LMFDB - Embedded newform 441.6.a.w.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.a (trivial)

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:	yes
Analytic conductor:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 97x^{2} + 7x + 294 $ x^4 - x^3 - 97x^2 + 7x + 294
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$10.1812$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.18123 q^{2} +52.2950 q^{4} -22.0716 q^{5} +186.333 q^{8} +O(q^{10})$	$q+9.18123 q^{2} +52.2950 q^{4} -22.0716 q^{5} +186.333 q^{8} -202.644 q^{10} -416.710 q^{11} +797.918 q^{13} +37.3245 q^{16} +1375.55 q^{17} +2313.03 q^{19} -1154.23 q^{20} -3825.91 q^{22} +955.402 q^{23} -2637.84 q^{25} +7325.87 q^{26} +7035.29 q^{29} +1261.19 q^{31} -5619.96 q^{32} +12629.2 q^{34} +9776.44 q^{37} +21236.4 q^{38} -4112.66 q^{40} +5400.95 q^{41} +19686.6 q^{43} -21791.8 q^{44} +8771.76 q^{46} -2056.56 q^{47} -24218.7 q^{50} +41727.1 q^{52} -18022.7 q^{53} +9197.45 q^{55} +64592.6 q^{58} -7435.68 q^{59} +3495.38 q^{61} +11579.3 q^{62} -52792.5 q^{64} -17611.3 q^{65} +15856.4 q^{67} +71934.4 q^{68} -58133.5 q^{71} +39110.7 q^{73} +89759.8 q^{74} +120960. q^{76} +9760.69 q^{79} -823.812 q^{80} +49587.4 q^{82} +70395.7 q^{83} -30360.6 q^{85} +180747. q^{86} -77646.6 q^{88} -144306. q^{89} +49962.7 q^{92} -18881.8 q^{94} -51052.2 q^{95} -79328.7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8}+O(q^{10}) $ 4 * q - 3 * q^2 + 69 * q^4 - 123 * q^8	$ 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + 283 q^{10} - 402 q^{11} + 462 q^{13} + 3273 q^{16} - 276 q^{17} + 510 q^{19} - 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} + 15138 q^{26} - 540 q^{29} - 6410 q^{31} - 15519 q^{32} + 21144 q^{34} + 15250 q^{37} + 41250 q^{38} - 8547 q^{40} - 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} + 15060 q^{47} + 7302 q^{50} - 47476 q^{52} - 13692 q^{53} + 73124 q^{55} + 52309 q^{58} - 34830 q^{59} - 5364 q^{61} - 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} + 58272 q^{68} - 89268 q^{71} + 59638 q^{73} + 185442 q^{74} + 21308 q^{76} - 44062 q^{79} + 33381 q^{80} + 57596 q^{82} + 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} + 77520 q^{89} - 158256 q^{92} - 73722 q^{94} + 221376 q^{95} - 188630 q^{97}+O(q^{100}) $ 4 * q - 3 * q^2 + 69 * q^4 - 123 * q^8 + 283 * q^10 - 402 * q^11 + 462 * q^13 + 3273 * q^16 - 276 * q^17 + 510 * q^19 - 4719 * q^20 + 1375 * q^22 - 6900 * q^23 + 2814 * q^25 + 15138 * q^26 - 540 * q^29 - 6410 * q^31 - 15519 * q^32 + 21144 * q^34 + 15250 * q^37 + 41250 * q^38 - 8547 * q^40 - 4308 * q^41 + 29198 * q^43 - 70743 * q^44 + 61800 * q^46 + 15060 * q^47 + 7302 * q^50 - 47476 * q^52 - 13692 * q^53 + 73124 * q^55 + 52309 * q^58 - 34830 * q^59 - 5364 * q^61 - 16029 * q^62 - 73487 * q^64 - 66864 * q^65 - 5994 * q^67 + 58272 * q^68 - 89268 * q^71 + 59638 * q^73 + 185442 * q^74 + 21308 * q^76 - 44062 * q^79 + 33381 * q^80 + 57596 * q^82 + 208446 * q^83 + 36324 * q^85 + 136968 * q^86 + 87597 * q^88 + 77520 * q^89 - 158256 * q^92 - 73722 * q^94 + 221376 * q^95 - 188630 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	9.18123	1.62303	0.811514	−	0.584333i	$-0.198644\pi$
$2$	9.18123	1.62303	0.811514	+	0.584333i	$0.198644\pi$
$3$	0	0
$3$	0	0
$4$	52.2950	1.63422
$5$	−22.0716	−0.394829	−0.197414	−	0.980320i	$-0.563254\pi$
$5$	−22.0716	−0.394829	−0.197414	+	0.980320i	$0.563254\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	186.333	1.02935
$9$	0	0
$10$	−202.644	−0.640818
$11$	−416.710	−1.03837	−0.519184	−	0.854662i	$-0.673764\pi$
$11$	−416.710	−1.03837	−0.519184	+	0.854662i	$0.673764\pi$
$12$	0	0
$13$	797.918	1.30948	0.654742	−	0.755853i	$-0.272777\pi$
$13$	797.918	1.30948	0.654742	+	0.755853i	$0.272777\pi$
$14$	0	0
$15$	0	0
$16$	37.3245	0.0364497
$17$	1375.55	1.15439	0.577197	−	0.816605i	$-0.304146\pi$
$17$	1375.55	1.15439	0.577197	+	0.816605i	$0.304146\pi$
$18$	0	0
$19$	2313.03	1.46993	0.734965	−	0.678105i	$-0.237199\pi$
$19$	2313.03	1.46993	0.734965	+	0.678105i	$0.237199\pi$
$20$	−1154.23	−0.645236
$21$	0	0
$22$	−3825.91	−1.68530
$23$	955.402	0.376588	0.188294	−	0.982113i	$-0.439704\pi$
$23$	955.402	0.376588	0.188294	+	0.982113i	$0.439704\pi$
$24$	0	0
$25$	−2637.84	−0.844110
$26$	7325.87	2.12533
$27$	0	0
$28$	0	0
$29$	7035.29	1.55341	0.776707	−	0.629862i	$-0.216889\pi$
$29$	7035.29	1.55341	0.776707	+	0.629862i	$0.216889\pi$
$30$	0	0
$31$	1261.19	0.235709	0.117855	−	0.993031i	$-0.462398\pi$
$31$	1261.19	0.235709	0.117855	+	0.993031i	$0.462398\pi$
$32$	−5619.96	−0.970194
$33$	0	0
$34$	12629.2	1.87361
$35$	0	0
$36$	0	0
$37$	9776.44	1.17402	0.587012	−	0.809579i	$-0.300304\pi$
$37$	9776.44	1.17402	0.587012	+	0.809579i	$0.300304\pi$
$38$	21236.4	2.38574
$39$	0	0
$40$	−4112.66	−0.406418
$41$	5400.95	0.501777	0.250888	−	0.968016i	$-0.419277\pi$
$41$	5400.95	0.501777	0.250888	+	0.968016i	$0.419277\pi$
$42$	0	0
$43$	19686.6	1.62367	0.811837	−	0.583885i	$-0.198468\pi$
$43$	19686.6	1.62367	0.811837	+	0.583885i	$0.198468\pi$
$44$	−21791.8	−1.69692
$45$	0	0
$46$	8771.76	0.611213
$47$	−2056.56	−0.135799	−0.0678997	−	0.997692i	$-0.521630\pi$
$47$	−2056.56	−0.135799	−0.0678997	+	0.997692i	$0.521630\pi$
$48$	0	0
$49$	0	0
$50$	−24218.7	−1.37001
$51$	0	0
