LMFDB - Embedded newform 225.6.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,6,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, 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("225.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	225.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:	$36.0863594579$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{70}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 70 $ x^2 - 70
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.36660$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.36660 q^{2} +38.0000 q^{4} -45.0000 q^{7} +50.1996 q^{8} +O(q^{10})$	$q+8.36660 q^{2} +38.0000 q^{4} -45.0000 q^{7} +50.1996 q^{8} -334.664 q^{11} -1045.00 q^{13} -376.497 q^{14} -796.000 q^{16} -1271.72 q^{17} +1159.00 q^{19} -2800.00 q^{22} +3815.17 q^{23} -8743.10 q^{26} -1710.00 q^{28} -3681.30 q^{29} +3633.00 q^{31} -8266.20 q^{32} -10640.0 q^{34} -3110.00 q^{37} +9696.89 q^{38} -17737.2 q^{41} +10355.0 q^{43} -12717.2 q^{44} +31920.0 q^{46} -13520.4 q^{47} -14782.0 q^{49} -39710.0 q^{52} +9972.99 q^{53} -2258.98 q^{56} -30800.0 q^{58} +28781.1 q^{59} -7613.00 q^{61} +30395.9 q^{62} -43688.0 q^{64} -50445.0 q^{67} -48325.5 q^{68} +80654.0 q^{71} -74710.0 q^{73} -26020.1 q^{74} +44042.0 q^{76} +15059.9 q^{77} +38316.0 q^{79} -148400. q^{82} -35742.1 q^{83} +86636.1 q^{86} -16800.0 q^{88} +120479. q^{89} +47025.0 q^{91} +144976. q^{92} -113120. q^{94} +71755.0 q^{97} -123675. q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 76 q^{4} - 90 q^{7}+O(q^{10}) $ 2 * q + 76 * q^4 - 90 * q^7	$ 2 q + 76 q^{4} - 90 q^{7} - 2090 q^{13} - 1592 q^{16} + 2318 q^{19} - 5600 q^{22} - 3420 q^{28} + 7266 q^{31} - 21280 q^{34} - 6220 q^{37} + 20710 q^{43} + 63840 q^{46} - 29564 q^{49} - 79420 q^{52} - 61600 q^{58} - 15226 q^{61} - 87376 q^{64} - 100890 q^{67} - 149420 q^{73} + 88084 q^{76} + 76632 q^{79} - 296800 q^{82} - 33600 q^{88} + 94050 q^{91} - 226240 q^{94} + 143510 q^{97}+O(q^{100}) $ 2 * q + 76 * q^4 - 90 * q^7 - 2090 * q^13 - 1592 * q^16 + 2318 * q^19 - 5600 * q^22 - 3420 * q^28 + 7266 * q^31 - 21280 * q^34 - 6220 * q^37 + 20710 * q^43 + 63840 * q^46 - 29564 * q^49 - 79420 * q^52 - 61600 * q^58 - 15226 * q^61 - 87376 * q^64 - 100890 * q^67 - 149420 * q^73 + 88084 * q^76 + 76632 * q^79 - 296800 * q^82 - 33600 * q^88 + 94050 * q^91 - 226240 * q^94 + 143510 * 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 $

$2$	8.36660	1.47902	0.739510	−	0.673146i	$-0.235057\pi$
$2$	8.36660	1.47902	0.739510	+	0.673146i	$0.235057\pi$
$3$	0	0
$3$	0	0
$4$	38.0000	1.18750
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−45.0000	−0.347110	−0.173555	−	0.984824i	$-0.555525\pi$
$7$	−45.0000	−0.347110	−0.173555	+	0.984824i	$0.555525\pi$
$8$	50.1996	0.277316
$9$	0	0
$10$	0	0
$11$	−334.664	−0.833926	−0.416963	−	0.908924i	$-0.636905\pi$
$11$	−334.664	−0.833926	−0.416963	+	0.908924i	$0.636905\pi$
$12$	0	0
$13$	−1045.00	−1.71498	−0.857488	−	0.514504i	$-0.827976\pi$
$13$	−1045.00	−1.71498	−0.857488	+	0.514504i	$0.827976\pi$
$14$	−376.497	−0.513383
$15$	0	0
$16$	−796.000	−0.777344
$17$	−1271.72	−1.06726	−0.533630	−	0.845718i	$-0.679173\pi$
$17$	−1271.72	−1.06726	−0.533630	+	0.845718i	$0.679173\pi$
$18$	0	0
