LMFDB - Embedded newform 441.6.a.n.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.54138$ 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)

$f(q)$	$=$	$q-7.08276 q^{2} +18.1655 q^{4} +79.8276 q^{5} +97.9863 q^{8} +O(q^{10})$	$q-7.08276 q^{2} +18.1655 q^{4} +79.8276 q^{5} +97.9863 q^{8} -565.400 q^{10} -351.904 q^{11} -291.683 q^{13} -1275.31 q^{16} +370.075 q^{17} +1504.93 q^{19} +1450.11 q^{20} +2492.45 q^{22} +425.711 q^{23} +3247.45 q^{25} +2065.92 q^{26} +7783.93 q^{29} -2575.18 q^{31} +5897.16 q^{32} -2621.16 q^{34} +739.618 q^{37} -10659.0 q^{38} +7822.01 q^{40} -7029.84 q^{41} +1835.23 q^{43} -6392.51 q^{44} -3015.21 q^{46} +1532.68 q^{47} -23000.9 q^{50} -5298.57 q^{52} +9537.46 q^{53} -28091.6 q^{55} -55131.8 q^{58} +29674.1 q^{59} -46510.8 q^{61} +18239.4 q^{62} -958.246 q^{64} -23284.3 q^{65} +26746.1 q^{67} +6722.61 q^{68} +14388.8 q^{71} -70095.1 q^{73} -5238.54 q^{74} +27337.8 q^{76} -27085.8 q^{79} -101805. q^{80} +49790.7 q^{82} +79755.4 q^{83} +29542.2 q^{85} -12998.5 q^{86} -34481.7 q^{88} -43577.3 q^{89} +7733.26 q^{92} -10855.6 q^{94} +120135. q^{95} +103374. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 12 q^{4} + 38 q^{5} - 96 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 12 * q^4 + 38 * q^5 - 96 * q^8	$ 2 q - 2 q^{2} + 12 q^{4} + 38 q^{5} - 96 q^{8} - 778 q^{10} - 424 q^{11} - 924 q^{13} - 2064 q^{16} + 2346 q^{17} - 360 q^{19} + 1708 q^{20} + 2126 q^{22} + 12 q^{23} + 1872 q^{25} - 1148 q^{26} + 7052 q^{29} + 3548 q^{31} + 8096 q^{32} + 7422 q^{34} + 11090 q^{37} - 20138 q^{38} + 15936 q^{40} - 3500 q^{41} - 12680 q^{43} - 5948 q^{44} - 5118 q^{46} + 22956 q^{47} - 29992 q^{50} - 1400 q^{52} - 3042 q^{53} - 25076 q^{55} - 58852 q^{58} + 65808 q^{59} - 42486 q^{61} + 49362 q^{62} + 35456 q^{64} + 3164 q^{65} + 42312 q^{67} - 5460 q^{68} + 2208 q^{71} - 50506 q^{73} + 47370 q^{74} + 38836 q^{76} + 9004 q^{79} - 68816 q^{80} + 67732 q^{82} + 104328 q^{83} - 53106 q^{85} - 86776 q^{86} - 20496 q^{88} + 26666 q^{89} + 10284 q^{92} + 98034 q^{94} + 198140 q^{95} + 209132 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 12 * q^4 + 38 * q^5 - 96 * q^8 - 778 * q^10 - 424 * q^11 - 924 * q^13 - 2064 * q^16 + 2346 * q^17 - 360 * q^19 + 1708 * q^20 + 2126 * q^22 + 12 * q^23 + 1872 * q^25 - 1148 * q^26 + 7052 * q^29 + 3548 * q^31 + 8096 * q^32 + 7422 * q^34 + 11090 * q^37 - 20138 * q^38 + 15936 * q^40 - 3500 * q^41 - 12680 * q^43 - 5948 * q^44 - 5118 * q^46 + 22956 * q^47 - 29992 * q^50 - 1400 * q^52 - 3042 * q^53 - 25076 * q^55 - 58852 * q^58 + 65808 * q^59 - 42486 * q^61 + 49362 * q^62 + 35456 * q^64 + 3164 * q^65 + 42312 * q^67 - 5460 * q^68 + 2208 * q^71 - 50506 * q^73 + 47370 * q^74 + 38836 * q^76 + 9004 * q^79 - 68816 * q^80 + 67732 * q^82 + 104328 * q^83 - 53106 * q^85 - 86776 * q^86 - 20496 * q^88 + 26666 * q^89 + 10284 * q^92 + 98034 * q^94 + 198140 * q^95 + 209132 * 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 $

$2$	−7.08276	−1.25207	−0.626034	−	0.779796i	$-0.715323\pi$
$2$	−7.08276	−1.25207	−0.626034	+	0.779796i	$0.715323\pi$
$3$	0	0
$3$	0	0
$4$	18.1655	0.567673
$5$	79.8276	1.42800	0.714000	−	0.700146i	$-0.246882\pi$
$5$	79.8276	1.42800	0.714000	+	0.700146i	$0.246882\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	97.9863	0.541303
$9$	0	0
$10$	−565.400	−1.78795
$11$	−351.904	−0.876884	−0.438442	−	0.898760i	$-0.644469\pi$
$11$	−351.904	−0.876884	−0.438442	+	0.898760i	$0.644469\pi$
$12$	0	0
$13$	−291.683	−0.478688	−0.239344	−	0.970935i	$-0.576932\pi$
$13$	−291.683	−0.478688	−0.239344	+	0.970935i	$0.576932\pi$
$14$	0	0
$15$	0	0
$16$	−1275.31	−1.24542
$17$	370.075	0.310576	0.155288	−	0.987869i	$-0.450369\pi$
$17$	370.075	0.310576	0.155288	+	0.987869i	$0.450369\pi$
$18$	0	0
$19$	1504.93	0.956381	0.478190	−	0.878256i	$-0.341293\pi$
$19$	1504.93	0.956381	0.478190	+	0.878256i	$0.341293\pi$
$20$	1450.11	0.810637
$21$	0	0
$22$	2492.45	1.09792
$23$	425.711	0.167801	0.0839006	−	0.996474i	$-0.473262\pi$
$23$	425.711	0.167801	0.0839006	+	0.996474i	$0.473262\pi$
$24$	0	0
$25$	3247.45	1.03918
$26$	2065.92	0.599349
$27$	0	0
$28$	0	0
$29$	7783.93	1.71872	0.859358	−	0.511374i	$-0.170863\pi$
$29$	7783.93	1.71872	0.859358	+	0.511374i	$0.170863\pi$
$30$	0	0
$31$	−2575.18	−0.481285	−0.240643	−	0.970614i	$-0.577358\pi$
$31$	−2575.18	−0.481285	−0.240643	+	0.970614i	$0.577358\pi$
$32$	5897.16	1.01805
$33$	0	0
$34$	−2621.16	−0.388862
$35$	0	0
$36$	0	0
$37$	739.618	0.0888184	0.0444092	−	0.999013i	$-0.485859\pi$
$37$	739.618	0.0888184	0.0444092	+	0.999013i	$0.485859\pi$
$38$	−10659.0	−1.19745
$39$	0	0
$40$	7822.01	0.772981
$41$	−7029.84	−0.653109	−0.326554	−	0.945178i	$-0.605888\pi$
$41$	−7029.84	−0.653109	−0.326554	+	0.945178i	$0.605888\pi$
$42$	0	0
$43$	1835.23	0.151363	0.0756816	−	0.997132i	$-0.475887\pi$
$43$	1835.23	0.151363	0.0756816	+	0.997132i	$0.475887\pi$
$44$	−6392.51	−0.497783
$45$	0	0
$46$	−3015.21	−0.210098
$47$	1532.68	0.101206	0.0506032	−	0.998719i	$-0.483886\pi$
$47$	1532.68	0.101206	0.0506032	+	0.998719i	$0.483886\pi$
$48$	0	0
$49$	0	0
$50$	−23000.9	−1.30113
$51$	0	0
$52$	−5298.57	−0.271738
$53$	9537.46	0.466383	0.233192	−	0.972431i	$-0.425083\pi$
$53$	9537.46	0.466383	0.233192	+	0.972431i	$0.425083\pi$