$52$	41727.1	2.13998
$53$	−18022.7	−0.881315	−0.440658	−	0.897675i	$-0.645255\pi$
$53$	−18022.7	−0.881315	−0.440658	+	0.897675i	$0.645255\pi$
$54$	0	0
$55$	9197.45	0.409978
$56$	0	0
$57$	0	0
$58$	64592.6	2.52123
$59$	−7435.68	−0.278093	−0.139047	−	0.990286i	$-0.544404\pi$
$59$	−7435.68	−0.278093	−0.139047	+	0.990286i	$0.544404\pi$
$60$	0	0
$61$	3495.38	0.120274	0.0601368	−	0.998190i	$-0.480846\pi$
$61$	3495.38	0.120274	0.0601368	+	0.998190i	$0.480846\pi$
$62$	11579.3	0.382563
$63$	0	0
$64$	−52792.5	−1.61110
$65$	−17611.3	−0.517022
$66$	0	0
$67$	15856.4	0.431537	0.215769	−	0.976445i	$-0.430774\pi$
$67$	15856.4	0.431537	0.215769	+	0.976445i	$0.430774\pi$
$68$	71934.4	1.88653
$69$	0	0
$70$	0	0
$71$	−58133.5	−1.36861	−0.684306	−	0.729195i	$-0.739895\pi$
$71$	−58133.5	−1.36861	−0.684306	+	0.729195i	$0.739895\pi$
$72$	0	0
$73$	39110.7	0.858990	0.429495	−	0.903069i	$-0.358692\pi$
$73$	39110.7	0.858990	0.429495	+	0.903069i	$0.358692\pi$
$74$	89759.8	1.90547
$75$	0	0
$76$	120960.	2.40218
$77$	0	0
$78$	0	0
$79$	9760.69	0.175960	0.0879798	−	0.996122i	$-0.471959\pi$
$79$	9760.69	0.175960	0.0879798	+	0.996122i	$0.471959\pi$
$80$	−823.812	−0.0143914
$81$	0	0
$82$	49587.4	0.814397
$83$	70395.7	1.12163	0.560816	−	0.827940i	$-0.310487\pi$
$83$	70395.7	1.12163	0.560816	+	0.827940i	$0.310487\pi$
$84$	0	0
$85$	−30360.6	−0.455788
$86$	180747.	2.63527
$87$	0	0
$88$	−77646.6	−1.06885
$89$	−144306.	−1.93112	−0.965562	−	0.260173i	$-0.916220\pi$
$89$	−144306.	−1.93112	−0.965562	+	0.260173i	$0.916220\pi$
$90$	0	0
$91$	0	0
$92$	49962.7	0.615427
$93$	0	0
$94$	−18881.8	−0.220406
$95$	−51052.2	−0.580371
$96$	0	0
$97$	−79328.7	−0.856053	−0.428027	−	0.903766i	$-0.640791\pi$
$97$	−79328.7	−0.856053	−0.428027	+	0.903766i	$0.640791\pi$
$98$	0	0
$99$	0	0
$100$	−137946.	−1.37946
$101$	84833.7	0.827495	0.413747	−	0.910392i	$-0.364220\pi$
$101$	84833.7	0.827495	0.413747	+	0.910392i	$0.364220\pi$
$102$	0	0
$103$	20332.3	0.188839	0.0944197	−	0.995532i	$-0.469900\pi$
$103$	20332.3	0.188839	0.0944197	+	0.995532i	$0.469900\pi$
$104$	148678.	1.34792
$105$	0	0
$106$	−165471.	−1.43040
$107$	−6962.19	−0.0587877	−0.0293938	−	0.999568i	$-0.509358\pi$
$107$	−6962.19	−0.0587877	−0.0293938	+	0.999568i	$0.509358\pi$
$108$	0	0
$109$	112651.	0.908177	0.454088	−	0.890957i	$-0.349965\pi$
$109$	112651.	0.908177	0.454088	+	0.890957i	$0.349965\pi$
$110$	84443.9	0.665406
$111$	0	0
$112$	0	0
$113$	−112005.	−0.825167	−0.412583	−	0.910920i	$-0.635373\pi$
$113$	−112005.	−0.825167	−0.412583	+	0.910920i	$0.635373\pi$
$114$	0	0
$115$	−21087.3	−0.148688
$116$	367910.	2.53862
$117$	0	0
$118$	−68268.7	−0.451353
$119$	0	0
$120$	0	0
$121$	12595.8	0.0782102
$122$	32091.9	0.195207
$123$	0	0
$124$	65954.0	0.385200
$125$	127195.	0.728108
$126$	0	0
$127$	82224.5	0.452368	0.226184	−	0.974085i	$-0.427375\pi$
$127$	82224.5	0.452368	0.226184	+	0.974085i	$0.427375\pi$
$128$	−304862.	−1.64467
$129$	0	0
$130$	−161694.	−0.839140
$131$	175812.	0.895097	0.447548	−	0.894260i	$-0.352297\pi$
$131$	175812.	0.895097	0.447548	+	0.894260i	$0.352297\pi$
$132$	0	0
$133$	0	0
$134$	145581.	0.700397
$135$	0	0
$136$	256310.	1.18828
$137$	−31330.3	−0.142614	−0.0713072	−	0.997454i	$-0.522717\pi$
$137$	−31330.3	−0.142614	−0.0713072	+	0.997454i	$0.522717\pi$
$138$	0	0
$139$	152234.	0.668305	0.334152	−	0.942519i	$-0.391550\pi$
$139$	152234.	0.668305	0.334152	+	0.942519i	$0.391550\pi$
$140$	0	0
$141$	0	0
$142$	−533737.	−2.22129
$143$	−332500.	−1.35973
$144$	0	0
$145$	−155280.	−0.613332
$146$	359084.	1.39416
$147$	0	0
$148$	511259.	1.91861
$149$	−362860.	−1.33898	−0.669489	−	0.742822i	$-0.733487\pi$
$149$	−362860.	−1.33898	−0.669489	+	0.742822i	$0.733487\pi$
$150$	0	0
$151$	205126.	0.732112	0.366056	−	0.930593i	$-0.380708\pi$
$151$	205126.	0.732112	0.366056	+	0.930593i	$0.380708\pi$
$152$	430992.	1.51308
$153$	0	0
$154$	0	0
$155$	−27836.5	−0.0930648
$156$	0	0
$157$	−77272.0	−0.250192	−0.125096	−	0.992145i	$-0.539924\pi$
$157$	−77272.0	−0.250192	−0.125096	+	0.992145i	$0.539924\pi$
$158$	89615.1	0.285587
$159$	0	0
$160$	124042.	0.383060
$161$	0	0
$162$	0	0
$163$	184931.	0.545182	0.272591	−	0.962130i	$-0.412119\pi$
$163$	184931.	0.545182	0.272591	+	0.962130i	$0.412119\pi$
$164$	282442.	0.820012
$165$	0	0
$166$	646319.	1.82044
$167$	−129262.	−0.358657	−0.179329	−	0.983789i	$-0.557393\pi$
$167$	−129262.	−0.358657	−0.179329	+	0.983789i	$0.557393\pi$
$168$	0	0
$169$	265380.	0.714746
$170$	−278748.	−0.739757
$171$	0	0
$172$	1.02951e6	2.65344
$173$	507867.	1.29013	0.645067	−	0.764126i	$-0.276829\pi$
$173$	507867.	1.29013	0.645067	+	0.764126i	$0.276829\pi$
$174$	0	0
$175$	0	0
$176$	−15553.5	−0.0378483
$177$	0	0
$178$	−1.32491e6	−3.13427
$179$	132589.	0.309296	0.154648	−	0.987970i	$-0.450576\pi$
$179$	132589.	0.309296	0.154648	+	0.987970i	$0.450576\pi$
$180$	0	0
$181$	−740060.	−1.67908	−0.839538	−	0.543301i	$-0.817174\pi$
$181$	−740060.	−1.67908	−0.839538	+	0.543301i	$0.817174\pi$
$182$	0	0
$183$	0	0
$184$	178023.	0.387642
$185$	−215782.	−0.463538
$186$	0	0
$187$	−573205.	−1.19869
$188$	−107548.	−0.221926
$189$	0	0
$190$	−468722.	−0.941957
$191$	582732.	1.15581	0.577904	−	0.816105i	$-0.303871\pi$
$191$	582732.	1.15581	0.577904	+	0.816105i	$0.303871\pi$
$192$	0	0
