LMFDB - Embedded newform 432.4.i.b.145.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,4,Mod(145,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("432.145");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	432.i (of order $3$, degree $2$, 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:	$25.4888251225$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-35})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} - 9x + 81 $ x^4 - x^3 - 8x^2 - 9x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			145.1
Root			$2.81174 - 1.04601i$ of defining polynomial
Character	$\chi$	$=$	432.145
Dual form			432.4.i.b.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-9.93521 + 17.2083i) q^{5} +(-2.93521 - 5.08394i) q^{7} +(-9.37043 - 16.2301i) q^{11} +(-22.9352 + 39.7250i) q^{13} -16.8704 q^{17} +10.3521 q^{19} +(-24.9352 + 43.1891i) q^{23} +(-134.917 - 233.683i) q^{25} +(5.45351 + 9.44575i) q^{29} +(75.8056 - 131.299i) q^{31} +116.648 q^{35} +346.186 q^{37} +(132.370 - 229.272i) q^{41} +(-205.945 - 356.707i) q^{43} +(-236.028 - 408.813i) q^{47} +(154.269 - 267.202i) q^{49} -290.186 q^{53} +372.389 q^{55} +(-26.6296 + 46.1238i) q^{59} +(146.972 + 254.563i) q^{61} +(-455.732 - 789.352i) q^{65} +(-199.277 + 345.159i) q^{67} -647.854 q^{71} -478.279 q^{73} +(-55.0084 + 95.2773i) q^{77} +(187.158 + 324.167i) q^{79} +(-466.639 - 808.243i) q^{83} +(167.611 - 290.311i) q^{85} -368.817 q^{89} +269.279 q^{91} +(-102.851 + 178.143i) q^{95} +(-137.075 - 237.420i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 9 q^{5} + 19 q^{7} + 24 q^{11} - 61 q^{13} - 6 q^{17} - 266 q^{19} - 69 q^{23} - 263 q^{25} + 237 q^{29} + 211 q^{31} + 774 q^{35} + 524 q^{37} + 468 q^{41} - 86 q^{43} - 483 q^{47} + 33 q^{49} - 300 q^{53}+ \cdots + 128 q^{97}+O(q^{100}) $ 4 * q - 9 * q^5 + 19 * q^7 + 24 * q^11 - 61 * q^13 - 6 * q^17 - 266 * q^19 - 69 * q^23 - 263 * q^25 + 237 * q^29 + 211 * q^31 + 774 * q^35 + 524 * q^37 + 468 * q^41 - 86 * q^43 - 483 * q^47 + 33 * q^49 - 300 * q^53 + 1674 * q^55 - 168 * q^59 + 1049 * q^61 - 747 * q^65 - 1166 * q^67 - 624 * q^71 - 622 * q^73 - 1173 * q^77 + 349 * q^79 - 1221 * q^83 + 486 * q^85 + 984 * q^89 - 214 * q^91 - 1764 * q^95 + 128 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−9.93521	+	17.2083i	−0.888632	+	1.53916i	−0.0471396	+	0.998888i	$0.515011\pi$
$5$	−9.93521	+	17.2083i	−0.888632	+	1.53916i	−0.841493	+	0.540268i	$0.818323\pi$
$6$		0			0
$7$	−2.93521	−	5.08394i	−0.158487	−	0.274507i	0.775837	−	0.630934i	$-0.217328\pi$
$7$	−2.93521	−	5.08394i	−0.158487	−	0.274507i	−0.934323	+	0.356427i	$0.883995\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9.37043	−	16.2301i	−0.256845	−	0.444868i	0.708550	−	0.705660i	$-0.249349\pi$
$11$	−9.37043	−	16.2301i	−0.256845	−	0.444868i	−0.965395	+	0.260792i	$0.916016\pi$
$12$		0			0
$13$	−22.9352	+	39.7250i	−0.489314	+	0.847517i	−0.999924	−	0.0122953i	$-0.996086\pi$
$13$	−22.9352	+	39.7250i	−0.489314	+	0.847517i	0.510610	+	0.859812i	$0.329420\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−16.8704			−0.240687			−0.120344	−	0.992732i	$-0.538400\pi$
$17$	−16.8704			−0.240687			−0.120344	+	0.992732i	$0.538400\pi$
$18$		0			0
$19$	10.3521			0.124997			0.0624985	−	0.998045i	$-0.480093\pi$
$19$	10.3521			0.124997			0.0624985	+	0.998045i	$0.480093\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−24.9352	+	43.1891i	−0.226059	+	0.391545i	−0.956637	−	0.291284i	$-0.905917\pi$
$23$	−24.9352	+	43.1891i	−0.226059	+	0.391545i	0.730578	+	0.682829i	$0.239251\pi$
$24$		0			0
$25$	−134.917	−	233.683i	−1.07934	−	1.86946i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.45351	+	9.44575i	0.0349204	+	0.0604839i	0.882957	−	0.469453i	$-0.155549\pi$
$29$	5.45351	+	9.44575i	0.0349204	+	0.0604839i	−0.848037	+	0.529937i	$0.822216\pi$
$30$		0			0
$31$	75.8056	−	131.299i	0.439197	−	0.760711i	−0.558431	−	0.829551i	$-0.688596\pi$
$31$	75.8056	−	131.299i	0.439197	−	0.760711i	0.997628	+	0.0688401i	$0.0219298\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	116.648			0.563345
$36$		0			0
$37$	346.186			1.53818			0.769089	−	0.639141i	$-0.220710\pi$
$37$	346.186			1.53818			0.769089	+	0.639141i	$0.220710\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	132.370	−	229.272i	0.504214	−	0.873325i	−0.495774	−	0.868452i	$-0.665115\pi$
$41$	132.370	−	229.272i	0.504214	−	0.873325i	0.999988	−	0.00487314i	$-0.00155118\pi$
$42$		0			0
$43$	−205.945	−	356.707i	−0.730380	−	1.26506i	−0.956721	−	0.291007i	$-0.906010\pi$
$43$	−205.945	−	356.707i	−0.730380	−	1.26506i	0.226341	−	0.974048i	$-0.427324\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−236.028	−	408.813i	−0.732516	−	1.26875i	−0.955805	−	0.294003i	$-0.905013\pi$
$47$	−236.028	−	408.813i	−0.732516	−	1.26875i	0.223289	−	0.974752i	$-0.428321\pi$
$48$		0			0
$49$	154.269	−	267.202i	0.449764	−	0.779014i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−290.186			−0.752078			−0.376039	−	0.926604i	$-0.622714\pi$
$53$	−290.186			−0.752078			−0.376039	+	0.926604i	$0.622714\pi$
$54$		0			0
$55$	372.389			0.912962
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−26.6296	+	46.1238i	−0.0587606	+	0.101776i	−0.893909	−	0.448248i	$-0.852048\pi$
$59$	−26.6296	+	46.1238i	−0.0587606	+	0.101776i	0.835149	+	0.550024i	$0.185382\pi$
$60$		0			0
$61$	146.972	+	254.563i	0.308489	+	0.534318i	0.978032	−	0.208455i	$-0.0668435\pi$
$61$	146.972	+	254.563i	0.308489	+	0.534318i	−0.669543	+	0.742773i	$0.733510\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−455.732	−	789.352i	−0.869641	−	1.50626i
$66$		0			0
$67$	−199.277	+	345.159i	−0.363367	+	0.629371i	−0.988513	−	0.151138i	$-0.951706\pi$
$67$	−199.277	+	345.159i	−0.363367	+	0.629371i	0.625145	+	0.780508i	$0.285040\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−647.854			−1.08290			−0.541451	−	0.840732i	$-0.682125\pi$