$19$	1159.00	0.736545	0.368273	−	0.929718i	$-0.379949\pi$
$19$	1159.00	0.736545	0.368273	+	0.929718i	$0.379949\pi$
$20$	0	0
$21$	0	0
$22$	−2800.00	−1.23339
$23$	3815.17	1.50381	0.751907	−	0.659269i	$-0.229134\pi$
$23$	3815.17	1.50381	0.751907	+	0.659269i	$0.229134\pi$
$24$	0	0
$25$	0	0
$26$	−8743.10	−2.53648
$27$	0	0
$28$	−1710.00	−0.412193
$29$	−3681.30	−0.812843	−0.406422	−	0.913686i	$-0.633224\pi$
$29$	−3681.30	−0.812843	−0.406422	+	0.913686i	$0.633224\pi$
$30$	0	0
$31$	3633.00	0.678987	0.339493	−	0.940608i	$-0.389744\pi$
$31$	3633.00	0.678987	0.339493	+	0.940608i	$0.389744\pi$
$32$	−8266.20	−1.42702
$33$	0	0
$34$	−10640.0	−1.57850
$35$	0	0
$36$	0	0
$37$	−3110.00	−0.373470	−0.186735	−	0.982410i	$-0.559791\pi$
$37$	−3110.00	−0.373470	−0.186735	+	0.982410i	$0.559791\pi$
$38$	9696.89	1.08937
$39$	0	0
$40$	0	0
$41$	−17737.2	−1.64788	−0.823939	−	0.566678i	$-0.808228\pi$
$41$	−17737.2	−1.64788	−0.823939	+	0.566678i	$0.808228\pi$
$42$	0	0
$43$	10355.0	0.854041	0.427021	−	0.904242i	$-0.359563\pi$
$43$	10355.0	0.854041	0.427021	+	0.904242i	$0.359563\pi$
$44$	−12717.2	−0.990287
$45$	0	0
$46$	31920.0	2.22417
$47$	−13520.4	−0.892783	−0.446391	−	0.894838i	$-0.647291\pi$
$47$	−13520.4	−0.892783	−0.446391	+	0.894838i	$0.647291\pi$
$48$	0	0
$49$	−14782.0	−0.879514
$50$	0	0
$51$	0	0
$52$	−39710.0	−2.03653
$53$	9972.99	0.487681	0.243840	−	0.969815i	$-0.421593\pi$
$53$	9972.99	0.487681	0.243840	+	0.969815i	$0.421593\pi$
$54$	0	0
$55$	0	0
$56$	−2258.98	−0.0962593
$57$	0	0
$58$	−30800.0	−1.20221
$59$	28781.1	1.07641	0.538205	−	0.842814i	$-0.319103\pi$
$59$	28781.1	1.07641	0.538205	+	0.842814i	$0.319103\pi$
$60$	0	0
$61$	−7613.00	−0.261958	−0.130979	−	0.991385i	$-0.541812\pi$
$61$	−7613.00	−0.261958	−0.130979	+	0.991385i	$0.541812\pi$
$62$	30395.9	1.00423
$63$	0	0
$64$	−43688.0	−1.33325
$65$	0	0
$66$	0	0
$67$	−50445.0	−1.37288	−0.686438	−	0.727189i	$-0.740827\pi$
$67$	−50445.0	−1.37288	−0.686438	+	0.727189i	$0.740827\pi$
$68$	−48325.5	−1.26737
$69$	0	0
$70$	0	0
$71$	80654.0	1.89880	0.949402	−	0.314063i	$-0.101690\pi$
$71$	80654.0	1.89880	0.949402	+	0.314063i	$0.101690\pi$
$72$	0	0
$73$	−74710.0	−1.64086	−0.820430	−	0.571747i	$-0.806266\pi$
$73$	−74710.0	−1.64086	−0.820430	+	0.571747i	$0.806266\pi$
$74$	−26020.1	−0.552370
$75$	0	0
$76$	44042.0	0.874647
$77$	15059.9	0.289464
$78$	0	0
$79$	38316.0	0.690737	0.345368	−	0.938467i	$-0.387754\pi$
$79$	38316.0	0.690737	0.345368	+	0.938467i	$0.387754\pi$
$80$	0	0
$81$	0	0
$82$	−148400.	−2.43725
$83$	−35742.1	−0.569489	−0.284744	−	0.958604i	$-0.591909\pi$
$83$	−35742.1	−0.569489	−0.284744	+	0.958604i	$0.591909\pi$
$84$	0	0
$85$	0	0
$86$	86636.1	1.26314
$87$	0	0
$88$	−16800.0	−0.231261
$89$	120479.	1.61227	0.806133	−	0.591735i	$-0.201557\pi$
$89$	120479.	1.61227	0.806133	+	0.591735i	$0.201557\pi$
$90$	0	0
$91$	47025.0	0.595286
$92$	144976.	1.78578
$93$	0	0
$94$	−113120.	−1.32044
$95$	0	0
$96$	0	0
$97$	71755.0	0.774324	0.387162	−	0.922012i	$-0.373455\pi$
$97$	71755.0	0.774324	0.387162	+	0.922012i	$0.373455\pi$
$98$	−123675.	−1.30082
$99$	0	0
$100$	0	0
$101$	93371.3	0.910772	0.455386	−	0.890294i	$-0.349501\pi$
$101$	93371.3	0.910772	0.455386	+	0.890294i	$0.349501\pi$
$102$	0	0
$103$	33820.0	0.314109	0.157055	−	0.987590i	$-0.449800\pi$