$193$	−400904.	−0.774725	−0.387362	−	0.921928i	$-0.626614\pi$
$193$	−400904.	−0.774725	−0.387362	+	0.921928i	$0.626614\pi$
$194$	−728335.	−1.38940
$195$	0	0
$196$	0	0
$197$	−671589.	−1.23293	−0.616464	−	0.787383i	$-0.711436\pi$
$197$	−671589.	−1.23293	−0.616464	+	0.787383i	$0.711436\pi$
$198$	0	0
$199$	−455022.	−0.814515	−0.407258	−	0.913313i	$-0.633515\pi$
$199$	−455022.	−0.814515	−0.407258	+	0.913313i	$0.633515\pi$
$200$	−491517.	−0.868887
$201$	0	0
$202$	778878.	1.34305
$203$	0	0
$204$	0	0
$205$	−119208.	−0.198116
$206$	186675.	0.306492
$207$	0	0
$208$	29781.9	0.0477303
$209$	−963860.	−1.52633
$210$	0	0
$211$	−1.19545e6	−1.84852	−0.924260	−	0.381764i	$-0.875317\pi$
$211$	−1.19545e6	−1.84852	−0.924260	+	0.381764i	$0.875317\pi$
$212$	−942499.	−1.44026
$213$	0	0
$214$	−63921.4	−0.0954140
$215$	−434514.	−0.641073
$216$	0	0
$217$	0	0
$218$	1.03428e6	1.47400
$219$	0	0
$220$	480980.	0.669993
$221$	1.09758e6	1.51166
$222$	0	0
$223$	296529.	0.399305	0.199653	−	0.979867i	$-0.436019\pi$
$223$	296529.	0.399305	0.199653	+	0.979867i	$0.436019\pi$
$224$	0	0
$225$	0	0
$226$	−1.02834e6	−1.33927
$227$	−218146.	−0.280985	−0.140492	−	0.990082i	$-0.544869\pi$
$227$	−218146.	−0.280985	−0.140492	+	0.990082i	$0.544869\pi$
$228$	0	0
$229$	−1.22920e6	−1.54894	−0.774471	−	0.632609i	$-0.781984\pi$
$229$	−1.22920e6	−1.54894	−0.774471	+	0.632609i	$0.781984\pi$
$230$	−193607.	−0.241324
$231$	0	0
$232$	1.31090e6	1.59901
$233$	62944.2	0.0759567	0.0379784	−	0.999279i	$-0.487908\pi$
$233$	62944.2	0.0759567	0.0379784	+	0.999279i	$0.487908\pi$
$234$	0	0
$235$	45391.7	0.0536175
$236$	−388849.	−0.454465
$237$	0	0
$238$	0	0
$239$	−219330.	−0.248372	−0.124186	−	0.992259i	$-0.539632\pi$
$239$	−219330.	−0.248372	−0.124186	+	0.992259i	$0.539632\pi$
$240$	0	0
$241$	433864.	0.481183	0.240592	−	0.970626i	$-0.422659\pi$
$241$	433864.	0.481183	0.240592	+	0.970626i	$0.422659\pi$
$242$	115645.	0.126937
$243$	0	0
$244$	182791.	0.196553
$245$	0	0
$246$	0	0
$247$	1.84561e6	1.92485
$248$	235001.	0.242628
$249$	0	0
$250$	1.16781e6	1.18174
$251$	1.71109e6	1.71431	0.857155	−	0.515059i	$-0.172230\pi$
$251$	1.71109e6	1.71431	0.857155	+	0.515059i	$0.172230\pi$
$252$	0	0
$253$	−398125.	−0.391037
$254$	754922.	0.734206
$255$	0	0
$256$	−1.10964e6	−1.05824
$257$	−984057.	−0.929368	−0.464684	−	0.885477i	$-0.653832\pi$
$257$	−984057.	−0.929368	−0.464684	+	0.885477i	$0.653832\pi$
$258$	0	0
$259$	0	0
$260$	−920984.	−0.844926
$261$	0	0
$262$	1.61417e6	1.45277
$263$	−547095.	−0.487724	−0.243862	−	0.969810i	$-0.578414\pi$
$263$	−547095.	−0.487724	−0.243862	+	0.969810i	$0.578414\pi$
$264$	0	0
$265$	397791.	0.347969
$266$	0	0
$267$	0	0
$268$	829211.	0.705226
$269$	641112.	0.540198	0.270099	−	0.962833i	$-0.412944\pi$
$269$	641112.	0.540198	0.270099	+	0.962833i	$0.412944\pi$
$270$	0	0
$271$	−548012.	−0.453280	−0.226640	−	0.973979i	$-0.572774\pi$
$271$	−548012.	−0.453280	−0.226640	+	0.973979i	$0.572774\pi$
$272$	51341.8	0.0420774
$273$	0	0
$274$	−287651.	−0.231467
$275$	1.09921e6	0.876498
$276$	0	0
$277$	1.67414e6	1.31097	0.655485	−	0.755208i	$-0.272464\pi$
$277$	1.67414e6	1.31097	0.655485	+	0.755208i	$0.272464\pi$
$278$	1.39769e6	1.08468
$279$	0	0
$280$	0	0
$281$	−1.81078e6	−1.36804	−0.684021	−	0.729462i	$-0.739770\pi$
$281$	−1.81078e6	−1.36804	−0.684021	+	0.729462i	$0.739770\pi$
$282$	0	0
$283$	2.51315e6	1.86531	0.932657	−	0.360764i	$-0.117484\pi$
$283$	2.51315e6	1.86531	0.932657	+	0.360764i	$0.117484\pi$
$284$	−3.04009e6	−2.23661
$285$	0	0
$286$	−3.05276e6	−2.20687
$287$	0	0
$288$	0	0
$289$	472283.	0.332627
$290$	−1.42566e6	−0.995455
$291$	0	0
$292$	2.04529e6	1.40378
$293$	−107228.	−0.0729691	−0.0364845	−	0.999334i	$-0.511616\pi$
$293$	−107228.	−0.0729691	−0.0364845	+	0.999334i	$0.511616\pi$
$294$	0	0
$295$	164117.	0.109799
$296$	1.82167e6	1.20848
$297$	0	0
$298$	−3.33150e6	−2.17320
$299$	762332.	0.493136
$300$	0	0
$301$	0	0
$302$	1.88330e6	1.18824
$303$	0	0
$304$	86332.6	0.0535785
$305$	−77148.7	−0.0474875
$306$	0	0
$307$	1.49622e6	0.906042	0.453021	−	0.891500i	$-0.350346\pi$
$307$	1.49622e6	0.906042	0.453021	+	0.891500i	$0.350346\pi$
$308$	0	0
$309$	0	0
$310$	−255573.	−0.151047
$311$	−861202.	−0.504899	−0.252449	−	0.967610i	$-0.581236\pi$
$311$	−861202.	−0.504899	−0.252449	+	0.967610i	$0.581236\pi$
$312$	0	0
$313$	−503937.	−0.290747	−0.145374	−	0.989377i	$-0.546438\pi$
$313$	−503937.	−0.290747	−0.145374	+	0.989377i	$0.546438\pi$
$314$	−709452.	−0.406068
$315$	0	0
$316$	510435.	0.287556
$317$	480009.	0.268288	0.134144	−	0.990962i	$-0.457172\pi$
$317$	480009.	0.268288	0.134144	+	0.990962i	$0.457172\pi$
$318$	0	0
$319$	−2.93167e6	−1.61302
$320$	1.16522e6	0.636109
$321$	0	0
$322$	0	0
$323$	3.18168e6	1.69688
$324$	0	0
$325$	−2.10478e6	−1.10535
$326$	1.69790e6	0.884845
$327$	0	0
$328$	1.00637e6	0.516505
$329$	0	0
$330$	0	0
$331$	−2.19922e6	−1.10331	−0.551656	−	0.834072i	$-0.686004\pi$
$331$	−2.19922e6	−1.10331	−0.551656	+	0.834072i	$0.686004\pi$
$332$	3.68134e6	1.83299
$333$	0	0
$334$	−1.18678e6	−0.582110
$335$	−349977.	−0.170383
$336$	0	0
$337$	−1.35725e6	−0.651008	−0.325504	−	0.945541i	$-0.605534\pi$
$337$	−1.35725e6	−0.651008	−0.325504	+	0.945541i	$0.605534\pi$
$338$	2.43652e6	1.16005
$339$	0	0
$340$	−1.58771e6	−0.744857