$71$	−647.854			−1.08290			−0.541451	+	0.840732i	$0.682125\pi$
$72$		0			0
$73$	−478.279			−0.766826			−0.383413	−	0.923577i	$-0.625251\pi$
$73$	−478.279			−0.766826			−0.383413	+	0.923577i	$0.625251\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−55.0084	+	95.2773i	−0.0814128	+	0.141011i
$78$		0			0
$79$	187.158	+	324.167i	0.266543	+	0.461666i	0.967967	−	0.251079i	$-0.0807852\pi$
$79$	187.158	+	324.167i	0.266543	+	0.461666i	−0.701424	+	0.712744i	$0.747452\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−466.639	−	808.243i	−0.617112	−	1.06887i	−0.990010	−	0.140998i	$-0.954969\pi$
$83$	−466.639	−	808.243i	−0.617112	−	1.06887i	0.372897	−	0.927873i	$-0.378364\pi$
$84$		0			0
$85$	167.611	−	290.311i	0.213882	−	0.370455i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−368.817			−0.439264			−0.219632	−	0.975583i	$-0.570486\pi$
$89$	−368.817			−0.439264			−0.219632	+	0.975583i	$0.570486\pi$
$90$		0			0
$91$	269.279			0.310199
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−102.851	+	178.143i	−0.111076	+	0.192390i
$96$		0			0
$97$	−137.075	−	237.420i	−0.143483	−	0.248519i	0.785323	−	0.619086i	$-0.212497\pi$
$97$	−137.075	−	237.420i	−0.143483	−	0.248519i	−0.928806	+	0.370567i	$0.879163\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.78584	−	8.28931i	−0.00471494	−	0.00816651i	0.863658	−	0.504078i	$-0.168167\pi$
$101$	−4.78584	−	8.28931i	−0.00471494	−	0.00816651i	−0.868373	+	0.495911i	$0.834834\pi$
$102$		0			0
$103$	985.752	−	1707.37i	0.943001	−	1.63332i	0.183295	−	0.983058i	$-0.441324\pi$
$103$	985.752	−	1707.37i	0.943001	−	1.63332i	0.759706	−	0.650267i	$-0.225343\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1441.13			1.30205			0.651025	−	0.759057i	$-0.274339\pi$
$107$	1441.13			1.30205			0.651025	+	0.759057i	$0.274339\pi$
$108$		0			0
$109$	−90.3323			−0.0793786			−0.0396893	−	0.999212i	$-0.512637\pi$
$109$	−90.3323			−0.0793786			−0.0396893	+	0.999212i	$0.512637\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−825.364	+	1429.57i	−0.687112	+	1.19011i	0.285656	+	0.958332i	$0.407789\pi$
$113$	−825.364	+	1429.57i	−0.687112	+	1.19011i	−0.972768	+	0.231781i	$0.925545\pi$
$114$		0			0
$115$	−495.473	−	858.185i	−0.401766	−	0.695880i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	49.5183	+	85.7682i	0.0381457	+	0.0660702i
$120$		0			0
$121$	489.890	−	848.515i	0.368062	−	0.637502i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2877.91			2.05926
$126$		0			0
$127$	−1997.45			−1.39563			−0.697814	−	0.716279i	$-0.745844\pi$
$127$	−1997.45			−1.39563			−0.697814	+	0.716279i	$0.745844\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	61.8224	−	107.080i	0.0412324	−	0.0714167i	−0.844673	−	0.535283i	$-0.820205\pi$
$131$	61.8224	−	107.080i	0.0412324	−	0.0714167i	0.885905	+	0.463866i	$0.153538\pi$
$132$		0			0
$133$	−30.3857	−	52.6296i	−0.0198103	−	0.0343125i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	56.7028	+	98.2121i	0.0353609	+	0.0612469i	0.883164	−	0.469064i	$-0.155409\pi$
$137$	56.7028	+	98.2121i	0.0353609	+	0.0612469i	−0.847803	+	0.530311i	$0.822075\pi$
$138$		0			0
$139$	−196.799	+	340.865i	−0.120088	+	0.207999i	−0.919802	−	0.392382i	$-0.871651\pi$
$139$	−196.799	+	340.865i	−0.120088	+	0.207999i	0.799714	+	0.600381i	$0.204984\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	859.651			0.502711
$144$		0			0
$145$	−216.727			−0.124126
$146$		0			0
$147$		0			0
$148$		0			0
$149$	753.606	−	1305.28i	0.414348	−	0.717671i	−0.581012	−	0.813895i	$-0.697343\pi$
$149$	753.606	−	1305.28i	0.414348	−	0.717671i	0.995360	+	0.0962237i	$0.0306764\pi$
$150$		0			0
$151$	1580.70	+	2737.85i	0.851890	+	1.47552i	0.879500	+	0.475899i	$0.157877\pi$
$151$	1580.70	+	2737.85i	0.851890	+	1.47552i	−0.0276097	+	0.999619i	$0.508790\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1506.29	+	2608.97i	0.780569	+	1.35198i
$156$		0			0
$157$	−687.806	+	1191.31i	−0.349636	+	0.605587i	−0.986185	−	0.165649i	$-0.947028\pi$
$157$	−687.806	+	1191.31i	−0.349636	+	0.605587i	0.636549	+	0.771237i	$0.280361\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	292.761			0.143309
$162$		0			0
$163$	−542.073			−0.260481			−0.130241	−	0.991482i	$-0.541575\pi$
$163$	−542.073			−0.260481			−0.130241	+	0.991482i	$0.541575\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1564.81	+	2710.33i	−0.725081	+	1.25588i	0.233860	+	0.972270i	$0.424864\pi$
$167$	−1564.81	+	2710.33i	−0.725081	+	1.25588i	−0.958941	+	0.283607i	$0.908469\pi$
$168$		0			0
$169$	46.4520	+	80.4572i	0.0211434	+	0.0366214i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1259.14	−	2180.89i	−0.553355	−	0.958440i	−0.998029	−	0.0627472i	$-0.980014\pi$
$173$	−1259.14	−	2180.89i	−0.553355	−	0.958440i	0.444674	−	0.895692i	$-0.353320\pi$
$174$		0			0
$175$	−792.020	+	1371.82i	−0.342120	+	0.592570i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2331.26			−0.973443			−0.486721	−	0.873557i	$-0.661807\pi$
$179$	−2331.26			−0.973443			−0.486721	+	0.873557i	$0.661807\pi$
$180$		0			0
$181$	734.969			0.301822			0.150911	−	0.988547i	$-0.451779\pi$
$181$	734.969			0.301822			0.150911	+	0.988547i	$0.451779\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3439.43	+	5957.27i	−1.36688	+	2.36750i
$186$		0			0
$187$	158.083	+	273.808i	0.0618191	+	0.107074i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1808.88	−	3133.07i	−0.685266	−	1.18692i	−0.973353	−	0.229312i	$-0.926352\pi$
$191$	−1808.88	−	3133.07i	−0.685266	−	1.18692i	0.288087	−	0.957604i	$-0.406981\pi$
$192$		0			0
$193$	−2170.56	+	3759.52i	−0.809536	+	1.40216i	0.103649	+	0.994614i	$0.466948\pi$
$193$	−2170.56	+	3759.52i	−0.809536	+	1.40216i	−0.913186	+	0.407544i	$0.866385\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3286.20			1.18849			0.594243	−	0.804285i	$-0.297452\pi$
$197$	3286.20			1.18849			0.594243	+	0.804285i	$0.297452\pi$
$198$		0			0