$54$	0	0
$55$	−28091.6	−1.25219
$56$	0	0
$57$	0	0
$58$	−55131.8	−2.15195
$59$	29674.1	1.10981	0.554903	−	0.831915i	$-0.312755\pi$
$59$	29674.1	1.10981	0.554903	+	0.831915i	$0.312755\pi$
$60$	0	0
$61$	−46510.8	−1.60040	−0.800201	−	0.599732i	$-0.795274\pi$
$61$	−46510.8	−1.60040	−0.800201	+	0.599732i	$0.795274\pi$
$62$	18239.4	0.602602
$63$	0	0
$64$	−958.246	−0.0292434
$65$	−23284.3	−0.683566
$66$	0	0
$67$	26746.1	0.727902	0.363951	−	0.931418i	$-0.381428\pi$
$67$	26746.1	0.727902	0.363951	+	0.931418i	$0.381428\pi$
$68$	6722.61	0.176305
$69$	0	0
$70$	0	0
$71$	14388.8	0.338748	0.169374	−	0.985552i	$-0.445825\pi$
$71$	14388.8	0.338748	0.169374	+	0.985552i	$0.445825\pi$
$72$	0	0
$73$	−70095.1	−1.53950	−0.769752	−	0.638343i	$-0.779620\pi$
$73$	−70095.1	−1.53950	−0.769752	+	0.638343i	$0.779620\pi$
$74$	−5238.54	−0.111207
$75$	0	0
$76$	27337.8	0.542911
$77$	0	0
$78$	0	0
$79$	−27085.8	−0.488285	−0.244143	−	0.969739i	$-0.578507\pi$
$79$	−27085.8	−0.488285	−0.244143	+	0.969739i	$0.578507\pi$
$80$	−101805.	−1.77846
$81$	0	0
$82$	49790.7	0.817736
$83$	79755.4	1.27076	0.635382	−	0.772198i	$-0.280843\pi$
$83$	79755.4	1.27076	0.635382	+	0.772198i	$0.280843\pi$
$84$	0	0
$85$	29542.2	0.443502
$86$	−12998.5	−0.189517
$87$	0	0
$88$	−34481.7	−0.474660
$89$	−43577.3	−0.583157	−0.291579	−	0.956547i	$-0.594180\pi$
$89$	−43577.3	−0.583157	−0.291579	+	0.956547i	$0.594180\pi$
$90$	0	0
$91$	0	0
$92$	7733.26	0.0952561
$93$	0	0
$94$	−10855.6	−0.126717
$95$	120135.	1.36571
$96$	0	0
$97$	103374.	1.11553	0.557765	−	0.829999i	$-0.311659\pi$
$97$	103374.	1.11553	0.557765	+	0.829999i	$0.311659\pi$
$98$	0	0
$99$	0	0
$100$	58991.6	0.589916
$101$	−28700.2	−0.279951	−0.139975	−	0.990155i	$-0.544702\pi$
$101$	−28700.2	−0.279951	−0.139975	+	0.990155i	$0.544702\pi$
$102$	0	0
$103$	29227.9	0.271459	0.135730	−	0.990746i	$-0.456662\pi$
$103$	29227.9	0.271459	0.135730	+	0.990746i	$0.456662\pi$
$104$	−28580.9	−0.259115
$105$	0	0
$106$	−67551.6	−0.583944
$107$	−87858.4	−0.741863	−0.370932	−	0.928660i	$-0.620962\pi$
$107$	−87858.4	−0.741863	−0.370932	+	0.928660i	$0.620962\pi$
$108$	0	0
$109$	220628.	1.77867	0.889333	−	0.457260i	$-0.151169\pi$
$109$	220628.	1.77867	0.889333	+	0.457260i	$0.151169\pi$
$110$	198966.	1.56783
$111$	0	0
$112$	0	0
$113$	−39665.6	−0.292225	−0.146113	−	0.989268i	$-0.546676\pi$
$113$	−39665.6	−0.292225	−0.146113	+	0.989268i	$0.546676\pi$
$114$	0	0
$115$	33983.5	0.239620
$116$	141399.	0.975668
$117$	0	0
$118$	−210174.	−1.38955
$119$	0	0
$120$	0	0
$121$	−37214.9	−0.231075
$122$	329425.	2.00381
$123$	0	0
$124$	−46779.4	−0.273212
$125$	9774.87	0.0559546
$126$	0	0
$127$	51740.3	0.284655	0.142328	−	0.989820i	$-0.454541\pi$
$127$	51740.3	0.284655	0.142328	+	0.989820i	$0.454541\pi$
$128$	−181922.	−0.981433
$129$	0	0
$130$	164917.	0.855871
$131$	166674.	0.848572	0.424286	−	0.905528i	$-0.360525\pi$
$131$	166674.	0.848572	0.424286	+	0.905528i	$0.360525\pi$
$132$	0	0
$133$	0	0
$134$	−189436.	−0.911382
$135$	0	0
$136$	36262.3	0.168116
$137$	−28259.4	−0.128636	−0.0643178	−	0.997929i	$-0.520487\pi$
$137$	−28259.4	−0.128636	−0.0643178	+	0.997929i	$0.520487\pi$
$138$	0	0
$139$	336393.	1.47676	0.738380	−	0.674384i	$-0.235591\pi$
$139$	336393.	1.47676	0.738380	+	0.674384i	$0.235591\pi$
$140$	0	0
$141$	0	0
$142$	−101912.	−0.424136
$143$	102644.	0.419753
$144$	0	0
$145$	621373.	2.45433
$146$	496467.	1.92756
$147$	0	0
$148$	13435.5	0.0504198
$149$	355381.	1.31138	0.655691	−	0.755030i	$-0.272378\pi$
$149$	355381.	1.31138	0.655691	+	0.755030i	$0.272378\pi$
$150$	0	0
$151$	−358797.	−1.28058	−0.640290	−	0.768133i	$-0.721186\pi$
$151$	−358797.	−1.28058	−0.640290	+	0.768133i	$0.721186\pi$
$152$	147462.	0.517692
$153$	0	0
$154$	0	0
$155$	−205570.	−0.687275
$156$	0	0
$157$	458911.	1.48586	0.742932	−	0.669367i	$-0.233435\pi$
$157$	458911.	1.48586	0.742932	+	0.669367i	$0.233435\pi$
$158$	191842.	0.611366
$159$	0	0
$160$	470756.	1.45377
$161$	0	0
$162$	0	0
$163$	502441.	1.48121	0.740603	−	0.671943i	$-0.234540\pi$
$163$	502441.	1.48121	0.740603	+	0.671943i	$0.234540\pi$
$164$	−127701.	−0.370752
$165$	0	0
$166$	−564889.	−1.59108
$167$	676652.	1.87748	0.938738	−	0.344632i	$-0.111996\pi$
$167$	676652.	1.87748	0.938738	+	0.344632i	$0.111996\pi$
$168$	0	0
$169$	−286214.	−0.770858
$170$	−209241.	−0.555295
$171$	0	0
$172$	33338.0	0.0859247
$173$	249160.	0.632941	0.316470	−	0.948602i	$-0.397502\pi$
$173$	249160.	0.632941	0.316470	+	0.948602i	$0.397502\pi$
$174$	0	0
$175$	0	0
$176$	448786.	1.09209
$177$	0	0
$178$	308648.	0.730152
$179$	−139258.	−0.324853	−0.162427	−	0.986721i	$-0.551932\pi$
$179$	−139258.	−0.324853	−0.162427	+	0.986721i	$0.551932\pi$
$180$	0	0
$181$	306246.	0.694823	0.347412	−	0.937713i	$-0.387061\pi$
$181$	306246.	0.694823	0.347412	+	0.937713i	$0.387061\pi$
$182$	0	0
$183$	0	0
$184$	41713.8	0.0908312
$185$	59041.9	0.126833
$186$	0	0
$187$	−130231.	−0.272339
$188$	27842.0	0.0574521
$189$	0	0
$190$	−850885.	−1.70996
$191$	−227494.	−0.451218	−0.225609	−	0.974218i	$-0.572437\pi$
$191$	−227494.	−0.451218	−0.225609	+	0.974218i	$0.572437\pi$
$192$	0	0
$193$	−672374.	−1.29933	−0.649663	−	0.760223i	$-0.725090\pi$
$193$	−672374.	−1.29933	−0.649663	+	0.760223i	$0.725090\pi$
$194$	−732172.	−1.39672
$195$	0	0
$196$	0	0