$341$	−525550.	−0.244753
$342$	0	0
$343$	0	0
$344$	3.66825e6	1.67133
$345$	0	0
$346$	4.66285e6	2.09392
$347$	−3.79941e6	−1.69392	−0.846959	−	0.531659i	$-0.821569\pi$
$347$	−3.79941e6	−1.69392	−0.846959	+	0.531659i	$0.821569\pi$
$348$	0	0
$349$	−1.31753e6	−0.579024	−0.289512	−	0.957174i	$-0.593493\pi$
$349$	−1.31753e6	−0.579024	−0.289512	+	0.957174i	$0.593493\pi$
$350$	0	0
$351$	0	0
$352$	2.34189e6	1.00742
$353$	3.42505e6	1.46295	0.731477	−	0.681866i	$-0.238832\pi$
$353$	3.42505e6	1.46295	0.731477	+	0.681866i	$0.238832\pi$
$354$	0	0
$355$	1.28310e6	0.540367
$356$	−7.54649e6	−3.15588
$357$	0	0
$358$	1.21733e6	0.501997
$359$	−1.47084e6	−0.602322	−0.301161	−	0.953573i	$-0.597374\pi$
$359$	−1.47084e6	−0.602322	−0.301161	+	0.953573i	$0.597374\pi$
$360$	0	0
$361$	2.87399e6	1.16069
$362$	−6.79466e6	−2.72519
$363$	0	0
$364$	0	0
$365$	−863235.	−0.339154
$366$	0	0
$367$	−4.98079e6	−1.93034	−0.965168	−	0.261630i	$-0.915740\pi$
$367$	−4.98079e6	−1.93034	−0.965168	+	0.261630i	$0.915740\pi$
$368$	35659.9	0.0137265
$369$	0	0
$370$	−1.98114e6	−0.752335
$371$	0	0
$372$	0	0
$373$	3.96047e6	1.47392	0.736962	−	0.675934i	$-0.236260\pi$
$373$	3.96047e6	1.47392	0.736962	+	0.675934i	$0.236260\pi$
$374$	−5.26273e6	−1.94550
$375$	0	0
$376$	−383205.	−0.139785
$377$	5.61359e6	2.03417
$378$	0	0
$379$	−1.75155e6	−0.626359	−0.313179	−	0.949694i	$-0.601394\pi$
$379$	−1.75155e6	−0.626359	−0.313179	+	0.949694i	$0.601394\pi$
$380$	−2.66977e6	−0.948452
$381$	0	0
$382$	5.35020e6	1.87591
$383$	−3.13834e6	−1.09321	−0.546604	−	0.837391i	$-0.684080\pi$
$383$	−3.13834e6	−1.09321	−0.546604	+	0.837391i	$0.684080\pi$
$384$	0	0
$385$	0	0
$386$	−3.68079e6	−1.25740
$387$	0	0
$388$	−4.14849e6	−1.39898
$389$	1.05252e6	0.352661	0.176331	−	0.984331i	$-0.443577\pi$
$389$	1.05252e6	0.352661	0.176331	+	0.984331i	$0.443577\pi$
$390$	0	0
$391$	1.31420e6	0.434731
$392$	0	0
$393$	0	0
$394$	−6.16601e6	−2.00108
$395$	−215434.	−0.0694739
$396$	0	0
$397$	454724.	0.144801	0.0724005	−	0.997376i	$-0.476934\pi$
$397$	454724.	0.144801	0.0724005	+	0.997376i	$0.476934\pi$
$398$	−4.17766e6	−1.32198
$399$	0	0
$400$	−98456.3	−0.0307676
$401$	2.88431e6	0.895739	0.447870	−	0.894099i	$-0.352183\pi$
$401$	2.88431e6	0.895739	0.447870	+	0.894099i	$0.352183\pi$
$402$	0	0
$403$	1.00633e6	0.308657
$404$	4.43638e6	1.35231
$405$	0	0
$406$	0	0
$407$	−4.07394e6	−1.21907
$408$	0	0
$409$	−225914.	−0.0667782	−0.0333891	−	0.999442i	$-0.510630\pi$
$409$	−225914.	−0.0667782	−0.0333891	+	0.999442i	$0.510630\pi$
$410$	−1.09447e6	−0.321548
$411$	0	0
$412$	1.06328e6	0.308605
$413$	0	0
$414$	0	0
$415$	−1.55375e6	−0.442853
$416$	−4.48427e6	−1.27045
$417$	0	0
$418$	−8.84942e6	−2.47727
$419$	−4.31027e6	−1.19941	−0.599707	−	0.800220i	$-0.704716\pi$
$419$	−4.31027e6	−1.19941	−0.599707	+	0.800220i	$0.704716\pi$
$420$	0	0
$421$	1.25088e6	0.343962	0.171981	−	0.985100i	$-0.444983\pi$
$421$	1.25088e6	0.343962	0.171981	+	0.985100i	$0.444983\pi$
$422$	−1.09757e7	−3.00020
$423$	0	0
$424$	−3.35823e6	−0.907184
$425$	−3.62849e6	−0.974436
$426$	0	0
$427$	0	0
$428$	−364087.	−0.0960719
$429$	0	0
$430$	−3.98937e6	−1.04048
$431$	4.40793e6	1.14299	0.571494	−	0.820606i	$-0.306364\pi$
$431$	4.40793e6	1.14299	0.571494	+	0.820606i	$0.306364\pi$
$432$	0	0
$433$	−1.60951e6	−0.412549	−0.206274	−	0.978494i	$-0.566134\pi$
$433$	−1.60951e6	−0.412549	−0.206274	+	0.978494i	$0.566134\pi$
$434$	0	0
$435$	0	0
$436$	5.89110e6	1.48416
$437$	2.20987e6	0.553558
$438$	0	0
$439$	−4.42452e6	−1.09573	−0.547867	−	0.836566i	$-0.684560\pi$
$439$	−4.42452e6	−1.09573	−0.547867	+	0.836566i	$0.684560\pi$
$440$	1.71379e6	0.422012
$441$	0	0
$442$	1.00771e7	2.45347
$443$	3.61438e6	0.875034	0.437517	−	0.899210i	$-0.355858\pi$
$443$	3.61438e6	0.875034	0.437517	+	0.899210i	$0.355858\pi$
$444$	0	0
$445$	3.18507e6	0.762464
$446$	2.72250e6	0.648084
$447$	0	0
$448$	0	0
$449$	467024.	0.109326	0.0546630	−	0.998505i	$-0.482592\pi$
$449$	467024.	0.109326	0.0546630	+	0.998505i	$0.482592\pi$
$450$	0	0
$451$	−2.25063e6	−0.521029
$452$	−5.85730e6	−1.34850
$453$	0	0
$454$	−2.00285e6	−0.456046
$455$	0	0
$456$	0	0
$457$	−601252.	−0.134668	−0.0673342	−	0.997730i	$-0.521449\pi$
$457$	−601252.	−0.134668	−0.0673342	+	0.997730i	$0.521449\pi$
$458$	−1.12856e7	−2.51398
$459$	0	0
$460$	−1.10276e6	−0.242988
$461$	−2.87193e6	−0.629392	−0.314696	−	0.949192i	$-0.601903\pi$
$461$	−2.87193e6	−0.629392	−0.314696	+	0.949192i	$0.601903\pi$
$462$	0	0
$463$	−2.91502e6	−0.631959	−0.315979	−	0.948766i	$-0.602333\pi$
$463$	−2.91502e6	−0.631959	−0.315979	+	0.948766i	$0.602333\pi$
$464$	262589.	0.0566215
$465$	0	0
$466$	577906.	0.123280
$467$	7.19376e6	1.52638	0.763192	−	0.646172i	$-0.223631\pi$
$467$	7.19376e6	1.52638	0.763192	+	0.646172i	$0.223631\pi$
$468$	0	0
$469$	0	0
$470$	416751.	0.0870227
$471$	0	0
$472$	−1.38551e6	−0.286256
$473$	−8.20358e6	−1.68597
$474$	0	0
$475$	−6.10140e6	−1.24078
$476$	0	0
$477$	0	0
$478$	−2.01372e6	−0.403115
$479$	2.79649e6	0.556896	0.278448	−	0.960451i	$-0.410180\pi$
$479$	2.79649e6	0.556896	0.278448	+	0.960451i	$0.410180\pi$
$480$	0	0
$481$	7.80080e6	1.53736
$482$	3.98340e6	0.780974
$483$	0	0
$484$	658698.	0.127812
$485$	1.75091e6	0.337995
$486$	0	0
$487$	−2.93247e6	−0.560289	−0.280144	−	0.959958i	$-0.590382\pi$