$199$	−332.265			−0.118360			−0.0591800	−	0.998247i	$-0.518849\pi$
$199$	−332.265			−0.118360			−0.0591800	+	0.998247i	$0.518849\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	32.0144	−	55.4506i	0.0110688	−	0.0191718i
$204$		0			0
$205$	2630.26	+	4555.74i	0.896122	+	1.55213i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−97.0039	−	168.016i	−0.0321048	−	0.0556071i
$210$		0			0
$211$	−2871.24	+	4973.13i	−0.936797	+	1.62258i	−0.165398	+	0.986227i	$0.552891\pi$
$211$	−2871.24	+	4973.13i	−0.936797	+	1.62258i	−0.771398	+	0.636353i	$0.780442\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8184.43			2.59616
$216$		0			0
$217$	−890.023			−0.278427
$218$		0			0
$219$		0			0
$220$		0			0
$221$	386.927	−	670.177i	0.117772	−	0.203986i
$222$		0			0
$223$	−1231.31	−	2132.69i	−0.369751	−	0.640427i	0.619776	−	0.784779i	$-0.287224\pi$
$223$	−1231.31	−	2132.69i	−0.369751	−	0.640427i	−0.989526	+	0.144352i	$0.953890\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1399.67	−	2424.30i	−0.409248	−	0.708838i	0.585558	−	0.810631i	$-0.300876\pi$
$227$	−1399.67	−	2424.30i	−0.409248	−	0.708838i	−0.994806	+	0.101792i	$0.967542\pi$
$228$		0			0
$229$	−706.423	+	1223.56i	−0.203850	+	0.353079i	−0.949766	−	0.312961i	$-0.898679\pi$
$229$	−706.423	+	1223.56i	−0.203850	+	0.353079i	0.745915	+	0.666041i	$0.232012\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2033.81			−0.571843			−0.285922	−	0.958253i	$-0.592300\pi$
$233$	−2033.81			−0.571843			−0.285922	+	0.958253i	$0.592300\pi$
$234$		0			0
$235$	9379.96			2.60375
$236$		0			0
$237$		0			0
$238$		0			0
$239$	260.806	−	451.729i	0.0705863	−	0.122259i	−0.828572	−	0.559882i	$-0.810846\pi$
$239$	260.806	−	451.729i	0.0705863	−	0.122259i	0.899158	+	0.437623i	$0.144180\pi$
$240$		0			0
$241$	−2957.82	−	5123.10i	−0.790582	−	1.36933i	−0.925607	−	0.378486i	$-0.876445\pi$
$241$	−2957.82	−	5123.10i	−0.790582	−	1.36933i	0.135025	−	0.990842i	$-0.456888\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3065.39	+	5309.41i	0.799350	+	1.38451i
$246$		0			0
$247$	−237.428	+	411.238i	−0.0611628	+	0.105937i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	710.127			0.178577			0.0892884	−	0.996006i	$-0.471541\pi$
$251$	710.127			0.178577			0.0892884	+	0.996006i	$0.471541\pi$
$252$		0			0
$253$	934.614			0.232248
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3150.50	−	5456.83i	0.764681	−	1.32447i	−0.175734	−	0.984438i	$-0.556230\pi$
$257$	3150.50	−	5456.83i	0.764681	−	1.32447i	0.940415	−	0.340028i	$-0.110437\pi$
$258$		0			0
$259$	−1016.13	−	1759.99i	−0.243781	−	0.422241i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2460.91	+	4262.42i	0.576981	+	0.999361i	0.995823	+	0.0913017i	$0.0291028\pi$
$263$	2460.91	+	4262.42i	0.576981	+	0.999361i	−0.418842	+	0.908059i	$0.637564\pi$
$264$		0			0
$265$	2883.06	−	4993.61i	0.668320	−	1.15757i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5454.50			1.23631			0.618154	−	0.786057i	$-0.287881\pi$
$269$	5454.50			1.23631			0.618154	+	0.786057i	$0.287881\pi$
$270$		0			0
$271$	−2797.10			−0.626981			−0.313491	−	0.949591i	$-0.601498\pi$
$271$	−2797.10			−0.626981			−0.313491	+	0.949591i	$0.601498\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2528.46	+	4379.42i	−0.554443	+	0.960323i
$276$		0			0
$277$	1072.59	+	1857.77i	0.232655	+	0.402970i	0.958589	−	0.284794i	$-0.0919254\pi$
$277$	1072.59	+	1857.77i	0.232655	+	0.402970i	−0.725934	+	0.687765i	$0.758592\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2754.88	−	4771.60i	−0.584849	−	1.01299i	−0.994894	−	0.100923i	$-0.967820\pi$
$281$	−2754.88	−	4771.60i	−0.584849	−	1.01299i	0.410045	−	0.912065i	$-0.365513\pi$
$282$		0			0
$283$	2003.92	−	3470.89i	0.420921	−	0.729056i	−0.575109	−	0.818077i	$-0.695040\pi$
$283$	2003.92	−	3470.89i	0.420921	−	0.729056i	0.996030	+	0.0890206i	$0.0283737\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1554.14			−0.319645
$288$		0			0
$289$	−4628.39			−0.942070
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2904.62	+	5030.95i	−0.579146	+	1.00311i	0.416432	+	0.909167i	$0.363280\pi$
$293$	−2904.62	+	5030.95i	−0.579146	+	1.00311i	−0.995578	+	0.0939429i	$0.970053\pi$
$294$		0			0
$295$	−529.141	−	916.499i	−0.104433	−	0.180884i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1143.79	−	1981.10i	−0.221227	−	0.383177i
$300$		0			0
$301$	−1208.99	+	2094.02i	−0.231511	+	0.400989i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5840.78			−1.09653
$306$		0			0
$307$	−8688.30			−1.61520			−0.807602	−	0.589728i	$-0.799235\pi$
$307$	−8688.30			−1.61520			−0.807602	+	0.589728i	$0.799235\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−716.005	+	1240.16i	−0.130550	+	0.226119i	−0.923889	−	0.382662i	$-0.875008\pi$
$311$	−716.005	+	1240.16i	−0.130550	+	0.226119i	0.793339	+	0.608780i	$0.208341\pi$
$312$		0			0
$313$	−3307.26	−	5728.34i	−0.597244	−	1.03446i	−0.993226	−	0.116199i	$-0.962929\pi$
$313$	−3307.26	−	5728.34i	−0.597244	−	1.03446i	0.395982	−	0.918258i	$-0.370404\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	712.045	+	1233.30i	0.126159	+	0.218514i	0.922185	−	0.386748i	$-0.126402\pi$
$317$	712.045	+	1233.30i	0.126159	+	0.218514i	−0.796026	+	0.605262i	$0.793068\pi$
$318$		0			0
$319$	102.203	−	177.021i	0.0179382	−	0.0310699i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−174.645			−0.0300851
$324$		0			0
$325$	12377.4			2.11254
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1385.59	+	2399.91i	−0.232188	+	0.402161i
$330$		0			0
$331$	1768.83	+	3063.70i	0.293726	+	0.508749i	0.974688	−	0.223570i	$-0.0717712\pi$
$331$	1768.83	+	3063.70i	0.293726	+	0.508749i	−0.680961	+	0.732319i	$0.738438\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3959.73	−	6858.45i	−0.645800	−	1.11856i
$336$		0			0
$337$	880.140	−	1524.45i	0.142268	−	0.246415i	−0.786082	−	0.618122i	$-0.787894\pi$
