$703$	415.692i	0.591312i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 69.2820i	− 0.0979944i
$708$	0	0
$709$	−902.000	−1.27221	−0.636107	−	0.771601i	$-0.719456\pi$
$709$	−902.000	−1.27221	−0.636107	+	0.771601i	$0.719456\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	336.000	0.471248
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	886.810i	1.23339i	0.787201	+	0.616697i	$0.211530\pi$
$719$	886.810i	1.23339i	−0.787201	+	0.616697i	$0.788470\pi$
$720$	0	0
$721$	−396.000	−0.549237
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1077.34i	− 1.48189i	−0.671565	−	0.740946i	$-0.734377\pi$
$727$	− 1077.34i	− 1.48189i	0.671565	−	0.740946i	$-0.265623\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1122.37i	− 1.53539i
$732$	0	0
$733$	−174.000	−0.237381	−0.118690	−	0.992931i	$-0.537870\pi$
$733$	−174.000	−0.237381	−0.118690	+	0.992931i	$0.537870\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	336.000	0.455902
$738$	0	0
$739$	595.825i	0.806259i	0.915143	+	0.403130i	$0.132078\pi$
$739$	595.825i	0.806259i	−0.915143	+	0.403130i	$0.867922\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	429.549i	0.578127i	0.957310	+	0.289064i	$0.0933439\pi$
$743$	429.549i	0.578127i	−0.957310	+	0.289064i	$0.906656\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−120.000	−0.160214
$750$	0	0
$751$	394.908i	0.525842i	0.964817	+	0.262921i	$0.0846859\pi$
$751$	394.908i	0.525842i	−0.964817	+	0.262921i	$0.915314\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	378.000	0.499339	0.249670	−	0.968331i	$-0.419678\pi$
$757$	378.000	0.499339	0.249670	+	0.968331i	$0.419678\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1150.00	−1.51117	−0.755585	−	0.655051i	$-0.772647\pi$
$761$	−1150.00	−1.51117	−0.755585	+	0.655051i	$0.772647\pi$
$762$	0	0
$763$	− 297.913i	− 0.390449i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 602.754i	− 0.785859i
$768$	0	0
$769$	34.0000	0.0442133	0.0221066	−	0.999756i	$-0.492963\pi$
$769$	34.0000	0.0442133	0.0221066	+	0.999756i	$0.492963\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−966.000	−1.24968	−0.624838	−	0.780754i	$-0.714835\pi$
$773$	−966.000	−1.24968	−0.624838	+	0.780754i	$0.714835\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	27.7128i	0.0355749i
$780$	0	0
$781$	24.0000	0.0307298
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 457.261i	− 0.581018i	−0.956872	−	0.290509i	$-0.906175\pi$
$787$	− 457.261i	− 0.581018i	0.956872	−	0.290509i	$-0.0938247\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 436.477i	− 0.551804i
$792$	0	0
$793$	−444.000	−0.559899
$794$	0	0
$795$	0	0
$796$	0	0
$797$	786.000	0.986198	0.493099	−	0.869973i	$-0.335864\pi$
$797$	786.000	0.986198	0.493099	+	0.869973i	$0.335864\pi$
$798$	0	0
$799$	997.661i	1.24864i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	457.261i	0.569441i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−386.000	−0.477132	−0.238566	−	0.971126i	$-0.576677\pi$
$809$	−386.000	−0.477132	−0.238566	+	0.971126i	$0.576677\pi$
$810$	0	0
$811$	− 1115.44i	− 1.37539i	−0.726000	−	0.687695i	$-0.758623\pi$
$811$	− 1115.44i	− 1.37539i	0.726000	−	0.687695i	$-0.241377\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−864.000	−1.05753
$818$	0	0
$819$	0	0
$820$	0	0
$821$	532.000	0.647990	0.323995	−	0.946059i	$-0.394974\pi$
$821$	532.000	0.647990	0.323995	+	0.946059i	$0.394974\pi$
$822$	0	0
$823$	17.3205i	0.0210456i	0.999945	+	0.0105228i	$0.00334957\pi$
$823$	17.3205i	0.0210456i	−0.999945	+	0.0105228i	$0.996650\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	284.056i	0.343478i	0.985142	+	0.171739i	$0.0549386\pi$