$487$	−2.93247e6	−0.560289	−0.280144	+	0.959958i	$0.590382\pi$
$488$	651305.	0.123804
$489$	0	0
$490$	0	0
$491$	3.06121e6	0.573046	0.286523	−	0.958073i	$-0.407501\pi$
$491$	3.06121e6	0.573046	0.286523	+	0.958073i	$0.407501\pi$
$492$	0	0
$493$	9.67740e6	1.79325
$494$	1.69449e7	3.12408
$495$	0	0
$496$	47073.4	0.00859154
$497$	0	0
$498$	0	0
$499$	−6.55154e6	−1.17786	−0.588928	−	0.808186i	$-0.700450\pi$
$499$	−6.55154e6	−1.17786	−0.588928	+	0.808186i	$0.700450\pi$
$500$	6.65167e6	1.18989
$501$	0	0
$502$	1.57099e7	2.78237
$503$	1.58524e6	0.279367	0.139684	−	0.990196i	$-0.455391\pi$
$503$	1.58524e6	0.279367	0.139684	+	0.990196i	$0.455391\pi$
$504$	0	0
$505$	−1.87242e6	−0.326719
$506$	−3.65528e6	−0.634664
$507$	0	0
$508$	4.29993e6	0.739268
$509$	−6.77981e6	−1.15991	−0.579953	−	0.814650i	$-0.696929\pi$
$509$	−6.77981e6	−1.15991	−0.579953	+	0.814650i	$0.696929\pi$
$510$	0	0
$511$	0	0
$512$	−432315.	−0.0728829
$513$	0	0
$514$	−9.03486e6	−1.50839
$515$	−448766.	−0.0745593
$516$	0	0
$517$	856990.	0.141010
$518$	0	0
$519$	0	0
$520$	−3.28157e6	−0.532198
$521$	1.01384e7	1.63634	0.818170	−	0.574977i	$-0.194989\pi$
$521$	1.01384e7	1.63634	0.818170	+	0.574977i	$0.194989\pi$
$522$	0	0
$523$	99997.3	0.0159858	0.00799289	−	0.999968i	$-0.497456\pi$
$523$	99997.3	0.0159858	0.00799289	+	0.999968i	$0.497456\pi$
$524$	9.19408e6	1.46278
$525$	0	0
$526$	−5.02301e6	−0.791589
$527$	1.73483e6	0.272102
$528$	0	0
$529$	−5.52355e6	−0.858181
$530$	3.65221e6	0.564763
$531$	0	0
$532$	0	0
$533$	4.30952e6	0.657068
$534$	0	0
$535$	153667.	0.0232111
$536$	2.95457e6	0.444204
$537$	0	0
$538$	5.88620e6	0.876756
$539$	0	0
$540$	0	0
$541$	−119743.	−0.0175896	−0.00879481	−	0.999961i	$-0.502800\pi$
$541$	−119743.	−0.0175896	−0.00879481	+	0.999961i	$0.502800\pi$
$542$	−5.03143e6	−0.735687
$543$	0	0
$544$	−7.73054e6	−1.11999
$545$	−2.48640e6	−0.358574
$546$	0	0
$547$	236568.	0.0338056	0.0169028	−	0.999857i	$-0.494619\pi$
$547$	236568.	0.0338056	0.0169028	+	0.999857i	$0.494619\pi$
$548$	−1.63842e6	−0.233063
$549$	0	0
$550$	1.00921e7	1.42258
$551$	1.62728e7	2.28341
$552$	0	0
$553$	0	0
$554$	1.53707e7	2.12774
$555$	0	0
$556$	7.96107e6	1.09216
$557$	4.83666e6	0.660553	0.330277	−	0.943884i	$-0.392858\pi$
$557$	4.83666e6	0.660553	0.330277	+	0.943884i	$0.392858\pi$
$558$	0	0
$559$	1.57083e7	2.12617
$560$	0	0
$561$	0	0
$562$	−1.66252e7	−2.22037
$563$	5.48295e6	0.729026	0.364513	−	0.931198i	$-0.381235\pi$
$563$	5.48295e6	0.729026	0.364513	+	0.931198i	$0.381235\pi$
$564$	0	0
$565$	2.47213e6	0.325800
$566$	2.30738e7	3.02746
$567$	0	0
$568$	−1.08322e7	−1.40878
$569$	6.97660e6	0.903364	0.451682	−	0.892179i	$-0.350824\pi$
$569$	6.97660e6	0.903364	0.451682	+	0.892179i	$0.350824\pi$
$570$	0	0
$571$	−1.17715e7	−1.51092	−0.755458	−	0.655197i	$-0.772585\pi$
$571$	−1.17715e7	−1.51092	−0.755458	+	0.655197i	$0.772585\pi$
$572$	−1.73881e7	−2.22209
$573$	0	0
$574$	0	0
$575$	−2.52020e6	−0.317882
$576$	0	0
$577$	6.77292e6	0.846909	0.423454	−	0.905917i	$-0.360817\pi$
$577$	6.77292e6	0.846909	0.423454	+	0.905917i	$0.360817\pi$
$578$	4.33614e6	0.539863
$579$	0	0
$580$	−8.12037e6	−1.00232
$581$	0	0
$582$	0	0
$583$	7.51025e6	0.915130
$584$	7.28760e6	0.884203
$585$	0	0
$586$	−984484.	−0.118431
$587$	1.05020e7	1.25799	0.628996	−	0.777408i	$-0.283466\pi$
$587$	1.05020e7	1.25799	0.628996	+	0.777408i	$0.283466\pi$
$588$	0	0
$589$	2.91717e6	0.346476
$590$	1.50680e6	0.178207
$591$	0	0
$592$	364901.	0.0427928
$593$	7.59074e6	0.886436	0.443218	−	0.896414i	$-0.353837\pi$
$593$	7.59074e6	0.886436	0.443218	+	0.896414i	$0.353837\pi$
$594$	0	0
$595$	0	0
$596$	−1.89758e7	−2.18818
$597$	0	0
$598$	6.99915e6	0.800373
$599$	1.30198e7	1.48265	0.741325	−	0.671146i	$-0.234198\pi$
$599$	1.30198e7	1.48265	0.741325	+	0.671146i	$0.234198\pi$
$600$	0	0
$601$	1.41821e7	1.60160	0.800801	−	0.598931i	$-0.204408\pi$
$601$	1.41821e7	1.60160	0.800801	+	0.598931i	$0.204408\pi$
$602$	0	0
$603$	0	0
$604$	1.07270e7	1.19643
$605$	−278010.	−0.0308796
$606$	0	0
$607$	1.20772e7	1.33044	0.665221	−	0.746646i	$-0.268337\pi$
$607$	1.20772e7	1.33044	0.665221	+	0.746646i	$0.268337\pi$
$608$	−1.29991e7	−1.42612
$609$	0	0
$610$	−708320.	−0.0770735
$611$	−1.64097e6	−0.177827
$612$	0	0
$613$	−2.72530e6	−0.292929	−0.146465	−	0.989216i	$-0.546789\pi$
$613$	−2.72530e6	−0.292929	−0.146465	+	0.989216i	$0.546789\pi$
$614$	1.37371e7	1.47053
$615$	0	0
$616$	0	0
$617$	−8.47094e6	−0.895816	−0.447908	−	0.894080i	$-0.647831\pi$
$617$	−8.47094e6	−0.895816	−0.447908	+	0.894080i	$0.647831\pi$
$618$	0	0
$619$	1.73835e7	1.82352	0.911759	−	0.410726i	$-0.134725\pi$
$619$	1.73835e7	1.82352	0.911759	+	0.410726i	$0.134725\pi$
$620$	−1.45571e6	−0.152088
$621$	0	0
$622$	−7.90690e6	−0.819464
$623$	0	0
$624$	0	0
$625$	5.43586e6	0.556632
$626$	−4.62676e6	−0.471891
$627$	0	0
$628$	−4.04093e6	−0.408868
$629$	1.34480e7	1.35529
$630$	0	0
$631$	−6.45149e6	−0.645040	−0.322520	−	0.946563i	$-0.604530\pi$
$631$	−6.45149e6	−0.645040	−0.322520	+	0.946563i	$0.604530\pi$
$632$	1.81874e6	0.181124
$633$	0	0
$634$	4.40707e6	0.435438
$635$	−1.81483e6	−0.178608
$636$	0	0
$637$	0	0
$638$	−2.69164e7	−2.61797
$639$	0	0
$640$	6.72879e6	0.649362
$641$	−1.77716e7	−1.70837	−0.854185	−	0.519969i	$-0.825943\pi$
$641$	−1.77716e7	−1.70837	−0.854185	+	0.519969i	$0.825943\pi$