$337$	880.140	−	1524.45i	0.142268	−	0.246415i	0.928350	+	0.371706i	$0.121227\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2841.32			−0.451221
$342$		0			0
$343$	−3824.81			−0.602099
$344$		0			0
$345$		0			0
$346$		0			0
$347$	802.720	−	1390.35i	0.124185	−	0.215095i	−0.797229	−	0.603677i	$-0.793702\pi$
$347$	802.720	−	1390.35i	0.124185	−	0.215095i	0.921414	+	0.388582i	$0.127035\pi$
$348$		0			0
$349$	3320.94	+	5752.04i	0.509358	+	0.882235i	0.999941	+	0.0108400i	$0.00345056\pi$
$349$	3320.94	+	5752.04i	0.509358	+	0.882235i	−0.490583	+	0.871395i	$0.663216\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1528.13	−	2646.81i	−0.230409	−	0.399080i	0.727520	−	0.686087i	$-0.240673\pi$
$353$	−1528.13	−	2646.81i	−0.230409	−	0.399080i	−0.957929	+	0.287007i	$0.907340\pi$
$354$		0			0
$355$	6436.56	−	11148.5i	0.962302	−	1.66676i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2489.46			−0.365985			−0.182993	−	0.983114i	$-0.558578\pi$
$359$	−2489.46			−0.365985			−0.182993	+	0.983114i	$0.558578\pi$
$360$		0			0
$361$	−6751.83			−0.984376
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4751.80	−	8230.36i	0.681427	−	1.18027i
$366$		0			0
$367$	1177.52	+	2039.53i	0.167483	+	0.290089i	0.937534	−	0.347893i	$-0.113103\pi$
$367$	1177.52	+	2039.53i	0.167483	+	0.290089i	−0.770051	+	0.637982i	$0.779769\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	851.758	+	1475.29i	0.119194	+	0.206450i
$372$		0			0
$373$	1378.51	−	2387.65i	0.191358	−	0.331443i	−0.754342	−	0.656481i	$-0.772044\pi$
$373$	1378.51	−	2387.65i	0.191358	−	0.331443i	0.945701	+	0.325039i	$0.105377\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−500.310			−0.0683481
$378$		0			0
$379$	−246.459			−0.0334030			−0.0167015	−	0.999861i	$-0.505317\pi$
$379$	−246.459			−0.0334030			−0.0167015	+	0.999861i	$0.505317\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−325.287	+	563.414i	−0.0433979	+	0.0751674i	−0.886908	−	0.461945i	$-0.847152\pi$
$383$	−325.287	+	563.414i	−0.0433979	+	0.0751674i	0.843511	+	0.537113i	$0.180485\pi$
$384$		0			0
$385$	−1093.04	−	1893.20i	−0.144692	−	0.250614i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5123.08	−	8873.44i	−0.667739	−	1.15656i	−0.978535	−	0.206082i	$-0.933929\pi$
$389$	−5123.08	−	8873.44i	−0.667739	−	1.15656i	0.310795	−	0.950477i	$-0.399405\pi$
$390$		0			0
$391$	420.668	−	728.618i	0.0544094	−	0.0942399i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7437.81			−0.947435
$396$		0			0
$397$	−9453.68			−1.19513			−0.597565	−	0.801820i	$-0.703865\pi$
$397$	−9453.68			−1.19513			−0.597565	+	0.801820i	$0.703865\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	135.361	−	234.453i	0.0168569	−	0.0291970i	−0.857474	−	0.514527i	$-0.827967\pi$
$401$	135.361	−	234.453i	0.0168569	−	0.0291970i	0.874331	+	0.485330i	$0.161301\pi$
$402$		0			0
$403$	3477.24	+	6022.75i	0.429810	+	0.744453i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3243.91	−	5618.62i	−0.395073	−	0.684286i
$408$		0			0
$409$	5793.08	−	10033.9i	0.700366	−	1.21307i	−0.267972	−	0.963427i	$-0.586354\pi$
$409$	5793.08	−	10033.9i	0.700366	−	1.21307i	0.968338	−	0.249642i	$-0.0803130\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	312.654			0.0372511
$414$		0			0
$415$	18544.7			2.19354
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8066.87	−	13972.2i	0.940554	−	1.62909i	0.176137	−	0.984366i	$-0.443640\pi$
$419$	8066.87	−	13972.2i	0.940554	−	1.62909i	0.764417	−	0.644722i	$-0.223027\pi$
$420$		0			0
$421$	−2495.96	−	4323.14i	−0.288945	−	0.500467i	0.684613	−	0.728907i	$-0.259971\pi$
$421$	−2495.96	−	4323.14i	−0.288945	−	0.500467i	−0.973558	+	0.228439i	$0.926638\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2276.11	+	3942.33i	0.259782	+	0.449956i
$426$		0			0
$427$	862.787	−	1494.39i	0.0977827	−	0.169365i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8184.74			0.914722			0.457361	−	0.889281i	$-0.348795\pi$
$431$	8184.74			0.914722			0.457361	+	0.889281i	$0.348795\pi$
$432$		0			0
$433$	8663.17			0.961490			0.480745	−	0.876860i	$-0.340366\pi$
$433$	8663.17			0.961490			0.480745	+	0.876860i	$0.340366\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−258.133	+	447.099i	−0.0282567	+	0.0489420i
$438$		0			0
$439$	−7932.15	−	13738.9i	−0.862371	−	1.49367i	−0.869634	−	0.493697i	$-0.835645\pi$
$439$	−7932.15	−	13738.9i	−0.862371	−	1.49367i	0.00726314	−	0.999974i	$-0.497688\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	199.820	+	346.099i	0.0214306	+	0.0371189i	0.876542	−	0.481326i	$-0.159845\pi$
$443$	199.820	+	346.099i	0.0214306	+	0.0371189i	−0.855111	+	0.518445i	$0.826511\pi$
$444$		0			0
$445$	3664.28	−	6346.71i	0.390345	−	0.676097i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5924.51			−0.622706			−0.311353	−	0.950294i	$-0.600782\pi$
$449$	−5924.51			−0.622706			−0.311353	+	0.950294i	$0.600782\pi$
$450$		0			0
$451$	−4961.47			−0.518019
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2675.34	+	4633.83i	−0.275653	+	0.477445i
$456$		0			0
$457$	3945.57	+	6833.93i	0.403864	+	0.699513i	0.994189	−	0.107653i	$-0.0343335\pi$
$457$	3945.57	+	6833.93i	0.403864	+	0.699513i	−0.590324	+	0.807166i	$0.701000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1631.09	−	2825.13i	−0.164789	−	0.285422i	0.771792	−	0.635876i	$-0.219361\pi$
$461$	−1631.09	−	2825.13i	−0.164789	−	0.285422i	−0.936580	+	0.350453i	$0.886028\pi$
$462$		0			0
$463$	1845.20	−	3195.98i	0.185213	−	0.320799i	−0.758435	−	0.651749i	$-0.774036\pi$
$463$	1845.20	−	3195.98i	0.185213	−	0.320799i	0.943648	+	0.330950i	$0.107369\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10193.2			−1.01003			−0.505017	−	0.863110i	$-0.668514\pi$
$467$	−10193.2			−1.01003			−0.505017	+	0.863110i	$0.668514\pi$
$468$		0			0
$469$	2339.69			0.230355
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3859.59	+	6685.00i	−0.375188	+	0.649845i
$474$		0			0