$197$	1282.76	0.00235493	0.00117747	−	0.999999i	$-0.499625\pi$
$197$	1282.76	0.00235493	0.00117747	+	0.999999i	$0.499625\pi$
$198$	0	0
$199$	368898.	0.660349	0.330175	−	0.943920i	$-0.392892\pi$
$199$	368898.	0.660349	0.330175	+	0.943920i	$0.392892\pi$
$200$	318206.	0.562513
$201$	0	0
$202$	203277.	0.350517
$203$	0	0
$204$	0	0
$205$	−561175.	−0.932640
$206$	−207014.	−0.339885
$207$	0	0
$208$	371986.	0.596167
$209$	−529589.	−0.838635
$210$	0	0
$211$	502168.	0.776503	0.388251	−	0.921553i	$-0.373079\pi$
$211$	502168.	0.776503	0.388251	+	0.921553i	$0.373079\pi$
$212$	173253.	0.264753
$213$	0	0
$214$	622280.	0.928863
$215$	146502.	0.216147
$216$	0	0
$217$	0	0
$218$	−1.56266e6	−2.22701
$219$	0	0
$220$	−510299.	−0.710834
$221$	−107945.	−0.148669
$222$	0	0
$223$	−1.17328e6	−1.57993	−0.789967	−	0.613149i	$-0.789902\pi$
$223$	−1.17328e6	−1.57993	−0.789967	+	0.613149i	$0.789902\pi$
$224$	0	0
$225$	0	0
$226$	280942.	0.365886
$227$	910159.	1.17234	0.586168	−	0.810189i	$-0.300636\pi$
$227$	910159.	1.17234	0.586168	+	0.810189i	$0.300636\pi$
$228$	0	0
$229$	521924.	0.657686	0.328843	−	0.944385i	$-0.393341\pi$
$229$	521924.	0.657686	0.328843	+	0.944385i	$0.393341\pi$
$230$	−240697.	−0.300020
$231$	0	0
$232$	762719.	0.930346
$233$	1.04279e6	1.25836	0.629182	−	0.777258i	$-0.283390\pi$
$233$	1.04279e6	1.25836	0.629182	+	0.777258i	$0.283390\pi$
$234$	0	0
$235$	122350.	0.144523
$236$	539045.	0.630006
$237$	0	0
$238$	0	0
$239$	1.53447e6	1.73766	0.868830	−	0.495110i	$-0.164872\pi$
$239$	1.53447e6	1.73766	0.868830	+	0.495110i	$0.164872\pi$
$240$	0	0
$241$	−1.00758e6	−1.11747	−0.558735	−	0.829346i	$-0.688713\pi$
$241$	−1.00758e6	−1.11747	−0.558735	+	0.829346i	$0.688713\pi$
$242$	263584.	0.289322
$243$	0	0
$244$	−844893.	−0.908505
$245$	0	0
$246$	0	0
$247$	−438961.	−0.457808
$248$	−252332.	−0.260521
$249$	0	0
$250$	−69233.1	−0.0700590
$251$	−8511.89	−0.00852789	−0.00426394	−	0.999991i	$-0.501357\pi$
$251$	−8511.89	−0.00852789	−0.00426394	+	0.999991i	$0.501357\pi$
$252$	0	0
$253$	−149809.	−0.147142
$254$	−366464.	−0.356408
$255$	0	0
$256$	1.31917e6	1.25806
$257$	527532.	0.498214	0.249107	−	0.968476i	$-0.419863\pi$
$257$	527532.	0.498214	0.249107	+	0.968476i	$0.419863\pi$
$258$	0	0
$259$	0	0
$260$	−422972.	−0.388042
$261$	0	0
$262$	−1.18051e6	−1.06247
$263$	352085.	0.313876	0.156938	−	0.987608i	$-0.449838\pi$
$263$	352085.	0.313876	0.156938	+	0.987608i	$0.449838\pi$
$264$	0	0
$265$	761353.	0.665996
$266$	0	0
$267$	0	0
$268$	485856.	0.413210
$269$	−479540.	−0.404058	−0.202029	−	0.979380i	$-0.564754\pi$
$269$	−479540.	−0.404058	−0.202029	+	0.979380i	$0.564754\pi$
$270$	0	0
$271$	−977611.	−0.808617	−0.404308	−	0.914623i	$-0.632488\pi$
$271$	−977611.	−0.808617	−0.404308	+	0.914623i	$0.632488\pi$
$272$	−471961.	−0.386798
$273$	0	0
$274$	200155.	0.161061
$275$	−1.14279e6	−0.911243
$276$	0	0
$277$	968723.	0.758578	0.379289	−	0.925278i	$-0.376169\pi$
$277$	968723.	0.758578	0.379289	+	0.925278i	$0.376169\pi$
$278$	−2.38259e6	−1.84900
$279$	0	0
$280$	0	0
$281$	318333.	0.240501	0.120250	−	0.992744i	$-0.461630\pi$
$281$	318333.	0.240501	0.120250	+	0.992744i	$0.461630\pi$
$282$	0	0
$283$	1.77210e6	1.31529	0.657646	−	0.753327i	$-0.271552\pi$
$283$	1.77210e6	1.31529	0.657646	+	0.753327i	$0.271552\pi$
$284$	261379.	0.192298
$285$	0	0
$286$	−727004.	−0.525559
$287$	0	0
$288$	0	0
$289$	−1.28290e6	−0.903543
$290$	−4.40104e6	−3.07298
$291$	0	0
$292$	−1.27331e6	−0.873934
$293$	−1.64148e6	−1.11703	−0.558516	−	0.829494i	$-0.688629\pi$
$293$	−1.64148e6	−1.11703	−0.558516	+	0.829494i	$0.688629\pi$
$294$	0	0
$295$	2.36881e6	1.58480
$296$	72472.4	0.0480777
$297$	0	0
$298$	−2.51708e6	−1.64194
$299$	−124172.	−0.0803243
$300$	0	0
$301$	0	0
$302$	2.54128e6	1.60337
$303$	0	0
$304$	−1.91925e6	−1.19110
$305$	−3.71285e6	−2.28537
$306$	0	0
$307$	−466930.	−0.282752	−0.141376	−	0.989956i	$-0.545153\pi$
$307$	−466930.	−0.282752	−0.141376	+	0.989956i	$0.545153\pi$
$308$	0	0
$309$	0	0
$310$	1.45600e6	0.860515
$311$	2.43796e6	1.42931	0.714654	−	0.699478i	$-0.246584\pi$
$311$	2.43796e6	1.42931	0.714654	+	0.699478i	$0.246584\pi$
$312$	0	0
$313$	2.42094e6	1.39676	0.698381	−	0.715726i	$-0.253904\pi$
$313$	2.42094e6	1.39676	0.698381	+	0.715726i	$0.253904\pi$
$314$	−3.25035e6	−1.86040
$315$	0	0
$316$	−492028.	−0.277186
$317$	−1.87611e6	−1.04860	−0.524301	−	0.851533i	$-0.675673\pi$
$317$	−1.87611e6	−1.04860	−0.524301	+	0.851533i	$0.675673\pi$
$318$	0	0
$319$	−2.73919e6	−1.50711
$320$	−76494.5	−0.0417595
$321$	0	0
$322$	0	0
$323$	556936.	0.297029
$324$	0	0
$325$	−947225.	−0.497445
$326$	−3.55867e6	−1.85457
$327$	0	0
$328$	−688828.	−0.353530
$329$	0	0
$330$	0	0
$331$	−1.08310e6	−0.543373	−0.271686	−	0.962386i	$-0.587581\pi$
$331$	−1.08310e6	−0.543373	−0.271686	+	0.962386i	$0.587581\pi$
$332$	1.44880e6	0.721378
$333$	0	0
$334$	−4.79257e6	−2.35073
$335$	2.13507e6	1.03944
$336$	0	0
$337$	−2.59465e6	−1.24453	−0.622263	−	0.782809i	$-0.713786\pi$
$337$	−2.59465e6	−1.24453	−0.622263	+	0.782809i	$0.713786\pi$
$338$	2.02719e6	0.965166
$339$	0	0
$340$	536650.	0.251764
$341$	906213.	0.422031
$342$	0	0
$343$	0	0
$344$	179828.	0.0819333
$345$	0	0
$346$	−1.76474e6	−0.792485
$347$	1.87051e6	0.833943	0.416972	−	0.908920i	$-0.363091\pi$