$642$	0	0
$643$	−9.34806e6	−0.891649	−0.445825	−	0.895120i	$-0.647090\pi$
$643$	−9.34806e6	−0.891649	−0.445825	+	0.895120i	$0.647090\pi$
$644$	0	0
$645$	0	0
$646$	2.92118e7	2.75408
$647$	−5.34386e6	−0.501874	−0.250937	−	0.968003i	$-0.580739\pi$
$647$	−5.34386e6	−0.501874	−0.250937	+	0.968003i	$0.580739\pi$
$648$	0	0
$649$	3.09852e6	0.288764
$650$	−1.93245e7	−1.79401
$651$	0	0
$652$	9.67098e6	0.890946
$653$	1.19701e6	0.109854	0.0549270	−	0.998490i	$-0.482507\pi$
$653$	1.19701e6	0.109854	0.0549270	+	0.998490i	$0.482507\pi$
$654$	0	0
$655$	−3.88045e6	−0.353410
$656$	201588.	0.0182896
$657$	0	0
$658$	0	0
$659$	1.17541e7	1.05433	0.527163	−	0.849764i	$-0.323256\pi$
$659$	1.17541e7	1.05433	0.527163	+	0.849764i	$0.323256\pi$
$660$	0	0
$661$	−1.80115e7	−1.60341	−0.801706	−	0.597719i	$-0.796074\pi$
$661$	−1.80115e7	−1.60341	−0.801706	+	0.597719i	$0.796074\pi$
$662$	−2.01915e7	−1.79071
$663$	0	0
$664$	1.31170e7	1.15456
$665$	0	0
$666$	0	0
$667$	6.72153e6	0.584997
$668$	−6.75975e6	−0.586124
$669$	0	0
$670$	−3.21322e6	−0.276537
$671$	−1.45656e6	−0.124888
$672$	0	0
$673$	1.40977e7	1.19981	0.599904	−	0.800072i	$-0.295206\pi$
$673$	1.40977e7	1.19981	0.599904	+	0.800072i	$0.295206\pi$
$674$	−1.24613e7	−1.05660
$675$	0	0
$676$	1.38780e7	1.16805
$677$	−7.65587e6	−0.641982	−0.320991	−	0.947082i	$-0.604016\pi$
$677$	−7.65587e6	−0.641982	−0.320991	+	0.947082i	$0.604016\pi$
$678$	0	0
$679$	0	0
$680$	−5.65718e6	−0.469167
$681$	0	0
$682$	−4.82520e6	−0.397241
$683$	1.08676e7	0.891415	0.445708	−	0.895179i	$-0.352952\pi$
$683$	1.08676e7	0.891415	0.445708	+	0.895179i	$0.352952\pi$
$684$	0	0
$685$	691510.	0.0563083
$686$	0	0
$687$	0	0
$688$	734791.	0.0591825
$689$	−1.43807e7	−1.15407
$690$	0	0
$691$	−1.46118e7	−1.16415	−0.582073	−	0.813136i	$-0.697758\pi$
$691$	−1.46118e7	−1.16415	−0.582073	+	0.813136i	$0.697758\pi$
$692$	2.65589e7	2.10836
$693$	0	0
$694$	−3.48832e7	−2.74927
$695$	−3.36005e6	−0.263866
$696$	0	0
$697$	7.42928e6	0.579248
$698$	−1.20965e7	−0.939772
$699$	0	0
$700$	0	0
$701$	−1.90104e7	−1.46115	−0.730577	−	0.682830i	$-0.760749\pi$
$701$	−1.90104e7	−1.46115	−0.730577	+	0.682830i	$0.760749\pi$
$702$	0	0
$703$	2.26132e7	1.72573
$704$	2.19992e7	1.67292
$705$	0	0
$706$	3.14462e7	2.37441
$707$	0	0
$708$	0	0
$709$	−1.15929e7	−0.866116	−0.433058	−	0.901366i	$-0.642565\pi$
$709$	−1.15929e7	−0.866116	−0.433058	+	0.901366i	$0.642565\pi$
$710$	1.17804e7	0.877031
$711$	0	0
$712$	−2.68890e7	−1.98781
$713$	1.20494e6	0.0887653
$714$	0	0
$715$	7.33881e6	0.536859
$716$	6.93374e6	0.505458
$717$	0	0
$718$	−1.35041e7	−0.977585
$719$	1.02861e7	0.742040	0.371020	−	0.928625i	$-0.379008\pi$
$719$	1.02861e7	0.742040	0.371020	+	0.928625i	$0.379008\pi$
$720$	0	0
$721$	0	0
$722$	2.63868e7	1.88384
$723$	0	0
$724$	−3.87014e7	−2.74398
$725$	−1.85580e7	−1.31125
$726$	0	0
$727$	1.00970e7	0.708526	0.354263	−	0.935146i	$-0.384732\pi$
$727$	1.00970e7	0.708526	0.354263	+	0.935146i	$0.384732\pi$
$728$	0	0
$729$	0	0
$730$	−7.92556e6	−0.550456
$731$	2.70799e7	1.87436
$732$	0	0
$733$	1.92402e6	0.132266	0.0661332	−	0.997811i	$-0.478934\pi$
$733$	1.92402e6	0.132266	0.0661332	+	0.997811i	$0.478934\pi$
$734$	−4.57298e7	−3.13299
$735$	0	0
$736$	−5.36932e6	−0.365363
$737$	−6.60752e6	−0.448095
$738$	0	0
$739$	4.20273e6	0.283087	0.141543	−	0.989932i	$-0.454793\pi$
$739$	4.20273e6	0.283087	0.141543	+	0.989932i	$0.454793\pi$
$740$	−1.12843e7	−0.757522
$741$	0	0
$742$	0	0
$743$	1.99659e7	1.32684	0.663418	−	0.748249i	$-0.269105\pi$
$743$	1.99659e7	1.32684	0.663418	+	0.748249i	$0.269105\pi$
$744$	0	0
$745$	8.00891e6	0.528667
$746$	3.63620e7	2.39222
$747$	0	0
$748$	−2.99757e7	−1.95892
$749$	0	0
$750$	0	0
$751$	−3.63975e6	−0.235490	−0.117745	−	0.993044i	$-0.537567\pi$
$751$	−3.63975e6	−0.235490	−0.117745	+	0.993044i	$0.537567\pi$
$752$	−76760.3	−0.00494985
$753$	0	0
$754$	5.15396e7	3.30151
$755$	−4.52745e6	−0.289059
$756$	0	0
$757$	1.73429e7	1.09997	0.549986	−	0.835174i	$-0.314633\pi$
$757$	1.73429e7	1.09997	0.549986	+	0.835174i	$0.314633\pi$
$758$	−1.60813e7	−1.01660
$759$	0	0
$760$	−9.51270e6	−0.597406
$761$	1.03568e7	0.648284	0.324142	−	0.946009i	$-0.394924\pi$
$761$	1.03568e7	0.648284	0.324142	+	0.946009i	$0.394924\pi$
$762$	0	0
$763$	0	0
$764$	3.04740e7	1.88884
$765$	0	0
$766$	−2.88138e7	−1.77431
$767$	−5.93306e6	−0.364159
$768$	0	0
$769$	−1.81548e7	−1.10707	−0.553534	−	0.832826i	$-0.686721\pi$
$769$	−1.81548e7	−1.10707	−0.553534	+	0.832826i	$0.686721\pi$
$770$	0	0
$771$	0	0
$772$	−2.09653e7	−1.26607
$773$	1.63566e7	0.984565	0.492282	−	0.870435i	$-0.336163\pi$
$773$	1.63566e7	0.984565	0.492282	+	0.870435i	$0.336163\pi$
$774$	0	0
$775$	−3.32683e6	−0.198965
$776$	−1.47815e7	−0.881181
$777$	0	0
$778$	9.66346e6	0.572379
$779$	1.24925e7	0.737576
$780$	0	0
$781$	2.42248e7	1.42112
$782$	1.20660e7	0.705581
$783$	0	0
$784$	0	0
$785$	1.70552e6	0.0987829
$786$	0	0
$787$	−4.44728e6	−0.255952	−0.127976	−	0.991777i	$-0.540848\pi$
$787$	−4.44728e6	−0.255952	−0.127976	+	0.991777i	$0.540848\pi$
$788$	−3.51207e7	−2.01487
$789$	0	0
$790$	−1.97795e6	−0.112758
$791$	0	0
$792$	0	0
$793$	2.78903e6	0.157496
$794$	4.17493e6	0.235016
$795$	0	0
$796$	−2.37953e7	−1.33110
$797$	9.15303e6	0.510410	0.255205	−	0.966887i	$-0.417857\pi$