$475$	−1396.68	−	2419.12i	−0.134914	−	0.233677i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−420.034	−	727.521i	−0.0400665	−	0.0693972i	0.845297	−	0.534297i	$-0.179424\pi$
$479$	−420.034	−	727.521i	−0.0400665	−	0.0693972i	−0.885363	+	0.464900i	$0.846090\pi$
$480$		0			0
$481$	−7939.85	+	13752.2i	−0.752653	+	1.30363i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5447.46			0.510014
$486$		0			0
$487$	3367.28			0.313319			0.156659	−	0.987653i	$-0.449928\pi$
$487$	3367.28			0.313319			0.156659	+	0.987653i	$0.449928\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9434.55	−	16341.1i	0.867160	−	1.50196i	0.00227283	−	0.999997i	$-0.499277\pi$
$491$	9434.55	−	16341.1i	0.867160	−	1.50196i	0.864887	−	0.501967i	$-0.167390\pi$
$492$		0			0
$493$	−92.0030	−	159.354i	−0.00840488	−	0.0145577i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1901.59	+	3293.65i	0.171626	+	0.297264i
$498$		0			0
$499$	−7283.00	+	12614.5i	−0.653371	+	1.13167i	0.328929	+	0.944355i	$0.393312\pi$
$499$	−7283.00	+	12614.5i	−0.653371	+	1.13167i	−0.982300	+	0.187317i	$0.940021\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−17361.6			−1.53900			−0.769499	−	0.638648i	$-0.779494\pi$
$503$	−17361.6			−1.53900			−0.769499	+	0.638648i	$0.779494\pi$
$504$		0			0
$505$	190.193			0.0167594
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6966.63	−	12066.6i	0.606661	−	1.05077i	−0.385126	−	0.922864i	$-0.625842\pi$
$509$	6966.63	−	12066.6i	0.606661	−	1.05077i	0.991787	−	0.127903i	$-0.0408247\pi$
$510$		0			0
$511$	1403.85	+	2431.54i	0.121532	+	0.210499i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19587.3	+	33926.2i	1.67596	+	2.90285i
$516$		0			0
$517$	−4423.37	+	7661.50i	−0.376285	+	0.651745i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5024.22			−0.422486			−0.211243	−	0.977434i	$-0.567751\pi$
$521$	−5024.22			−0.422486			−0.211243	+	0.977434i	$0.567751\pi$
$522$		0			0
$523$	−16008.7			−1.33845			−0.669226	−	0.743059i	$-0.733374\pi$
$523$	−16008.7			−1.33845			−0.669226	+	0.743059i	$0.733374\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1278.87	+	2215.07i	−0.105709	+	0.183093i
$528$		0			0
$529$	4839.97	+	8383.07i	0.397795	+	0.689001i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6071.89	+	10516.8i	0.493438	+	0.854660i
$534$		0			0
$535$	−14317.9	+	24799.4i	−1.15704	+	2.00406i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5782.27			−0.462078
$540$		0			0
$541$	−10094.1			−0.802179			−0.401089	−	0.916039i	$-0.631368\pi$
$541$	−10094.1			−0.802179			−0.401089	+	0.916039i	$0.631368\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	897.471	−	1554.47i	0.0705384	−	0.122176i
$546$		0			0
$547$	−3030.27	−	5248.58i	−0.236865	−	0.410262i	0.722948	−	0.690902i	$-0.242787\pi$
$547$	−3030.27	−	5248.58i	−0.236865	−	0.410262i	−0.959813	+	0.280640i	$0.909453\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	56.4554	+	97.7837i	0.00436494	+	0.00756030i
$552$		0			0
$553$	1098.70	−	1903.00i	0.0844870	−	0.146336i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13688.4			−1.04128			−0.520642	−	0.853775i	$-0.674307\pi$
$557$	−13688.4			−1.04128			−0.520642	+	0.853775i	$0.674307\pi$
$558$		0			0
$559$	18893.6			1.42954
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−11173.3	+	19352.8i	−0.836411	+	1.44871i	0.0564662	+	0.998405i	$0.482017\pi$
$563$	−11173.3	+	19352.8i	−0.836411	+	1.44871i	−0.892877	+	0.450301i	$0.851317\pi$
$564$		0			0
$565$	−16400.3	−	28406.2i	−1.22118	−	2.11515i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4058.97	+	7030.34i	0.299052	+	0.517974i	0.975919	−	0.218131i	$-0.0699961\pi$
$569$	4058.97	+	7030.34i	0.299052	+	0.517974i	−0.676867	+	0.736105i	$0.736663\pi$
$570$		0			0
$571$	3491.49	−	6047.43i	0.255892	−	0.443217i	−0.709246	−	0.704961i	$-0.750964\pi$
$571$	3491.49	−	6047.43i	0.255892	−	0.443217i	0.965137	+	0.261744i	$0.0842976\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	13456.7			0.975973
$576$		0			0
$577$	13972.4			1.00811			0.504055	−	0.863671i	$-0.331841\pi$
$577$	13972.4			1.00811			0.504055	+	0.863671i	$0.331841\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2739.37	+	4744.73i	−0.195608	+	0.338803i
$582$		0			0
$583$	2719.17	+	4709.73i	0.193167	+	0.334575i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	754.756	+	1307.28i	0.0530701	+	0.0919200i	0.891340	−	0.453335i	$-0.149766\pi$
$587$	754.756	+	1307.28i	0.0530701	+	0.0919200i	−0.838270	+	0.545255i	$0.816433\pi$
$588$		0			0
$589$	784.750	−	1359.23i	0.0548982	−	0.0950865i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	21495.3			1.48854			0.744270	−	0.667879i	$-0.232798\pi$
$593$	21495.3			1.48854			0.744270	+	0.667879i	$0.232798\pi$
$594$		0			0
$595$	−1967.90			−0.135590
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−5052.44	+	8751.08i	−0.344636	+	0.596927i	−0.985288	−	0.170905i	$-0.945331\pi$
$599$	−5052.44	+	8751.08i	−0.344636	+	0.596927i	0.640652	+	0.767832i	$0.278664\pi$
$600$		0			0
$601$	5504.09	+	9533.37i	0.373572	+	0.647045i	0.990112	−	0.140278i	$-0.0447997\pi$
$601$	5504.09	+	9533.37i	0.373572	+	0.647045i	−0.616540	+	0.787323i	$0.711466\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9734.33	+	16860.3i	0.654143	+	1.13301i
$606$		0			0
$607$	7340.08	−	12713.4i	0.490815	−	0.850116i	−0.509129	−	0.860690i	$-0.670033\pi$
$607$	7340.08	−	12713.4i	0.490815	−	0.850116i	0.999944	+	0.0105740i	$0.00336587\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	21653.4			1.43372
$612$		0			0
$613$	235.863			0.0155407			0.00777033	−	0.999970i	$-0.497527\pi$
$613$	235.863			0.0155407			0.00777033	+	0.999970i	$0.497527\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9156.53	−	15859.6i	0.597452	−	1.03482i	−0.395744	−	0.918361i	$-0.629513\pi$
$617$	9156.53	−	15859.6i	0.597452	−	1.03482i	0.993196	−	0.116457i	$-0.0371536\pi$
$618$		0			0
$619$	3424.73	+	5931.81i	0.222377	+	0.385169i	0.955529	−	0.294896i	$-0.0952850\pi$