$347$	1.87051e6	0.833943	0.416972	+	0.908920i	$0.363091\pi$
$348$	0	0
$349$	−1.61685e6	−0.710568	−0.355284	−	0.934758i	$-0.615616\pi$
$349$	−1.61685e6	−0.710568	−0.355284	+	0.934758i	$0.615616\pi$
$350$	0	0
$351$	0	0
$352$	−2.07523e6	−0.892709
$353$	578305.	0.247013	0.123507	−	0.992344i	$-0.460586\pi$
$353$	578305.	0.247013	0.123507	+	0.992344i	$0.460586\pi$
$354$	0	0
$355$	1.14862e6	0.483733
$356$	−791605.	−0.331042
$357$	0	0
$358$	986330.	0.406738
$359$	1.96818e6	0.805988	0.402994	−	0.915203i	$-0.367970\pi$
$359$	1.96818e6	0.805988	0.402994	+	0.915203i	$0.367970\pi$
$360$	0	0
$361$	−211299.	−0.0853355
$362$	−2.16907e6	−0.869965
$363$	0	0
$364$	0	0
$365$	−5.59553e6	−2.19841
$366$	0	0
$367$	2.17452e6	0.842749	0.421375	−	0.906887i	$-0.361548\pi$
$367$	2.17452e6	0.842749	0.421375	+	0.906887i	$0.361548\pi$
$368$	−542913.	−0.208983
$369$	0	0
$370$	−418180.	−0.158803
$371$	0	0
$372$	0	0
$373$	1.38476e6	0.515349	0.257675	−	0.966232i	$-0.417044\pi$
$373$	1.38476e6	0.515349	0.257675	+	0.966232i	$0.417044\pi$
$374$	922394.	0.340987
$375$	0	0
$376$	150182.	0.0547833
$377$	−2.27044e6	−0.822728
$378$	0	0
$379$	3.37190e6	1.20580	0.602902	−	0.797815i	$-0.294011\pi$
$379$	3.37190e6	1.20580	0.602902	+	0.797815i	$0.294011\pi$
$380$	2.18231e6	0.775277
$381$	0	0
$382$	1.61128e6	0.564955
$383$	−3.28060e6	−1.14276	−0.571382	−	0.820685i	$-0.693592\pi$
$383$	−3.28060e6	−1.14276	−0.571382	+	0.820685i	$0.693592\pi$
$384$	0	0
$385$	0	0
$386$	4.76227e6	1.62684
$387$	0	0
$388$	1.87784e6	0.633256
$389$	2.94810e6	0.987797	0.493899	−	0.869520i	$-0.335571\pi$
$389$	2.94810e6	0.987797	0.493899	+	0.869520i	$0.335571\pi$
$390$	0	0
$391$	157545.	0.0521150
$392$	0	0
$393$	0	0
$394$	−9085.46	−0.00294854
$395$	−2.16219e6	−0.697271
$396$	0	0
$397$	−69270.2	−0.0220582	−0.0110291	−	0.999939i	$-0.503511\pi$
$397$	−69270.2	−0.0220582	−0.0110291	+	0.999939i	$0.503511\pi$
$398$	−2.61282e6	−0.826802
$399$	0	0
$400$	−4.14151e6	−1.29422
$401$	3.34786e6	1.03970	0.519848	−	0.854259i	$-0.325988\pi$
$401$	3.34786e6	1.03970	0.519848	+	0.854259i	$0.325988\pi$
$402$	0	0
$403$	751134.	0.230385
$404$	−521354.	−0.158920
$405$	0	0
$406$	0	0
$407$	−260274.	−0.0778834
$408$	0	0
$409$	2.91217e6	0.860812	0.430406	−	0.902636i	$-0.358370\pi$
$409$	2.91217e6	0.860812	0.430406	+	0.902636i	$0.358370\pi$
$410$	3.97467e6	1.16773
$411$	0	0
$412$	530940.	0.154100
$413$	0	0
$414$	0	0
$415$	6.36669e6	1.81465
$416$	−1.72010e6	−0.487327
$417$	0	0
$418$	3.75095e6	1.05003
$419$	4.62361e6	1.28661	0.643304	−	0.765611i	$-0.277563\pi$
$419$	4.62361e6	1.28661	0.643304	+	0.765611i	$0.277563\pi$
$420$	0	0
$421$	−2.63042e6	−0.723303	−0.361652	−	0.932313i	$-0.617787\pi$
$421$	−2.63042e6	−0.723303	−0.361652	+	0.932313i	$0.617787\pi$
$422$	−3.55674e6	−0.972234
$423$	0	0
$424$	934541.	0.252455
$425$	1.20180e6	0.322746
$426$	0	0
$427$	0	0
$428$	−1.59599e6	−0.421135
$429$	0	0
$430$	−1.03764e6	−0.270630
$431$	−7.54128e6	−1.95547	−0.977736	−	0.209837i	$-0.932707\pi$
$431$	−7.54128e6	−1.95547	−0.977736	+	0.209837i	$0.932707\pi$
$432$	0	0
$433$	5.83558e6	1.49577	0.747883	−	0.663830i	$-0.231070\pi$
$433$	5.83558e6	1.49577	0.747883	+	0.663830i	$0.231070\pi$
$434$	0	0
$435$	0	0
$436$	4.00782e6	1.00970
$437$	640663.	0.160482
$438$	0	0
$439$	168104.	0.0416310	0.0208155	−	0.999783i	$-0.493374\pi$
$439$	168104.	0.0416310	0.0208155	+	0.999783i	$0.493374\pi$
$440$	−2.75259e6	−0.677814
$441$	0	0
$442$	764546.	0.186143
$443$	2.84151e6	0.687924	0.343962	−	0.938984i	$-0.388231\pi$
$443$	2.84151e6	0.687924	0.343962	+	0.938984i	$0.388231\pi$
$444$	0	0
$445$	−3.47867e6	−0.832748
$446$	8.31005e6	1.97818
$447$	0	0
$448$	0	0
$449$	1.41567e6	0.331396	0.165698	−	0.986177i	$-0.447012\pi$
$449$	1.41567e6	0.331396	0.165698	+	0.986177i	$0.447012\pi$
$450$	0	0
$451$	2.47382e6	0.572701
$452$	−720547.	−0.165888
$453$	0	0
$454$	−6.44644e6	−1.46784
$455$	0	0
$456$	0	0
$457$	1.55727e6	0.348799	0.174399	−	0.984675i	$-0.444202\pi$
$457$	1.55727e6	0.348799	0.174399	+	0.984675i	$0.444202\pi$
$458$	−3.69667e6	−0.823468
$459$	0	0
$460$	617328.	0.136026
$461$	4.45345e6	0.975987	0.487994	−	0.872847i	$-0.337729\pi$
$461$	4.45345e6	0.975987	0.487994	+	0.872847i	$0.337729\pi$
$462$	0	0
$463$	4.92263e6	1.06720	0.533599	−	0.845738i	$-0.320839\pi$
$463$	4.92263e6	1.06720	0.533599	+	0.845738i	$0.320839\pi$
$464$	−9.92693e6	−2.14052
$465$	0	0
$466$	−7.38582e6	−1.57556
$467$	−5.09090e6	−1.08020	−0.540098	−	0.841602i	$-0.681613\pi$
$467$	−5.09090e6	−1.08020	−0.540098	+	0.841602i	$0.681613\pi$
$468$	0	0
$469$	0	0
$470$	−866579.	−0.180952
$471$	0	0
$472$	2.90765e6	0.600741
$473$	−645825.	−0.132728
$474$	0	0
$475$	4.88717e6	0.993856
$476$	0	0
$477$	0	0
$478$	−1.08683e7	−2.17567
$479$	8.30085e6	1.65304	0.826521	−	0.562907i	$-0.190317\pi$
$479$	8.30085e6	1.65304	0.826521	+	0.562907i	$0.190317\pi$
$480$	0	0
$481$	−215734.	−0.0425163
$482$	7.13644e6	1.39915
$483$	0	0
$484$	−676028.	−0.131175
$485$	8.25208e6	1.59298
$486$	0	0
$487$	8.63401e6	1.64964	0.824822	−	0.565392i	$-0.191275\pi$
$487$	8.63401e6	1.64964	0.824822	+	0.565392i	$0.191275\pi$
$488$	−4.55742e6	−0.866303
$489$	0	0
$490$	0	0
$491$	−95039.5	−0.0177910	−0.00889550	−	0.999960i	$-0.502832\pi$
$491$	−95039.5	−0.0177910	−0.00889550	+	0.999960i	$0.502832\pi$
$492$	0	0