$797$	9.15303e6	0.510410	0.255205	+	0.966887i	$0.417857\pi$
$798$	0	0
$799$	−2.82891e6	−0.156766
$800$	1.48246e7	0.818950
$801$	0	0
$802$	2.64815e7	1.45381
$803$	−1.62978e7	−0.891948
$804$	0	0
$805$	0	0
$806$	9.23932e6	0.500959
$807$	0	0
$808$	1.58073e7	0.851784
$809$	−2.51923e7	−1.35331	−0.676654	−	0.736301i	$-0.736571\pi$
$809$	−2.51923e7	−1.35331	−0.676654	+	0.736301i	$0.736571\pi$
$810$	0	0
$811$	−8.90585e6	−0.475470	−0.237735	−	0.971330i	$-0.576405\pi$
$811$	−8.90585e6	−0.475470	−0.237735	+	0.971330i	$0.576405\pi$
$812$	0	0
$813$	0	0
$814$	−3.74038e7	−1.97858
$815$	−4.08173e6	−0.215254
$816$	0	0
$817$	4.55355e7	2.38669
$818$	−2.07417e6	−0.108383
$819$	0	0
$820$	−6.23396e6	−0.323765
$821$	−3.35490e7	−1.73709	−0.868543	−	0.495613i	$-0.834943\pi$
$821$	−3.35490e7	−1.73709	−0.868543	+	0.495613i	$0.834943\pi$
$822$	0	0
$823$	−1.46001e7	−0.751376	−0.375688	−	0.926746i	$-0.622593\pi$
$823$	−1.46001e7	−0.751376	−0.375688	+	0.926746i	$0.622593\pi$
$824$	3.78857e6	0.194382
$825$	0	0
$826$	0	0
$827$	−1.97530e7	−1.00431	−0.502156	−	0.864777i	$-0.667460\pi$
$827$	−1.97530e7	−1.00431	−0.502156	+	0.864777i	$0.667460\pi$
$828$	0	0
$829$	−4.06255e6	−0.205311	−0.102655	−	0.994717i	$-0.532734\pi$
$829$	−4.06255e6	−0.205311	−0.102655	+	0.994717i	$0.532734\pi$
$830$	−1.42653e7	−0.718762
$831$	0	0
$832$	−4.21241e7	−2.10971
$833$	0	0
$834$	0	0
$835$	2.85302e6	0.141608
$836$	−5.04050e7	−2.49435
$837$	0	0
$838$	−3.95735e7	−1.94668
$839$	1.60509e7	0.787217	0.393609	−	0.919278i	$-0.371227\pi$
$839$	1.60509e7	0.787217	0.393609	+	0.919278i	$0.371227\pi$
$840$	0	0
$841$	2.89842e7	1.41309
$842$	1.14846e7	0.558260
$843$	0	0
$844$	−6.25159e7	−3.02088
$845$	−5.85737e6	−0.282202
$846$	0	0
$847$	0	0
$848$	−672690.	−0.0321237
$849$	0	0
$850$	−3.33140e7	−1.58154
$851$	9.34043e6	0.442123
$852$	0	0
$853$	−1.23887e7	−0.582979	−0.291489	−	0.956574i	$-0.594151\pi$
$853$	−1.23887e7	−0.582979	−0.291489	+	0.956574i	$0.594151\pi$
$854$	0	0
$855$	0	0
$856$	−1.29728e6	−0.0605133
$857$	−2.63519e7	−1.22563	−0.612816	−	0.790226i	$-0.709963\pi$
$857$	−2.63519e7	−1.22563	−0.612816	+	0.790226i	$0.709963\pi$
$858$	0	0
$859$	−1.21249e7	−0.560654	−0.280327	−	0.959905i	$-0.590443\pi$
$859$	−1.21249e7	−0.560654	−0.280327	+	0.959905i	$0.590443\pi$
$860$	−2.27229e7	−1.04765
$861$	0	0
$862$	4.04702e7	1.85510
$863$	−2.55952e6	−0.116985	−0.0584927	−	0.998288i	$-0.518629\pi$
$863$	−2.55952e6	−0.116985	−0.0584927	+	0.998288i	$0.518629\pi$
$864$	0	0
$865$	−1.12094e7	−0.509382
$866$	−1.47773e7	−0.669578
$867$	0	0
$868$	0	0
$869$	−4.06737e6	−0.182711
$870$	0	0
$871$	1.26521e7	0.565091
$872$	2.09906e7	0.934834
$873$	0	0
$874$	2.02893e7	0.898439
$875$	0	0
$876$	0	0
$877$	2.65820e7	1.16705	0.583523	−	0.812096i	$-0.301674\pi$
$877$	2.65820e7	1.16705	0.583523	+	0.812096i	$0.301674\pi$
$878$	−4.06225e7	−1.77841
$879$	0	0
$880$	343290.	0.0149436
$881$	8.32262e6	0.361260	0.180630	−	0.983551i	$-0.442186\pi$
$881$	8.32262e6	0.361260	0.180630	+	0.983551i	$0.442186\pi$
$882$	0	0
$883$	1.81133e7	0.781798	0.390899	−	0.920434i	$-0.372164\pi$
$883$	1.81133e7	0.781798	0.390899	+	0.920434i	$0.372164\pi$
$884$	5.73977e7	2.47038
$885$	0	0
$886$	3.31845e7	1.42020
$887$	2.04255e7	0.871694	0.435847	−	0.900021i	$-0.356449\pi$
$887$	2.04255e7	0.871694	0.435847	+	0.900021i	$0.356449\pi$
$888$	0	0
$889$	0	0
$890$	2.92429e7	1.23750
$891$	0	0
$892$	1.55070e7	0.652552
$893$	−4.75689e6	−0.199616
$894$	0	0
$895$	−2.92645e6	−0.122119
$896$	0	0
$897$	0	0
$898$	4.28785e6	0.177439
$899$	8.87285e6	0.366154
$900$	0	0
$901$	−2.47912e7	−1.01739
$902$	−2.06635e7	−0.845645
$903$	0	0
$904$	−2.08702e7	−0.849388
$905$	1.63343e7	0.662948
$906$	0	0
$907$	−2.01197e7	−0.812089	−0.406044	−	0.913853i	$-0.633092\pi$
$907$	−2.01197e7	−0.812089	−0.406044	+	0.913853i	$0.633092\pi$
$908$	−1.14079e7	−0.459190
$909$	0	0
$910$	0	0
$911$	−3.17075e7	−1.26580	−0.632902	−	0.774232i	$-0.718137\pi$
$911$	−3.17075e7	−1.26580	−0.632902	+	0.774232i	$0.718137\pi$
$912$	0	0
$913$	−2.93345e7	−1.16467
$914$	−5.52023e6	−0.218571
$915$	0	0
$916$	−6.42812e7	−2.53131
$917$	0	0
$918$	0	0
$919$	−8.70727e6	−0.340090	−0.170045	−	0.985436i	$-0.554391\pi$
$919$	−8.70727e6	−0.340090	−0.170045	+	0.985436i	$0.554391\pi$
$920$	−3.92924e6	−0.153052
$921$	0	0
$922$	−2.63678e7	−1.02152
$923$	−4.63857e7	−1.79217
$924$	0	0
$925$	−2.57887e7	−0.991005
$926$	−2.67634e7	−1.02569
$927$	0	0
$928$	−3.95381e7	−1.50711
$929$	−3.24042e7	−1.23186	−0.615931	−	0.787800i	$-0.711220\pi$
$929$	−3.24042e7	−1.23186	−0.615931	+	0.787800i	$0.711220\pi$
$930$	0	0
$931$	0	0
$932$	3.29167e6	0.124130
$933$	0	0
$934$	6.60475e7	2.47736
$935$	1.26516e7	0.473277
$936$	0	0
$937$	−2.19155e7	−0.815458	−0.407729	−	0.913103i	$-0.633679\pi$
$937$	−2.19155e7	−0.815458	−0.407729	+	0.913103i	$0.633679\pi$
$938$	0	0
$939$	0	0
$940$	2.37376e6	0.0876227
$941$	−1.46019e7	−0.537571	−0.268785	−	0.963200i	$-0.586622\pi$
$941$	−1.46019e7	−0.537571	−0.268785	+	0.963200i	$0.586622\pi$
$942$	0	0
$943$	5.16008e6	0.188963
$944$	−277533.	−0.0101364
$945$	0	0
$946$	−7.53189e7	−2.73638
$947$	2.08378e7	0.755053	0.377527	−	0.925999i	$-0.376775\pi$
$947$	2.08378e7	0.755053	0.377527	+	0.925999i	$0.376775\pi$
$948$	0	0
$949$	3.12071e7	1.12483