$619$	3424.73	+	5931.81i	0.222377	+	0.385169i	−0.733152	+	0.680065i	$0.761952\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1082.56	+	1875.04i	0.0696175	+	0.120581i
$624$		0			0
$625$	−11728.0	+	20313.6i	−0.750594	+	1.30007i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5840.30			−0.370220
$630$		0			0
$631$	−22464.3			−1.41726			−0.708629	−	0.705581i	$-0.750686\pi$
$631$	−22464.3			−1.41726			−0.708629	+	0.705581i	$0.750686\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	19845.1	−	34372.7i	1.24020	−	2.14809i
$636$		0			0
$637$	7076.39	+	12256.7i	0.440152	+	0.762365i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9444.39	+	16358.2i	0.581952	+	1.00797i	0.995248	+	0.0973730i	$0.0310440\pi$
$641$	9444.39	+	16358.2i	0.581952	+	1.00797i	−0.413296	+	0.910597i	$0.635623\pi$
$642$		0			0
$643$	3396.98	−	5883.74i	0.208342	−	0.360858i	−0.742851	−	0.669457i	$-0.766527\pi$
$643$	3396.98	−	5883.74i	0.208342	−	0.360858i	0.951192	+	0.308599i	$0.0998601\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5277.92			0.320706			0.160353	−	0.987060i	$-0.448737\pi$
$647$	5277.92			0.320706			0.160353	+	0.987060i	$0.448737\pi$
$648$		0			0
$649$	998.122			0.0603694
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8282.09	+	14345.0i	−0.496330	+	0.859668i	−0.999991	−	0.00423291i	$-0.998653\pi$
$653$	−8282.09	+	14345.0i	−0.496330	+	0.859668i	0.503661	+	0.863901i	$0.331986\pi$
$654$		0			0
$655$	1228.44	+	2127.72i	0.0732810	+	0.126926i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1650.51	+	2858.77i	0.0975641	+	0.168986i	0.910676	−	0.413122i	$-0.135562\pi$
$659$	1650.51	+	2858.77i	0.0975641	+	0.168986i	−0.813112	+	0.582108i	$0.802228\pi$
$660$		0			0
$661$	−6495.17	+	11250.0i	−0.382198	+	0.661986i	−0.991376	−	0.131047i	$-0.958166\pi$
$661$	−6495.17	+	11250.0i	−0.382198	+	0.661986i	0.609178	+	0.793033i	$0.291499\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1207.55			0.0704164
$666$		0			0
$667$	−543.938			−0.0315762
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2754.38	−	4770.72i	0.158467	−	0.274473i
$672$		0			0
$673$	−9257.15	−	16033.9i	−0.530219	−	0.918365i	−0.999378	−	0.0352522i	$-0.988777\pi$
$673$	−9257.15	−	16033.9i	−0.530219	−	0.918365i	0.469160	−	0.883113i	$-0.344557\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3050.00	−	5282.75i	−0.173148	−	0.299900i	0.766371	−	0.642398i	$-0.222060\pi$
$677$	−3050.00	−	5282.75i	−0.173148	−	0.299900i	−0.939519	+	0.342498i	$0.888727\pi$
$678$		0			0
$679$	−804.687	+	1393.76i	−0.0454802	+	0.0787740i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−22630.0			−1.26781			−0.633905	−	0.773411i	$-0.718549\pi$
$683$	−22630.0			−1.26781			−0.633905	+	0.773411i	$0.718549\pi$
$684$		0			0
$685$	−2253.42			−0.125691
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6655.48	−	11527.6i	0.368002	−	0.637398i
$690$		0			0
$691$	11493.4	+	19907.2i	0.632750	+	1.09595i	0.986987	+	0.160799i	$0.0514072\pi$
$691$	11493.4	+	19907.2i	0.632750	+	1.09595i	−0.354237	+	0.935156i	$0.615259\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3910.48	−	6773.14i	−0.213428	−	0.369669i
$696$		0			0
$697$	−2233.15	+	3867.92i	−0.121358	+	0.210198i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−27015.5			−1.45558			−0.727790	−	0.685800i	$-0.759452\pi$
$701$	−27015.5			−1.45558			−0.727790	+	0.685800i	$0.759452\pi$
$702$		0			0
$703$	3583.76			0.192268
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−28.0949	+	48.6618i	−0.00149451	+	0.00258857i
$708$		0			0
$709$	9588.67	+	16608.1i	0.507912	+	0.879730i	0.999958	+	0.00916077i	$0.00291601\pi$
$709$	9588.67	+	16608.1i	0.507912	+	0.879730i	−0.492046	+	0.870569i	$0.663751\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3780.46	+	6547.95i	0.198568	+	0.343931i
$714$		0			0
$715$	−8540.81	+	14793.1i	−0.446725	+	0.773750i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	25931.8			1.34505			0.672526	−	0.740073i	$-0.265209\pi$
$719$	25931.8			1.34505			0.672526	+	0.740073i	$0.265209\pi$
$720$		0			0
$721$	−11573.6			−0.597812
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1471.54	−	2548.78i	0.0753816	−	0.130565i
$726$		0			0
$727$	2915.25	+	5049.36i	0.148722	+	0.257594i	0.930755	−	0.365643i	$-0.119151\pi$
$727$	2915.25	+	5049.36i	0.148722	+	0.257594i	−0.782034	+	0.623236i	$0.785817\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3474.38	+	6017.81i	0.175793	+	0.304482i
$732$		0			0
$733$	11577.3	−	20052.4i	0.583379	−	1.01044i	−0.411696	−	0.911321i	$-0.635064\pi$
$733$	11577.3	−	20052.4i	0.583379	−	1.01044i	0.995075	−	0.0991211i	$-0.0316031\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7469.26			0.373316
$738$		0			0
$739$	27085.8			1.34827			0.674133	−	0.738610i	$-0.264517\pi$
$739$	27085.8			1.34827			0.674133	+	0.738610i	$0.264517\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16093.8	+	27875.3i	−0.794650	+	1.37637i	0.128412	+	0.991721i	$0.459012\pi$
$743$	−16093.8	+	27875.3i	−0.794650	+	1.37637i	−0.923061	+	0.384653i	$0.874321\pi$
$744$		0			0
$745$	14974.5	+	25936.5i	0.736406	+	1.27549i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4230.02	−	7326.61i	−0.206357	−	0.357421i
$750$		0			0
$751$	−1366.46	+	2366.78i	−0.0663954	+	0.115000i	−0.897312	−	0.441397i	$-0.854483\pi$
$751$	−1366.46	+	2366.78i	−0.0663954	+	0.115000i	0.830917	+	0.556397i	$0.187817\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−62818.3			−3.02807
$756$		0			0
$757$	−6315.62			−0.303230			−0.151615	−	0.988440i	$-0.548447\pi$
$757$	−6315.62			−0.303230			−0.151615	+	0.988440i	$0.548447\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15740.7	−	27263.6i	0.749801	−	1.29869i	−0.198116	−	0.980179i	$-0.563482\pi$
$761$	15740.7	−	27263.6i	0.749801	−	1.29869i	0.947918	−	0.318516i	$-0.103184\pi$
$762$		0			0