$493$	2.88064e6	0.533792
$494$	3.10905e6	0.573206
$495$	0	0
$496$	3.28415e6	0.599402
$497$	0	0
$498$	0	0
$499$	2.14203e6	0.385101	0.192551	−	0.981287i	$-0.438324\pi$
$499$	2.14203e6	0.385101	0.192551	+	0.981287i	$0.438324\pi$
$500$	177566.	0.0317639
$501$	0	0
$502$	60287.7	0.0106775
$503$	−5.24794e6	−0.924844	−0.462422	−	0.886660i	$-0.653019\pi$
$503$	−5.24794e6	−0.924844	−0.462422	+	0.886660i	$0.653019\pi$
$504$	0	0
$505$	−2.29107e6	−0.399769
$506$	1.06106e6	0.184232
$507$	0	0
$508$	939889.	0.161591
$509$	1.05891e7	1.81160	0.905802	−	0.423702i	$-0.139270\pi$
$509$	1.05891e7	1.81160	0.905802	+	0.423702i	$0.139270\pi$
$510$	0	0
$511$	0	0
$512$	−3.52190e6	−0.593747
$513$	0	0
$514$	−3.73639e6	−0.623798
$515$	2.33319e6	0.387644
$516$	0	0
$517$	−539357.	−0.0887462
$518$	0	0
$519$	0	0
$520$	−2.28155e6	−0.370016
$521$	4.54465e6	0.733510	0.366755	−	0.930318i	$-0.380469\pi$
$521$	4.54465e6	0.733510	0.366755	+	0.930318i	$0.380469\pi$
$522$	0	0
$523$	−5.27197e6	−0.842789	−0.421394	−	0.906877i	$-0.638459\pi$
$523$	−5.27197e6	−0.842789	−0.421394	+	0.906877i	$0.638459\pi$
$524$	3.02772e6	0.481711
$525$	0	0
$526$	−2.49373e6	−0.392993
$527$	−953009.	−0.149476
$528$	0	0
$529$	−6.25511e6	−0.971843
$530$	−5.39248e6	−0.833871
$531$	0	0
$532$	0	0
$533$	2.05048e6	0.312635
$534$	0	0
$535$	−7.01353e6	−1.05938
$536$	2.62075e6	0.394015
$537$	0	0
$538$	3.39647e6	0.505908
$539$	0	0
$540$	0	0
$541$	5.93445e6	0.871741	0.435871	−	0.900009i	$-0.356441\pi$
$541$	5.93445e6	0.871741	0.435871	+	0.900009i	$0.356441\pi$
$542$	6.92418e6	1.01244
$543$	0	0
$544$	2.18239e6	0.316181
$545$	1.76122e7	2.53994
$546$	0	0
$547$	−8.82017e6	−1.26040	−0.630200	−	0.776433i	$-0.717027\pi$
$547$	−8.82017e6	−1.26040	−0.630200	+	0.776433i	$0.717027\pi$
$548$	−513347.	−0.0730230
$549$	0	0
$550$	8.09410e6	1.14094
$551$	1.17142e7	1.64375
$552$	0	0
$553$	0	0
$554$	−6.86124e6	−0.949791
$555$	0	0
$556$	6.11076e6	0.838317
$557$	1.18224e6	0.161461	0.0807304	−	0.996736i	$-0.474275\pi$
$557$	1.18224e6	0.161461	0.0807304	+	0.996736i	$0.474275\pi$
$558$	0	0
$559$	−535306.	−0.0724556
$560$	0	0
$561$	0	0
$562$	−2.25468e6	−0.301123
$563$	−4.07741e6	−0.542142	−0.271071	−	0.962559i	$-0.587378\pi$
$563$	−4.07741e6	−0.542142	−0.271071	+	0.962559i	$0.587378\pi$
$564$	0	0
$565$	−3.16641e6	−0.417298
$566$	−1.25514e7	−1.64684
$567$	0	0
$568$	1.40990e6	0.183366
$569$	−8.17615e6	−1.05869	−0.529344	−	0.848407i	$-0.677562\pi$
$569$	−8.17615e6	−1.05869	−0.529344	+	0.848407i	$0.677562\pi$
$570$	0	0
$571$	3.30615e6	0.424357	0.212179	−	0.977231i	$-0.431944\pi$
$571$	3.30615e6	0.424357	0.212179	+	0.977231i	$0.431944\pi$
$572$	1.86458e6	0.238282
$573$	0	0
$574$	0	0
$575$	1.38247e6	0.174376
$576$	0	0
$577$	−7.14994e6	−0.894052	−0.447026	−	0.894521i	$-0.647517\pi$
$577$	−7.14994e6	−0.894052	−0.447026	+	0.894521i	$0.647517\pi$
$578$	9.08648e6	1.13130
$579$	0	0
$580$	1.12876e7	1.39325
$581$	0	0
$582$	0	0
$583$	−3.35627e6	−0.408964
$584$	−6.86836e6	−0.833338
$585$	0	0
$586$	1.16262e7	1.39860
$587$	−9.69191e6	−1.16095	−0.580476	−	0.814277i	$-0.697133\pi$
$587$	−9.69191e6	−1.16095	−0.580476	+	0.814277i	$0.697133\pi$
$588$	0	0
$589$	−3.87545e6	−0.460292
$590$	−1.67777e7	−1.98428
$591$	0	0
$592$	−943242.	−0.110616
$593$	−6.63960e6	−0.775363	−0.387682	−	0.921793i	$-0.626724\pi$
$593$	−6.63960e6	−0.775363	−0.387682	+	0.921793i	$0.626724\pi$
$594$	0	0
$595$	0	0
$596$	6.45569e6	0.744435
$597$	0	0
$598$	879484.	0.100571
$599$	3.24191e6	0.369177	0.184588	−	0.982816i	$-0.440905\pi$
$599$	3.24191e6	0.369177	0.184588	+	0.982816i	$0.440905\pi$
$600$	0	0
$601$	−5.65076e6	−0.638147	−0.319074	−	0.947730i	$-0.603372\pi$
$601$	−5.65076e6	−0.638147	−0.319074	+	0.947730i	$0.603372\pi$
$602$	0	0
$603$	0	0
$604$	−6.51774e6	−0.726950
$605$	−2.97078e6	−0.329975
$606$	0	0
$607$	235674.	0.0259621	0.0129811	−	0.999916i	$-0.495868\pi$
$607$	235674.	0.0259621	0.0129811	+	0.999916i	$0.495868\pi$
$608$	8.87478e6	0.973641
$609$	0	0
$610$	2.62972e7	2.86144
$611$	−447057.	−0.0484462
$612$	0	0
$613$	788877.	0.0847926	0.0423963	−	0.999101i	$-0.486501\pi$
$613$	788877.	0.0847926	0.0423963	+	0.999101i	$0.486501\pi$
$614$	3.30716e6	0.354025
$615$	0	0
$616$	0	0
$617$	−1.67739e7	−1.77387	−0.886935	−	0.461894i	$-0.847170\pi$
$617$	−1.67739e7	−1.77387	−0.886935	+	0.461894i	$0.847170\pi$
$618$	0	0
$619$	8.22300e6	0.862588	0.431294	−	0.902211i	$-0.358057\pi$
$619$	8.22300e6	0.862588	0.431294	+	0.902211i	$0.358057\pi$
$620$	−3.73429e6	−0.390147
$621$	0	0
$622$	−1.72675e7	−1.78959
$623$	0	0
$624$	0	0
$625$	−9.36798e6	−0.959281
$626$	−1.71469e7	−1.74884
$627$	0	0
$628$	8.33635e6	0.843484
$629$	273714.	0.0275849
$630$	0	0
$631$	−5.94507e6	−0.594406	−0.297203	−	0.954814i	$-0.596054\pi$
$631$	−5.94507e6	−0.594406	−0.297203	+	0.954814i	$0.596054\pi$
$632$	−2.65404e6	−0.264310
$633$	0	0
$634$	1.32881e7	1.31292
$635$	4.13030e6	0.406488
$636$	0	0
$637$	0	0
$638$	1.94011e7	1.88701
$639$	0	0
$640$	−1.45224e7	−1.40149
$641$	1.06761e7	1.02628	0.513141	−	0.858304i	$-0.328482\pi$
$641$	1.06761e7	1.02628	0.513141	+	0.858304i	$0.328482\pi$
$642$	0	0
$643$	−3.13159e6	−0.298701	−0.149351	−	0.988784i	$-0.547718\pi$
$643$	−3.13159e6	−0.298701	−0.149351	+	0.988784i	$0.547718\pi$
$644$	0	0
$645$	0	0
$646$	−3.94464e6	−0.371900