$827$	284.056i	0.343478i	−0.985142	+	0.171739i	$0.945061\pi$
$828$	0	0
$829$	−994.000	−1.19903	−0.599517	−	0.800362i	$-0.704641\pi$
$829$	−994.000	−1.19903	−0.599517	+	0.800362i	$0.704641\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	666.000	0.799520
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1441.07i	1.71760i	0.512311	+	0.858800i	$0.328789\pi$
$839$	1441.07i	1.71760i	−0.512311	+	0.858800i	$0.671211\pi$
$840$	0	0
$841$	−57.0000	−0.0677765
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 377.587i	− 0.445793i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	207.846i	0.244237i
$852$	0	0
$853$	−1614.00	−1.89215	−0.946073	−	0.323954i	$-0.894988\pi$
$853$	−1614.00	−1.89215	−0.946073	+	0.323954i	$0.894988\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1170.00	1.36523	0.682614	−	0.730779i	$-0.260843\pi$
$857$	1170.00	1.36523	0.682614	+	0.730779i	$0.260843\pi$
$858$	0	0
$859$	464.190i	0.540384i	0.962807	+	0.270192i	$0.0870871\pi$
$859$	464.190i	0.540384i	−0.962807	+	0.270192i	$0.912913\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	131.636i	0.152533i	0.997087	+	0.0762664i	$0.0243000\pi$
$863$	131.636i	0.152533i	−0.997087	+	0.0762664i	$0.975700\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−360.000	−0.414269
$870$	0	0
$871$	581.969i	0.668162i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−690.000	−0.786773	−0.393387	−	0.919373i	$-0.628697\pi$
$877$	−690.000	−0.786773	−0.393387	+	0.919373i	$0.628697\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1310.00	−1.48695	−0.743473	−	0.668766i	$-0.766823\pi$
$881$	−1310.00	−1.48695	−0.743473	+	0.668766i	$0.766823\pi$
$882$	0	0
$883$	1108.51i	1.25539i	0.778458	+	0.627697i	$0.216002\pi$
$883$	1108.51i	1.25539i	−0.778458	+	0.627697i	$0.783998\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	727.461i	0.820137i	0.912055	+	0.410068i	$0.134495\pi$
$887$	727.461i	0.820137i	−0.912055	+	0.410068i	$0.865505\pi$
$888$	0	0
$889$	−636.000	−0.715411
$890$	0	0
$891$	0	0
$892$	0	0
$893$	768.000	0.860022
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1357.93i	1.51049i
$900$	0	0
$901$	1836.00	2.03774
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 90.0666i	− 0.0993017i	−0.998767	−	0.0496509i	$-0.984189\pi$
$907$	− 90.0666i	− 0.0993017i	0.998767	−	0.0496509i	$-0.0158109\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	637.395i	0.699665i	0.936812	+	0.349832i	$0.113761\pi$
$911$	637.395i	0.699665i	−0.936812	+	0.349832i	$0.886239\pi$
$912$	0	0
$913$	408.000	0.446878
$914$	0	0
$915$	0	0
$916$	0	0
$917$	156.000	0.170120
$918$	0	0
$919$	117.779i	0.128160i	0.997945	+	0.0640802i	$0.0204114\pi$
$919$	117.779i	0.128160i	−0.997945	+	0.0640802i	$0.979589\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	41.5692i	0.0450371i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1222.00	−1.31539	−0.657696	−	0.753283i	$-0.728469\pi$
$929$	−1222.00	−1.31539	−0.657696	+	0.753283i	$0.728469\pi$
$930$	0	0
$931$	− 512.687i	− 0.550684i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	984.000	1.05016	0.525080	−	0.851053i	$-0.324035\pi$
$937$	984.000	1.05016	0.525080	+	0.851053i	$0.324035\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	104.000	0.110521	0.0552604	−	0.998472i	$-0.482401\pi$
$941$	104.000	0.110521	0.0552604	+	0.998472i	$0.482401\pi$
$942$	0	0
$943$	13.8564i	0.0146940i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	907.595i	0.958389i	0.877709	+	0.479195i	$0.159071\pi$
$947$	907.595i	0.958389i	−0.877709	+	0.479195i	$0.840929\pi$
$948$	0	0
$949$	−792.000	−0.834563