$763$	265.145	+	459.244i	0.0125804	+	0.0217900i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1221.51	−	2115.72i	−0.0575048	−	0.0996012i
$768$		0			0
$769$	13648.3	−	23639.6i	0.640014	−	1.10854i	−0.345415	−	0.938450i	$-0.612262\pi$
$769$	13648.3	−	23639.6i	0.640014	−	1.10854i	0.985429	−	0.170087i	$-0.0544048\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−24025.2			−1.11789			−0.558943	−	0.829206i	$-0.688793\pi$
$773$	−24025.2			−1.11789			−0.558943	+	0.829206i	$0.688793\pi$
$774$		0			0
$775$	−40909.9			−1.89616
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1370.32	−	2373.46i	0.0630252	−	0.109163i
$780$		0			0
$781$	6070.66	+	10514.7i	0.278138	+	0.481748i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13667.0	−	23671.9i	−0.621396	−	1.07629i
$786$		0			0
$787$	−5279.44	+	9144.26i	−0.239125	+	0.414177i	−0.960464	−	0.278406i	$-0.910194\pi$
$787$	−5279.44	+	9144.26i	−0.239125	+	0.414177i	0.721338	+	0.692583i	$0.243527\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9690.47			0.435592
$792$		0			0
$793$	−13483.3			−0.603792
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13444.6	+	23286.7i	−0.597529	+	1.03495i	0.395656	+	0.918399i	$0.370517\pi$
$797$	−13444.6	+	23286.7i	−0.597529	+	1.03495i	−0.993185	+	0.116552i	$0.962816\pi$
$798$		0			0
$799$	3981.90	+	6896.85i	0.176307	+	0.305373i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4481.68	+	7762.49i	0.196955	+	0.341136i
$804$		0			0
$805$	−2908.64	+	5037.91i	−0.127349	+	0.220575i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18643.8			0.810236			0.405118	−	0.914264i	$-0.367230\pi$
$809$	18643.8			0.810236			0.405118	+	0.914264i	$0.367230\pi$
$810$		0			0
$811$	23870.8			1.03356			0.516781	−	0.856118i	$-0.327130\pi$
$811$	23870.8			1.03356			0.516781	+	0.856118i	$0.327130\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5385.61	−	9328.15i	0.231472	−	0.400921i
$816$		0			0
$817$	−2131.97	−	3692.68i	−0.0912952	−	0.158128i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7266.51	+	12586.0i	0.308895	+	0.535022i	0.978121	−	0.208037i	$-0.0667074\pi$
$821$	7266.51	+	12586.0i	0.308895	+	0.535022i	−0.669226	+	0.743059i	$0.733374\pi$
$822$		0			0
$823$	4434.21	−	7680.28i	0.187809	−	0.325295i	−0.756710	−	0.653750i	$-0.773195\pi$
$823$	4434.21	−	7680.28i	0.187809	−	0.325295i	0.944520	+	0.328455i	$0.106528\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25059.3			1.05369			0.526843	−	0.849963i	$-0.323376\pi$
$827$	25059.3			1.05369			0.526843	+	0.849963i	$0.323376\pi$
$828$		0			0
$829$	−22556.3			−0.945009			−0.472505	−	0.881328i	$-0.656650\pi$
$829$	−22556.3			−0.945009			−0.472505	+	0.881328i	$0.656650\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2602.58	+	4507.81i	−0.108252	+	0.187499i
$834$		0			0
$835$	−31093.4	−	53855.4i	−1.28866	−	2.23203i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−10604.8	−	18368.0i	−0.436374	−	0.755822i	0.561033	−	0.827794i	$-0.310404\pi$
$839$	−10604.8	−	18368.0i	−0.436374	−	0.755822i	−0.997407	+	0.0719718i	$0.977071\pi$
$840$		0			0
$841$	12135.0	−	21018.5i	0.497561	−	0.861801i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1846.04			−0.0751548
$846$		0			0
$847$	−5751.73			−0.233331
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8632.22	+	14951.4i	−0.347719	+	0.602267i
$852$		0			0
$853$	−14058.1	−	24349.4i	−0.564292	−	0.977382i	−0.997115	−	0.0759030i	$-0.975816\pi$
$853$	−14058.1	−	24349.4i	−0.564292	−	0.977382i	0.432824	−	0.901479i	$-0.357517\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4738.82	−	8207.87i	−0.188886	−	0.327159i	0.755993	−	0.654579i	$-0.227154\pi$
$857$	−4738.82	−	8207.87i	−0.188886	−	0.327159i	−0.944879	+	0.327420i	$0.893821\pi$
$858$		0			0
$859$	14772.9	−	25587.4i	0.586780	−	1.01633i	−0.407871	−	0.913040i	$-0.633729\pi$
$859$	14772.9	−	25587.4i	0.586780	−	1.01633i	0.994651	−	0.103293i	$-0.0329380\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9932.53			0.391781			0.195891	−	0.980626i	$-0.437240\pi$
$863$	9932.53			0.391781			0.195891	+	0.980626i	$0.437240\pi$
$864$		0			0
$865$	50039.2			1.96692
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3507.50	−	6075.16i	0.136920	−	0.237153i
$870$		0			0
$871$	−9140.94	−	15832.6i	−0.355602	−	0.615920i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8447.28	−	14631.1i	−0.326366	−	0.565282i
$876$		0			0
$877$	13512.7	−	23404.7i	0.520288	−	0.901165i	−0.479434	−	0.877578i	$-0.659158\pi$
$877$	13512.7	−	23404.7i	0.520288	−	0.901165i	0.999722	−	0.0235871i	$-0.00750870\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12798.1			0.489422			0.244711	−	0.969596i	$-0.421307\pi$
$881$	12798.1			0.489422			0.244711	+	0.969596i	$0.421307\pi$
$882$		0			0
$883$	−27016.7			−1.02965			−0.514826	−	0.857294i	$-0.672144\pi$
$883$	−27016.7			−1.02965			−0.514826	+	0.857294i	$0.672144\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10075.1	−	17450.5i	0.381384	−	0.660576i	−0.609876	−	0.792497i	$-0.708781\pi$
$887$	10075.1	−	17450.5i	0.381384	−	0.660576i	0.991260	+	0.131920i	$0.0421143\pi$
$888$		0			0
$889$	5862.94	+	10154.9i	0.221188	+	0.383110i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2443.39	−	4232.08i	−0.0915622	−	0.158590i
$894$		0			0
$895$	23161.5	−	40116.9i	0.865033	−	1.49828i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1653.63			0.0613477
$900$		0			0
$901$	4895.56			0.181015
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7302.08	+	12647.6i	−0.268209	+	0.464552i
$906$		0			0
$907$	4639.96	+	8036.64i	0.169865	+	0.294214i	0.938372	−	0.345627i	$-0.112334\pi$
$907$	4639.96	+	8036.64i	0.169865	+	0.294214i	−0.768508	+	0.639841i	$0.779000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8200.30	−	14203.3i	−0.298231	−	0.516551i	0.677501	−	0.735522i	$-0.263063\pi$
$911$	−8200.30	−	14203.3i	−0.298231	−	0.516551i	−0.975731	+	0.218972i	$0.929730\pi$
$912$		0			0