$647$	4.93457e6	0.463435	0.231717	−	0.972783i	$-0.425566\pi$
$647$	4.93457e6	0.463435	0.231717	+	0.972783i	$0.425566\pi$
$648$	0	0
$649$	−1.04424e7	−0.973170
$650$	6.70897e6	0.622834
$651$	0	0
$652$	9.12710e6	0.840841
$653$	−5.72224e6	−0.525150	−0.262575	−	0.964912i	$-0.584572\pi$
$653$	−5.72224e6	−0.525150	−0.262575	+	0.964912i	$0.584572\pi$
$654$	0	0
$655$	1.33052e7	1.21176
$656$	8.96523e6	0.813395
$657$	0	0
$658$	0	0
$659$	−362477.	−0.0325137	−0.0162569	−	0.999868i	$-0.505175\pi$
$659$	−362477.	−0.0325137	−0.0162569	+	0.999868i	$0.505175\pi$
$660$	0	0
$661$	−1.91211e7	−1.70219	−0.851096	−	0.525011i	$-0.824061\pi$
$661$	−1.91211e7	−1.70219	−0.851096	+	0.525011i	$0.824061\pi$
$662$	7.67133e6	0.680339
$663$	0	0
$664$	7.81494e6	0.687868
$665$	0	0
$666$	0	0
$667$	3.31370e6	0.288403
$668$	1.22917e7	1.06579
$669$	0	0
$670$	−1.51222e7	−1.30145
$671$	1.63673e7	1.40337
$672$	0	0
$673$	−573374.	−0.0487978	−0.0243989	−	0.999702i	$-0.507767\pi$
$673$	−573374.	−0.0487978	−0.0243989	+	0.999702i	$0.507767\pi$
$674$	1.83773e7	1.55823
$675$	0	0
$676$	−5.19923e6	−0.437595
$677$	−1.16903e7	−0.980291	−0.490146	−	0.871641i	$-0.663056\pi$
$677$	−1.16903e7	−0.980291	−0.490146	+	0.871641i	$0.663056\pi$
$678$	0	0
$679$	0	0
$680$	2.89473e6	0.240069
$681$	0	0
$682$	−6.41849e6	−0.528411
$683$	−1.83674e7	−1.50659	−0.753297	−	0.657681i	$-0.771538\pi$
$683$	−1.83674e7	−1.50659	−0.753297	+	0.657681i	$0.771538\pi$
$684$	0	0
$685$	−2.25588e6	−0.183692
$686$	0	0
$687$	0	0
$688$	−2.34049e6	−0.188511
$689$	−2.78191e6	−0.223252
$690$	0	0
$691$	2.35611e7	1.87716	0.938579	−	0.345066i	$-0.112143\pi$
$691$	2.35611e7	1.87716	0.938579	+	0.345066i	$0.112143\pi$
$692$	4.52612e6	0.359303
$693$	0	0
$694$	−1.32484e7	−1.04415
$695$	2.68535e7	2.10881
$696$	0	0
$697$	−2.60157e6	−0.202840
$698$	1.14517e7	0.889679
$699$	0	0
$700$	0	0
$701$	−1.32980e7	−1.02210	−0.511048	−	0.859552i	$-0.670743\pi$
$701$	−1.32980e7	−1.02210	−0.511048	+	0.859552i	$0.670743\pi$
$702$	0	0
$703$	1.11307e6	0.0849442
$704$	337210.	0.0256430
$705$	0	0
$706$	−4.09600e6	−0.309277
$707$	0	0
$708$	0	0
$709$	−6.85353e6	−0.512034	−0.256017	−	0.966672i	$-0.582410\pi$
$709$	−6.85353e6	−0.512034	−0.256017	+	0.966672i	$0.582410\pi$
$710$	−8.13540e6	−0.605666
$711$	0	0
$712$	−4.26998e6	−0.315665
$713$	−1.09628e6	−0.0807602
$714$	0	0
$715$	8.19384e6	0.599408
$716$	−2.52969e6	−0.184410
$717$	0	0
$718$	−1.39402e7	−1.00915
$719$	−2.65729e7	−1.91698	−0.958490	−	0.285127i	$-0.907964\pi$
$719$	−2.65729e7	−1.91698	−0.958490	+	0.285127i	$0.907964\pi$
$720$	0	0
$721$	0	0
$722$	1.49658e6	0.106846
$723$	0	0
$724$	5.56312e6	0.394432
$725$	2.52779e7	1.78606
$726$	0	0
$727$	2.16991e6	0.152267	0.0761335	−	0.997098i	$-0.475742\pi$
$727$	2.16991e6	0.152267	0.0761335	+	0.997098i	$0.475742\pi$
$728$	0	0
$729$	0	0
$730$	3.96318e7	2.75256
$731$	679174.	0.0470097
$732$	0	0
$733$	−1.74653e7	−1.20065	−0.600324	−	0.799757i	$-0.704962\pi$
$733$	−1.74653e7	−1.20065	−0.600324	+	0.799757i	$0.704962\pi$
$734$	−1.54016e7	−1.05518
$735$	0	0
$736$	2.51048e6	0.170829
$737$	−9.41203e6	−0.638285
$738$	0	0
$739$	−1.36461e7	−0.919172	−0.459586	−	0.888133i	$-0.652002\pi$
$739$	−1.36461e7	−0.919172	−0.459586	+	0.888133i	$0.652002\pi$
$740$	1.07253e6	0.0719994
$741$	0	0
$742$	0	0
$743$	−1.48965e7	−0.989944	−0.494972	−	0.868909i	$-0.664822\pi$
$743$	−1.48965e7	−0.989944	−0.494972	+	0.868909i	$0.664822\pi$
$744$	0	0
$745$	2.83692e7	1.87265
$746$	−9.80791e6	−0.645252
$747$	0	0
$748$	−2.36571e6	−0.154599
$749$	0	0
$750$	0	0
$751$	−2.53463e7	−1.63989	−0.819944	−	0.572443i	$-0.805996\pi$
$751$	−2.53463e7	−1.63989	−0.819944	+	0.572443i	$0.805996\pi$
$752$	−1.95465e6	−0.126044
$753$	0	0
$754$	1.60810e7	1.03011
$755$	−2.86419e7	−1.82867
$756$	0	0
$757$	−2.66725e7	−1.69170	−0.845852	−	0.533417i	$-0.820908\pi$
$757$	−2.66725e7	−1.69170	−0.845852	+	0.533417i	$0.820908\pi$
$758$	−2.38824e7	−1.50975
$759$	0	0
$760$	1.17715e7	0.739264
$761$	−579829.	−0.0362943	−0.0181471	−	0.999835i	$-0.505777\pi$
$761$	−579829.	−0.0362943	−0.0181471	+	0.999835i	$0.505777\pi$
$762$	0	0
$763$	0	0
$764$	−4.13255e6	−0.256144
$765$	0	0
$766$	2.32357e7	1.43082
$767$	−8.65541e6	−0.531250
$768$	0	0
$769$	1.52438e7	0.929562	0.464781	−	0.885426i	$-0.346133\pi$
$769$	1.52438e7	0.929562	0.464781	+	0.885426i	$0.346133\pi$
$770$	0	0
$771$	0	0
$772$	−1.22140e7	−0.737591
$773$	1.94926e7	1.17333	0.586665	−	0.809830i	$-0.300441\pi$
$773$	1.94926e7	1.17333	0.586665	+	0.809830i	$0.300441\pi$
$774$	0	0
$775$	−8.36275e6	−0.500144
$776$	1.01292e7	0.603839
$777$	0	0
$778$	−2.08807e7	−1.23679
$779$	−1.05794e7	−0.624621
$780$	0	0
$781$	−5.06345e6	−0.297043
$782$	−1.11585e6	−0.0652515
$783$	0	0
$784$	0	0
$785$	3.66337e7	2.12181
$786$	0	0
$787$	1.36277e7	0.784305	0.392153	−	0.919900i	$-0.371730\pi$
$787$	1.36277e7	0.784305	0.392153	+	0.919900i	$0.371730\pi$
$788$	23302.0	0.00133683
$789$	0	0
$790$	1.53143e7	0.873031
$791$	0	0
$792$	0	0
$793$	1.35664e7	0.766093
$794$	490624.	0.0276184
$795$	0	0
$796$	6.70123e6	0.374862
$797$	−3.62853e6	−0.202341	−0.101171	−	0.994869i	$-0.532259\pi$
$797$	−3.62853e6	−0.202341	−0.101171	+	0.994869i	$0.532259\pi$
$798$	0	0
$799$	567208.	0.0314323
$800$	1.91507e7	1.05794
$801$	0	0
$802$	−2.37121e7	−1.30177