$950$	−5.60184e7	−2.01382
$951$	0	0
$952$	0	0
$953$	942012.	0.0335988	0.0167994	−	0.999859i	$-0.494652\pi$
$953$	942012.	0.0335988	0.0167994	+	0.999859i	$0.494652\pi$
$954$	0	0
$955$	−1.28618e7	−0.456346
$956$	−1.14698e7	−0.405894
$957$	0	0
$958$	2.56752e7	0.903857
$959$	0	0
$960$	0	0
$961$	−2.70385e7	−0.944441
$962$	7.16209e7	2.49518
$963$	0	0
$964$	2.26889e7	0.786359
$965$	8.84860e6	0.305884
$966$	0	0
$967$	−2.56570e7	−0.882346	−0.441173	−	0.897422i	$-0.645438\pi$
$967$	−2.56570e7	−0.882346	−0.441173	+	0.897422i	$0.645438\pi$
$968$	2.34701e6	0.0805059
$969$	0	0
$970$	1.60755e7	0.548575
$971$	2.14274e7	0.729324	0.364662	−	0.931140i	$-0.381184\pi$
$971$	2.14274e7	0.729324	0.364662	+	0.931140i	$0.381184\pi$
$972$	0	0
$973$	0	0
$974$	−2.69237e7	−0.909364
$975$	0	0
$976$	130464.	0.00438394
$977$	2.04841e7	0.686562	0.343281	−	0.939233i	$-0.388462\pi$
$977$	2.04841e7	0.686562	0.343281	+	0.939233i	$0.388462\pi$
$978$	0	0
$979$	6.01338e7	2.00522
$980$	0	0
$981$	0	0
$982$	2.81057e7	0.930069
$983$	−2.77650e7	−0.916459	−0.458230	−	0.888834i	$-0.651516\pi$
$983$	−2.77650e7	−0.916459	−0.458230	+	0.888834i	$0.651516\pi$
$984$	0	0
$985$	1.48230e7	0.486796
$986$	8.88504e7	2.91050
$987$	0	0
$988$	9.65159e7	3.14562
$989$	1.88086e7	0.611456
$990$	0	0
$991$	3.46832e7	1.12185	0.560926	−	0.827866i	$-0.310445\pi$
$991$	3.46832e7	1.12185	0.560926	+	0.827866i	$0.310445\pi$
$992$	−7.08785e6	−0.228684
$993$	0	0
$994$	0	0
$995$	1.00431e7	0.321594
$996$	0	0
$997$	−1.48572e7	−0.473369	−0.236685	−	0.971587i	$-0.576061\pi$
$997$	−1.48572e7	−0.473369	−0.236685	+	0.971587i	$0.576061\pi$
$998$	−6.01512e7	−1.91169
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.w.1.4		4
3.2	odd	2		147.6.a.m.1.1		4
7.2	even	3		63.6.e.e.46.1		8
7.4	even	3		63.6.e.e.37.1		8
7.6	odd	2		441.6.a.v.1.4		4
21.2	odd	6		21.6.e.c.4.4	✓	8
21.5	even	6		147.6.e.o.67.4		8
21.11	odd	6		21.6.e.c.16.4	yes	8
21.17	even	6		147.6.e.o.79.4		8
21.20	even	2		147.6.a.l.1.1		4
84.11	even	6		336.6.q.j.289.3		8
84.23	even	6		336.6.q.j.193.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.e.c.4.4	✓	8	21.2	odd	6
21.6.e.c.16.4	yes	8	21.11	odd	6
63.6.e.e.37.1		8	7.4	even	3
63.6.e.e.46.1		8	7.2	even	3
147.6.a.l.1.1		4	21.20	even	2
147.6.a.m.1.1		4	3.2	odd	2
147.6.e.o.67.4		8	21.5	even	6
147.6.e.o.79.4		8	21.17	even	6
336.6.q.j.193.3		8	84.23	even	6
336.6.q.j.289.3		8	84.11	even	6
441.6.a.v.1.4		4	7.6	odd	2
441.6.a.w.1.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+9.18123 q^{2} +52.2950 q^{4} -22.0716 q^{5} +186.333 q^{8} +O(q^{10})\)	\(q+9.18123 q^{2} +52.2950 q^{4} -22.0716 q^{5} +186.333 q^{8} -202.644 q^{10} -416.710 q^{11} +797.918 q^{13} +37.3245 q^{16} +1375.55 q^{17} +2313.03 q^{19} -1154.23 q^{20} -3825.91 q^{22} +955.402 q^{23} -2637.84 q^{25} +7325.87 q^{26} +7035.29 q^{29} +1261.19 q^{31} -5619.96 q^{32} +12629.2 q^{34} +9776.44 q^{37} +21236.4 q^{38} -4112.66 q^{40} +5400.95 q^{41} +19686.6 q^{43} -21791.8 q^{44} +8771.76 q^{46} -2056.56 q^{47} -24218.7 q^{50} +41727.1 q^{52} -18022.7 q^{53} +9197.45 q^{55} +64592.6 q^{58} -7435.68 q^{59} +3495.38 q^{61} +11579.3 q^{62} -52792.5 q^{64} -17611.3 q^{65} +15856.4 q^{67} +71934.4 q^{68} -58133.5 q^{71} +39110.7 q^{73} +89759.8 q^{74} +120960. q^{76} +9760.69 q^{79} -823.812 q^{80} +49587.4 q^{82} +70395.7 q^{83} -30360.6 q^{85} +180747. q^{86} -77646.6 q^{88} -144306. q^{89} +49962.7 q^{92} -18881.8 q^{94} -51052.2 q^{95} -79328.7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8}+O(q^{10}) \) 4 * q - 3 * q^2 + 69 * q^4 - 123 * q^8	\( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + 283 q^{10} - 402 q^{11} + 462 q^{13} + 3273 q^{16} - 276 q^{17} + 510 q^{19} - 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} + 15138 q^{26} - 540 q^{29} - 6410 q^{31} - 15519 q^{32} + 21144 q^{34} + 15250 q^{37} + 41250 q^{38} - 8547 q^{40} - 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} + 15060 q^{47} + 7302 q^{50} - 47476 q^{52} - 13692 q^{53} + 73124 q^{55} + 52309 q^{58} - 34830 q^{59} - 5364 q^{61} - 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} + 58272 q^{68} - 89268 q^{71} + 59638 q^{73} + 185442 q^{74} + 21308 q^{76} - 44062 q^{79} + 33381 q^{80} + 57596 q^{82} + 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} + 77520 q^{89} - 158256 q^{92} - 73722 q^{94} + 221376 q^{95} - 188630 q^{97}+O(q^{100}) \) 4 * q - 3 * q^2 + 69 * q^4 - 123 * q^8 + 283 * q^10 - 402 * q^11 + 462 * q^13 + 3273 * q^16 - 276 * q^17 + 510 * q^19 - 4719 * q^20 + 1375 * q^22 - 6900 * q^23 + 2814 * q^25 + 15138 * q^26 - 540 * q^29 - 6410 * q^31 - 15519 * q^32 + 21144 * q^34 + 15250 * q^37 + 41250 * q^38 - 8547 * q^40 - 4308 * q^41 + 29198 * q^43 - 70743 * q^44 + 61800 * q^46 + 15060 * q^47 + 7302 * q^50 - 47476 * q^52 - 13692 * q^53 + 73124 * q^55 + 52309 * q^58 - 34830 * q^59 - 5364 * q^61 - 16029 * q^62 - 73487 * q^64 - 66864 * q^65 - 5994 * q^67 + 58272 * q^68 - 89268 * q^71 + 59638 * q^73 + 185442 * q^74 + 21308 * q^76 - 44062 * q^79 + 33381 * q^80 + 57596 * q^82 + 208446 * q^83 + 36324 * q^85 + 136968 * q^86 + 87597 * q^88 + 77520 * q^89 - 158256 * q^92 - 73722 * q^94 + 221376 * q^95 - 188630 * q^97

Introduction

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