$913$	−8745.22	+	15147.2i	−0.317004	+	0.549067i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−725.848			−0.0261392
$918$		0			0
$919$	11704.6			0.420128			0.210064	−	0.977688i	$-0.432633\pi$
$919$	11704.6			0.420128			0.210064	+	0.977688i	$0.432633\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	14858.7	−	25736.0i	0.529880	−	0.917778i
$924$		0			0
$925$	−46706.3	−	80897.8i	−1.66021	−	2.87557i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8162.22	+	14137.4i	0.288260	+	0.499281i	0.973395	−	0.229136i	$-0.0735899\pi$
$929$	8162.22	+	14137.4i	0.288260	+	0.499281i	−0.685134	+	0.728417i	$0.740257\pi$
$930$		0			0
$931$	1597.01	−	2766.11i	0.0562191	−	0.0973744i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6282.36			−0.219738
$936$		0			0
$937$	−7397.79			−0.257925			−0.128962	−	0.991649i	$-0.541165\pi$
$937$	−7397.79			−0.257925			−0.128962	+	0.991649i	$0.541165\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3713.44	−	6431.87i	0.128645	−	0.222819i	−0.794507	−	0.607255i	$-0.792271\pi$
$941$	3713.44	−	6431.87i	0.128645	−	0.222819i	0.923152	+	0.384436i	$0.125604\pi$
$942$		0			0
$943$	6601.37	+	11433.9i	0.227964	+	0.394845i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11526.0	+	19963.6i	0.395505	+	0.685035i	0.993166	−	0.116714i	$-0.0372362\pi$
$947$	11526.0	+	19963.6i	0.395505	+	0.685035i	−0.597660	+	0.801749i	$0.703903\pi$
$948$		0			0
$949$	10969.4	−	18999.6i	0.375219	−	0.649898i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	23445.7			0.796936			0.398468	−	0.917182i	$-0.369542\pi$
$953$	23445.7			0.796936			0.398468	+	0.917182i	$0.369542\pi$
$954$		0			0
$955$	71886.4			2.43580
$956$		0			0
$957$		0			0
$958$		0			0
$959$	332.869	−	576.547i	0.0112085	−	0.0194136i
$960$		0			0
$961$	3402.51	+	5893.32i	0.114213	+	0.197822i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−43130.0	−	74703.4i	−1.43876	−	2.49201i
$966$		0			0
$967$	−8370.95	+	14498.9i	−0.278378	+	0.482165i	−0.970982	−	0.239153i	$-0.923130\pi$
$967$	−8370.95	+	14498.9i	−0.278378	+	0.482165i	0.692604	+	0.721318i	$0.256464\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−43037.7			−1.42240			−0.711198	−	0.702992i	$-0.751847\pi$
$971$	−43037.7			−1.42240			−0.711198	+	0.702992i	$0.751847\pi$
$972$		0			0
$973$	2310.59			0.0761295
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8949.91	+	15501.7i	−0.293074	+	0.507619i	−0.974535	−	0.224236i	$-0.928011\pi$
$977$	−8949.91	+	15501.7i	−0.293074	+	0.507619i	0.681461	+	0.731854i	$0.261345\pi$
$978$		0			0
$979$	3455.97	+	5985.92i	0.112823	+	0.195415i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	23150.1	+	40097.2i	0.751144	+	1.30102i	0.947269	+	0.320441i	$0.103831\pi$
$983$	23150.1	+	40097.2i	0.751144	+	1.30102i	−0.196124	+	0.980579i	$0.562836\pi$
$984$		0			0
$985$	−32649.1	+	56549.9i	−1.05613	+	1.82927i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	20541.1			0.660435
$990$		0			0
$991$	−4621.74			−0.148148			−0.0740739	−	0.997253i	$-0.523600\pi$
$991$	−4621.74			−0.148148			−0.0740739	+	0.997253i	$0.523600\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3301.13	−	5717.72i	0.105179	−	0.182175i
$996$		0			0
$997$	−9972.43	−	17272.8i	−0.316780	−	0.548680i	0.663034	−	0.748589i	$-0.269268\pi$
$997$	−9972.43	−	17272.8i	−0.316780	−	0.548680i	−0.979814	+	0.199910i	$0.935935\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.4.i.b.145.1		4
3.2	odd	2		144.4.i.b.49.2		4
4.3	odd	2		54.4.c.b.37.1		4
9.2	odd	6		144.4.i.b.97.2		4
9.4	even	3		1296.4.a.r.1.2		2
9.5	odd	6		1296.4.a.l.1.1		2
9.7	even	3	inner	432.4.i.b.289.1		4
12.11	even	2		18.4.c.b.13.1	yes	4
36.7	odd	6		54.4.c.b.19.1		4
36.11	even	6		18.4.c.b.7.1	✓	4
36.23	even	6		162.4.a.f.1.1		2
36.31	odd	6		162.4.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.4.c.b.7.1	✓	4	36.11	even	6
18.4.c.b.13.1	yes	4	12.11	even	2
54.4.c.b.19.1		4	36.7	odd	6
54.4.c.b.37.1		4	4.3	odd	2
144.4.i.b.49.2		4	3.2	odd	2
144.4.i.b.97.2		4	9.2	odd	6
162.4.a.f.1.1		2	36.23	even	6
162.4.a.g.1.2		2	36.31	odd	6
432.4.i.b.145.1		4	1.1	even	1	trivial
432.4.i.b.289.1		4	9.7	even	3	inner
1296.4.a.l.1.1		2	9.5	odd	6
1296.4.a.r.1.2		2	9.4	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	432.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-9.93521 + 17.2083i) q^{5} +(-2.93521 - 5.08394i) q^{7} +(-9.37043 - 16.2301i) q^{11} +(-22.9352 + 39.7250i) q^{13} -16.8704 q^{17} +10.3521 q^{19} +(-24.9352 + 43.1891i) q^{23} +(-134.917 - 233.683i) q^{25} +(5.45351 + 9.44575i) q^{29} +(75.8056 - 131.299i) q^{31} +116.648 q^{35} +346.186 q^{37} +(132.370 - 229.272i) q^{41} +(-205.945 - 356.707i) q^{43} +(-236.028 - 408.813i) q^{47} +(154.269 - 267.202i) q^{49} -290.186 q^{53} +372.389 q^{55} +(-26.6296 + 46.1238i) q^{59} +(146.972 + 254.563i) q^{61} +(-455.732 - 789.352i) q^{65} +(-199.277 + 345.159i) q^{67} -647.854 q^{71} -478.279 q^{73} +(-55.0084 + 95.2773i) q^{77} +(187.158 + 324.167i) q^{79} +(-466.639 - 808.243i) q^{83} +(167.611 - 290.311i) q^{85} -368.817 q^{89} +269.279 q^{91} +(-102.851 + 178.143i) q^{95} +(-137.075 - 237.420i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 9 q^{5} + 19 q^{7} + 24 q^{11} - 61 q^{13} - 6 q^{17} - 266 q^{19} - 69 q^{23} - 263 q^{25} + 237 q^{29} + 211 q^{31} + 774 q^{35} + 524 q^{37} + 468 q^{41} - 86 q^{43} - 483 q^{47} + 33 q^{49} - 300 q^{53}+ \cdots + 128 q^{97}+O(q^{100}) \) 4 * q - 9 * q^5 + 19 * q^7 + 24 * q^11 - 61 * q^13 - 6 * q^17 - 266 * q^19 - 69 * q^23 - 263 * q^25 + 237 * q^29 + 211 * q^31 + 774 * q^35 + 524 * q^37 + 468 * q^41 - 86 * q^43 - 483 * q^47 + 33 * q^49 - 300 * q^53 + 1674 * q^55 - 168 * q^59 + 1049 * q^61 - 747 * q^65 - 1166 * q^67 - 624 * q^71 - 622 * q^73 - 1173 * q^77 + 349 * q^79 - 1221 * q^83 + 486 * q^85 + 984 * q^89 - 214 * q^91 - 1764 * q^95 + 128 * q^97

Introduction

L-functions

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