$803$	2.46667e7	1.34997
$804$	0	0
$805$	0	0
$806$	−5.32010e6	−0.288458
$807$	0	0
$808$	−2.81223e6	−0.151538
$809$	−2.98780e7	−1.60502	−0.802509	−	0.596640i	$-0.796502\pi$
$809$	−2.98780e7	−1.60502	−0.802509	+	0.596640i	$0.796502\pi$
$810$	0	0
$811$	−1.02643e6	−0.0547995	−0.0273998	−	0.999625i	$-0.508723\pi$
$811$	−1.02643e6	−0.0547995	−0.0273998	+	0.999625i	$0.508723\pi$
$812$	0	0
$813$	0	0
$814$	1.84346e6	0.0975153
$815$	4.01086e7	2.11516
$816$	0	0
$817$	2.76189e6	0.144761
$818$	−2.06262e7	−1.07779
$819$	0	0
$820$	−1.01940e7	−0.529434
$821$	1.15062e7	0.595766	0.297883	−	0.954602i	$-0.403719\pi$
$821$	1.15062e7	0.595766	0.297883	+	0.954602i	$0.403719\pi$
$822$	0	0
$823$	−2.51210e7	−1.29282	−0.646408	−	0.762992i	$-0.723730\pi$
$823$	−2.51210e7	−1.29282	−0.646408	+	0.762992i	$0.723730\pi$
$824$	2.86393e6	0.146942
$825$	0	0
$826$	0	0
$827$	2.14447e6	0.109032	0.0545162	−	0.998513i	$-0.482638\pi$
$827$	2.14447e6	0.109032	0.0545162	+	0.998513i	$0.482638\pi$
$828$	0	0
$829$	818065.	0.0413430	0.0206715	−	0.999786i	$-0.493420\pi$
$829$	818065.	0.0413430	0.0206715	+	0.999786i	$0.493420\pi$
$830$	−4.50937e7	−2.27207
$831$	0	0
$832$	279504.	0.0139984
$833$	0	0
$834$	0	0
$835$	5.40155e7	2.68104
$836$	−9.62025e6	−0.476070
$837$	0	0
$838$	−3.27480e7	−1.61092
$839$	2.11279e7	1.03622	0.518110	−	0.855314i	$-0.326636\pi$
$839$	2.11279e7	1.03622	0.518110	+	0.855314i	$0.326636\pi$
$840$	0	0
$841$	4.00785e7	1.95398
$842$	1.86307e7	0.905624
$843$	0	0
$844$	9.12215e6	0.440799
$845$	−2.28478e7	−1.10079
$846$	0	0
$847$	0	0
$848$	−1.21632e7	−0.580844
$849$	0	0
$850$	−8.51207e6	−0.404099
$851$	314863.	0.0149038
$852$	0	0
$853$	−1.89000e7	−0.889386	−0.444693	−	0.895683i	$-0.646687\pi$
$853$	−1.89000e7	−0.889386	−0.444693	+	0.895683i	$0.646687\pi$
$854$	0	0
$855$	0	0
$856$	−8.60892e6	−0.401573
$857$	−3.72286e7	−1.73151	−0.865753	−	0.500471i	$-0.833160\pi$
$857$	−3.72286e7	−1.73151	−0.865753	+	0.500471i	$0.833160\pi$
$858$	0	0
$859$	3.02064e6	0.139674	0.0698371	−	0.997558i	$-0.477752\pi$
$859$	3.02064e6	0.139674	0.0698371	+	0.997558i	$0.477752\pi$
$860$	2.66129e6	0.122700
$861$	0	0
$862$	5.34131e7	2.44838
$863$	2.71843e7	1.24248	0.621242	−	0.783619i	$-0.286629\pi$
$863$	2.71843e7	1.24248	0.621242	+	0.783619i	$0.286629\pi$
$864$	0	0
$865$	1.98899e7	0.903839
$866$	−4.13320e7	−1.87280
$867$	0	0
$868$	0	0
$869$	9.53158e6	0.428169
$870$	0	0
$871$	−7.80136e6	−0.348438
$872$	2.16185e7	0.962797
$873$	0	0
$874$	−4.53766e6	−0.200934
$875$	0	0
$876$	0	0
$877$	1.73868e7	0.763344	0.381672	−	0.924298i	$-0.375348\pi$
$877$	1.73868e7	0.763344	0.381672	+	0.924298i	$0.375348\pi$
$878$	−1.19064e6	−0.0521248
$879$	0	0
$880$	3.58255e7	1.55950
$881$	8.14472e6	0.353538	0.176769	−	0.984252i	$-0.443435\pi$
$881$	8.14472e6	0.353538	0.176769	+	0.984252i	$0.443435\pi$
$882$	0	0
$883$	−3.10298e7	−1.33930	−0.669649	−	0.742678i	$-0.733555\pi$
$883$	−3.10298e7	−1.33930	−0.669649	+	0.742678i	$0.733555\pi$
$884$	−1.96087e6	−0.0843953
$885$	0	0
$886$	−2.01258e7	−0.861327
$887$	1.47028e7	0.627465	0.313733	−	0.949511i	$-0.398420\pi$
$887$	1.47028e7	0.627465	0.313733	+	0.949511i	$0.398420\pi$
$888$	0	0
$889$	0	0
$890$	2.46386e7	1.04266
$891$	0	0
$892$	−2.13132e7	−0.896885
$893$	2.30657e6	0.0967918
$894$	0	0
$895$	−1.11166e7	−0.463890
$896$	0	0
$897$	0	0
$898$	−1.00269e7	−0.414930
$899$	−2.00450e7	−0.827193
$900$	0	0
$901$	3.52958e6	0.144848
$902$	−1.75215e7	−0.717060
$903$	0	0
$904$	−3.88669e6	−0.158183
$905$	2.44469e7	0.992208
$906$	0	0
$907$	1.30940e7	0.528512	0.264256	−	0.964453i	$-0.414874\pi$
$907$	1.30940e7	0.528512	0.264256	+	0.964453i	$0.414874\pi$
$908$	1.65335e7	0.665504
$909$	0	0
$910$	0	0
$911$	−2.68695e7	−1.07266	−0.536332	−	0.844007i	$-0.680191\pi$
$911$	−2.68695e7	−1.07266	−0.536332	+	0.844007i	$0.680191\pi$
$912$	0	0
$913$	−2.80662e7	−1.11431
$914$	−1.10298e7	−0.436719
$915$	0	0
$916$	9.48103e6	0.373351
$917$	0	0
$918$	0	0
$919$	1.27317e6	0.0497277	0.0248638	−	0.999691i	$-0.492085\pi$
$919$	1.27317e6	0.0497277	0.0248638	+	0.999691i	$0.492085\pi$
$920$	3.32991e6	0.129707
$921$	0	0
$922$	−3.15427e7	−1.22200
$923$	−4.19695e6	−0.162155
$924$	0	0
$925$	2.40187e6	0.0922986
$926$	−3.48658e7	−1.33620
$927$	0	0
$928$	4.59031e7	1.74973
$929$	9.31705e6	0.354192	0.177096	−	0.984194i	$-0.443330\pi$
$929$	9.31705e6	0.354192	0.177096	+	0.984194i	$0.443330\pi$
$930$	0	0
$931$	0	0
$932$	1.89428e7	0.714338
$933$	0	0
$934$	3.60577e7	1.35248
$935$	−1.03960e7	−0.388900
$936$	0	0
$937$	1.18158e7	0.439657	0.219829	−	0.975539i	$-0.429450\pi$
$937$	1.18158e7	0.439657	0.219829	+	0.975539i	$0.429450\pi$
$938$	0	0
$939$	0	0
$940$	2.22256e6	0.0820416
$941$	2.53529e7	0.933371	0.466685	−	0.884423i	$-0.345448\pi$
$941$	2.53529e7	0.933371	0.466685	+	0.884423i	$0.345448\pi$
$942$	0	0
$943$	−2.99268e6	−0.109592
$944$	−3.78436e7	−1.38217
$945$	0	0
$946$	4.57422e6	0.166184
$947$	−2.64941e7	−0.960008	−0.480004	−	0.877266i	$-0.659365\pi$
$947$	−2.64941e7	−0.960008	−0.480004	+	0.877266i	$0.659365\pi$
$948$	0	0
$949$	2.04455e7	0.736941
$950$	−3.46147e7	−1.24437
$951$	0	0
$952$	0	0
$953$	1.88335e7	0.671735	0.335868	−	0.941909i	$-0.390970\pi$
$953$	1.88335e7	0.671735	0.335868	+	0.941909i	$0.390970\pi$
$954$	0	0
$955$	−1.81603e7	−0.644339
$956$	2.78745e7	0.986423