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1446.00	1.51731	0.758657	−	0.651490i	$-0.225856\pi$
$953$	1446.00	1.51731	0.758657	+	0.651490i	$0.225856\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 353.338i	− 0.368445i
$960$	0	0
$961$	−1391.00	−1.44745
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	869.490i	0.899162i	0.893240	+	0.449581i	$0.148427\pi$
$967$	869.490i	0.899162i	−0.893240	+	0.449581i	$0.851573\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	723.997i	0.745620i	0.927908	+	0.372810i	$0.121606\pi$
$971$	723.997i	0.745620i	−0.927908	+	0.372810i	$0.878394\pi$
$972$	0	0
$973$	−312.000	−0.320658
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−294.000	−0.300921	−0.150461	−	0.988616i	$-0.548076\pi$
$977$	−294.000	−0.300921	−0.150461	+	0.988616i	$0.548076\pi$
$978$	0	0
$979$	48.4974i	0.0495377i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	803.672i	0.817570i	0.912631	+	0.408785i	$0.134047\pi$
$983$	803.672i	0.817570i	−0.912631	+	0.408785i	$0.865953\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−432.000	−0.436805
$990$	0	0
$991$	− 1822.12i	− 1.83867i	−0.393481	−	0.919333i	$-0.628729\pi$
$991$	− 1822.12i	− 1.83867i	0.393481	−	0.919333i	$-0.371271\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	270.000	0.270812	0.135406	−	0.990790i	$-0.456766\pi$
$997$	270.000	0.270812	0.135406	+	0.990790i	$0.456766\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.e.q.3151.1		2
3.2	odd	2		1200.3.e.e.751.2		2
4.3	odd	2	inner	3600.3.e.q.3151.2		2
5.2	odd	4		720.3.j.d.559.4		4
5.3	odd	4		720.3.j.d.559.1		4
5.4	even	2		3600.3.e.m.3151.2		2
12.11	even	2		1200.3.e.e.751.1		2
15.2	even	4		240.3.j.c.79.3	yes	4
15.8	even	4		240.3.j.c.79.2	yes	4
15.14	odd	2		1200.3.e.d.751.1		2
20.3	even	4		720.3.j.d.559.2		4
20.7	even	4		720.3.j.d.559.3		4
20.19	odd	2		3600.3.e.m.3151.1		2
60.23	odd	4		240.3.j.c.79.4	yes	4
60.47	odd	4		240.3.j.c.79.1	✓	4
60.59	even	2		1200.3.e.d.751.2		2
120.53	even	4		960.3.j.a.319.3		4
120.77	even	4		960.3.j.a.319.2		4
120.83	odd	4		960.3.j.a.319.1		4
120.107	odd	4		960.3.j.a.319.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.j.c.79.1	✓	4	60.47	odd	4
240.3.j.c.79.2	yes	4	15.8	even	4
240.3.j.c.79.3	yes	4	15.2	even	4
240.3.j.c.79.4	yes	4	60.23	odd	4
720.3.j.d.559.1		4	5.3	odd	4
720.3.j.d.559.2		4	20.3	even	4
720.3.j.d.559.3		4	20.7	even	4
720.3.j.d.559.4		4	5.2	odd	4
960.3.j.a.319.1		4	120.83	odd	4
960.3.j.a.319.2		4	120.77	even	4
960.3.j.a.319.3		4	120.53	even	4
960.3.j.a.319.4		4	120.107	odd	4
1200.3.e.d.751.1		2	15.14	odd	2
1200.3.e.d.751.2		2	60.59	even	2
1200.3.e.e.751.1		2	12.11	even	2
1200.3.e.e.751.2		2	3.2	odd	2
3600.3.e.m.3151.1		2	20.19	odd	2
3600.3.e.m.3151.2		2	5.4	even	2
3600.3.e.q.3151.1		2	1.1	even	1	trivial
3600.3.e.q.3151.2		2	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.46410i q^{7} -3.46410i q^{11} +6.00000 q^{13} +18.0000 q^{17} -13.8564i q^{19} -6.92820i q^{23} +28.0000 q^{29} +48.4974i q^{31} -30.0000 q^{37} -2.00000 q^{41} -62.3538i q^{43} +55.4256i q^{47} +37.0000 q^{49} +102.000 q^{53} -100.459i q^{59} -74.0000 q^{61} +96.9948i q^{67} +6.92820i q^{71} -132.000 q^{73} -12.0000 q^{77} -103.923i q^{79} +117.779i q^{83} -14.0000 q^{89} -20.7846i q^{91} -24.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{13} + 36 q^{17} + 56 q^{29} - 60 q^{37} - 4 q^{41} + 74 q^{49} + 204 q^{53} - 148 q^{61} - 264 q^{73} - 24 q^{77} - 28 q^{89} - 48 q^{97}+O(q^{100}) \) 2 * q + 12 * q^13 + 36 * q^17 + 56 * q^29 - 60 * q^37 - 4 * q^41 + 74 * q^49 + 204 * q^53 - 148 * q^61 - 264 * q^73 - 24 * q^77 - 28 * q^89 - 48 * q^97

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