$957$	0	0
$958$	−5.87929e7	−2.06972
$959$	0	0
$960$	0	0
$961$	−2.19976e7	−0.768365
$962$	1.52799e6	0.0532332
$963$	0	0
$964$	−1.83032e7	−0.634357
$965$	−5.36740e7	−1.85544
$966$	0	0
$967$	−3.14956e7	−1.08314	−0.541569	−	0.840656i	$-0.682169\pi$
$967$	−3.14956e7	−1.08314	−0.541569	+	0.840656i	$0.682169\pi$
$968$	−3.64655e6	−0.125082
$969$	0	0
$970$	−5.84475e7	−1.99451
$971$	6.85669e6	0.233381	0.116691	−	0.993168i	$-0.462771\pi$
$971$	6.85669e6	0.233381	0.116691	+	0.993168i	$0.462771\pi$
$972$	0	0
$973$	0	0
$974$	−6.11527e7	−2.06547
$975$	0	0
$976$	5.93157e7	1.99317
$977$	−2.81471e7	−0.943402	−0.471701	−	0.881759i	$-0.656360\pi$
$977$	−2.81471e7	−0.943402	−0.471701	+	0.881759i	$0.656360\pi$
$978$	0	0
$979$	1.53350e7	0.511361
$980$	0	0
$981$	0	0
$982$	673142.	0.0222755
$983$	−2.34916e7	−0.775406	−0.387703	−	0.921784i	$-0.626731\pi$
$983$	−2.34916e7	−0.775406	−0.387703	+	0.921784i	$0.626731\pi$
$984$	0	0
$985$	102399.	0.00336285
$986$	−2.04029e7	−0.668343
$987$	0	0
$988$	−7.97395e6	−0.259885
$989$	781278.	0.0253989
$990$	0	0
$991$	−2.14412e6	−0.0693530	−0.0346765	−	0.999399i	$-0.511040\pi$
$991$	−2.14412e6	−0.0693530	−0.0346765	+	0.999399i	$0.511040\pi$
$992$	−1.51862e7	−0.489971
$993$	0	0
$994$	0	0
$995$	2.94483e7	0.942979
$996$	0	0
$997$	2.50872e7	0.799307	0.399654	−	0.916666i	$-0.369130\pi$
$997$	2.50872e7	0.799307	0.399654	+	0.916666i	$0.369130\pi$
$998$	−1.51715e7	−0.482173
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.n.1.1		2
3.2	odd	2		49.6.a.d.1.2		2
7.2	even	3		63.6.e.d.46.2		4
7.4	even	3		63.6.e.d.37.2		4
7.6	odd	2		441.6.a.m.1.1		2
12.11	even	2		784.6.a.ba.1.2		2
21.2	odd	6		7.6.c.a.4.1	yes	4
21.5	even	6		49.6.c.f.18.1		4
21.11	odd	6		7.6.c.a.2.1	✓	4
21.17	even	6		49.6.c.f.30.1		4
21.20	even	2		49.6.a.e.1.2		2
84.11	even	6		112.6.i.c.65.1		4
84.23	even	6		112.6.i.c.81.1		4
84.83	odd	2		784.6.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.c.a.2.1	✓	4	21.11	odd	6
7.6.c.a.4.1	yes	4	21.2	odd	6
49.6.a.d.1.2		2	3.2	odd	2
49.6.a.e.1.2		2	21.20	even	2
49.6.c.f.18.1		4	21.5	even	6
49.6.c.f.30.1		4	21.17	even	6
63.6.e.d.37.2		4	7.4	even	3
63.6.e.d.46.2		4	7.2	even	3
112.6.i.c.65.1		4	84.11	even	6
112.6.i.c.81.1		4	84.23	even	6
441.6.a.m.1.1		2	7.6	odd	2
441.6.a.n.1.1		2	1.1	even	1	trivial
784.6.a.t.1.1		2	84.83	odd	2
784.6.a.ba.1.2		2	12.11	even	2

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)

\(f(q)\)	\(=\)	\(q-7.08276 q^{2} +18.1655 q^{4} +79.8276 q^{5} +97.9863 q^{8} +O(q^{10})\)	\(q-7.08276 q^{2} +18.1655 q^{4} +79.8276 q^{5} +97.9863 q^{8} -565.400 q^{10} -351.904 q^{11} -291.683 q^{13} -1275.31 q^{16} +370.075 q^{17} +1504.93 q^{19} +1450.11 q^{20} +2492.45 q^{22} +425.711 q^{23} +3247.45 q^{25} +2065.92 q^{26} +7783.93 q^{29} -2575.18 q^{31} +5897.16 q^{32} -2621.16 q^{34} +739.618 q^{37} -10659.0 q^{38} +7822.01 q^{40} -7029.84 q^{41} +1835.23 q^{43} -6392.51 q^{44} -3015.21 q^{46} +1532.68 q^{47} -23000.9 q^{50} -5298.57 q^{52} +9537.46 q^{53} -28091.6 q^{55} -55131.8 q^{58} +29674.1 q^{59} -46510.8 q^{61} +18239.4 q^{62} -958.246 q^{64} -23284.3 q^{65} +26746.1 q^{67} +6722.61 q^{68} +14388.8 q^{71} -70095.1 q^{73} -5238.54 q^{74} +27337.8 q^{76} -27085.8 q^{79} -101805. q^{80} +49790.7 q^{82} +79755.4 q^{83} +29542.2 q^{85} -12998.5 q^{86} -34481.7 q^{88} -43577.3 q^{89} +7733.26 q^{92} -10855.6 q^{94} +120135. q^{95} +103374. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 12 q^{4} + 38 q^{5} - 96 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 12 * q^4 + 38 * q^5 - 96 * q^8	\( 2 q - 2 q^{2} + 12 q^{4} + 38 q^{5} - 96 q^{8} - 778 q^{10} - 424 q^{11} - 924 q^{13} - 2064 q^{16} + 2346 q^{17} - 360 q^{19} + 1708 q^{20} + 2126 q^{22} + 12 q^{23} + 1872 q^{25} - 1148 q^{26} + 7052 q^{29} + 3548 q^{31} + 8096 q^{32} + 7422 q^{34} + 11090 q^{37} - 20138 q^{38} + 15936 q^{40} - 3500 q^{41} - 12680 q^{43} - 5948 q^{44} - 5118 q^{46} + 22956 q^{47} - 29992 q^{50} - 1400 q^{52} - 3042 q^{53} - 25076 q^{55} - 58852 q^{58} + 65808 q^{59} - 42486 q^{61} + 49362 q^{62} + 35456 q^{64} + 3164 q^{65} + 42312 q^{67} - 5460 q^{68} + 2208 q^{71} - 50506 q^{73} + 47370 q^{74} + 38836 q^{76} + 9004 q^{79} - 68816 q^{80} + 67732 q^{82} + 104328 q^{83} - 53106 q^{85} - 86776 q^{86} - 20496 q^{88} + 26666 q^{89} + 10284 q^{92} + 98034 q^{94} + 198140 q^{95} + 209132 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 12 * q^4 + 38 * q^5 - 96 * q^8 - 778 * q^10 - 424 * q^11 - 924 * q^13 - 2064 * q^16 + 2346 * q^17 - 360 * q^19 + 1708 * q^20 + 2126 * q^22 + 12 * q^23 + 1872 * q^25 - 1148 * q^26 + 7052 * q^29 + 3548 * q^31 + 8096 * q^32 + 7422 * q^34 + 11090 * q^37 - 20138 * q^38 + 15936 * q^40 - 3500 * q^41 - 12680 * q^43 - 5948 * q^44 - 5118 * q^46 + 22956 * q^47 - 29992 * q^50 - 1400 * q^52 - 3042 * q^53 - 25076 * q^55 - 58852 * q^58 + 65808 * q^59 - 42486 * q^61 + 49362 * q^62 + 35456 * q^64 + 3164 * q^65 + 42312 * q^67 - 5460 * q^68 + 2208 * q^71 - 50506 * q^73 + 47370 * q^74 + 38836 * q^76 + 9004 * q^79 - 68816 * q^80 + 67732 * q^82 + 104328 * q^83 - 53106 * q^85 - 86776 * q^86 - 20496 * q^88 + 26666 * q^89 + 10284 * q^92 + 98034 * q^94 + 198140 * q^95 + 209132 * q^97

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