$103$	33820.0	0.314109	0.157055	+	0.987590i	$0.449800\pi$
$104$	−52458.6	−0.475591
$105$	0	0
$106$	83440.0	0.721290
$107$	−79717.0	−0.673118	−0.336559	−	0.941662i	$-0.609263\pi$
$107$	−79717.0	−0.673118	−0.336559	+	0.941662i	$0.609263\pi$
$108$	0	0
$109$	−114019.	−0.919202	−0.459601	−	0.888125i	$-0.652008\pi$
$109$	−114019.	−0.919202	−0.459601	+	0.888125i	$0.652008\pi$
$110$	0	0
$111$	0	0
$112$	35820.0	0.269824
$113$	199727.	1.47144	0.735719	−	0.677287i	$-0.236845\pi$
$113$	199727.	1.47144	0.735719	+	0.677287i	$0.236845\pi$
$114$	0	0
$115$	0	0
$116$	−139890.	−0.965251
$117$	0	0
$118$	240800.	1.59203
$119$	57227.5	0.370457
$120$	0	0
$121$	−49051.0	−0.304568
$122$	−63694.9	−0.387441
$123$	0	0
$124$	138054.	0.806297
$125$	0	0
$126$	0	0
$127$	−118180.	−0.650182	−0.325091	−	0.945683i	$-0.605395\pi$
$127$	−118180.	−0.650182	−0.325091	+	0.945683i	$0.605395\pi$
$128$	−101002.	−0.544883
$129$	0	0
$130$	0	0
$131$	−113451.	−0.577604	−0.288802	−	0.957389i	$-0.593257\pi$
$131$	−113451.	−0.577604	−0.288802	+	0.957389i	$0.593257\pi$
$132$	0	0
$133$	−52155.0	−0.255662
$134$	−422053.	−2.03051
$135$	0	0
$136$	−63840.0	−0.295969
$137$	−242966.	−1.10597	−0.552986	−	0.833190i	$-0.686512\pi$
$137$	−242966.	−1.10597	−0.552986	+	0.833190i	$0.686512\pi$
$138$	0	0
$139$	212144.	0.931309	0.465654	−	0.884967i	$-0.345819\pi$
$139$	212144.	0.931309	0.465654	+	0.884967i	$0.345819\pi$
$140$	0	0
$141$	0	0
$142$	674800.	2.80837
$143$	349724.	1.43016
$144$	0	0
$145$	0	0
$146$	−625069.	−2.42686
$147$	0	0
$148$	−118180.	−0.443496
$149$	437071.	1.61282	0.806411	−	0.591355i	$-0.201407\pi$
$149$	437071.	1.61282	0.806411	+	0.591355i	$0.201407\pi$
$150$	0	0
$151$	−519137.	−1.85285	−0.926424	−	0.376483i	$-0.877133\pi$
$151$	−519137.	−1.85285	−0.926424	+	0.376483i	$0.877133\pi$
$152$	58181.3	0.204256
$153$	0	0
$154$	126000.	0.428123
$155$	0	0
$156$	0	0
$157$	−71975.0	−0.233041	−0.116521	−	0.993188i	$-0.537174\pi$
$157$	−71975.0	−0.233041	−0.116521	+	0.993188i	$0.537174\pi$
$158$	320575.	1.02161
$159$	0	0
$160$	0	0
$161$	−171683.	−0.521989
$162$	0	0
$163$	−436175.	−1.28585	−0.642927	−	0.765927i	$-0.722280\pi$
$163$	−436175.	−1.28585	−0.642927	+	0.765927i	$0.722280\pi$
$164$	−674013.	−1.95686
$165$	0	0
$166$	−299040.	−0.842285
$167$	−610226.	−1.69317	−0.846583	−	0.532256i	$-0.821344\pi$
$167$	−610226.	−1.69317	−0.846583	+	0.532256i	$0.821344\pi$
$168$	0	0
$169$	720732.	1.94114
$170$	0	0
$171$	0	0
$172$	393490.	1.01417
$173$	−73291.4	−0.186182	−0.0930910	−	0.995658i	$-0.529675\pi$
$173$	−73291.4	−0.186182	−0.0930910	+	0.995658i	$0.529675\pi$
$174$	0	0
$175$	0	0
$176$	266393.	0.648247
$177$	0	0
$178$	1.00800e6	2.38457
$179$	223556.	0.521498	0.260749	−	0.965407i	$-0.416030\pi$
$179$	223556.	0.521498	0.260749	+	0.965407i	$0.416030\pi$
$180$	0	0
$181$	141113.	0.320163	0.160081	−	0.987104i	$-0.448824\pi$
$181$	141113.	0.320163	0.160081	+	0.987104i	$0.448824\pi$
$182$	393439.	0.880439
$183$	0	0
$184$	191520.	0.417032
$185$	0	0
$186$	0	0
$187$	425600.	0.890016
$188$	−513776.	−1.06018
$189$	0	0
$190$	0	0
$191$	269405.	0.534345	0.267172	−	0.963649i	$-0.413911\pi$
$191$	269405.	0.534345	0.267172	+	0.963649i	$0.413911\pi$
$192$	0	0
$193$	−93575.0	−0.180828	−0.0904142	−	0.995904i	$-0.528819\pi$
$193$	−93575.0	−0.180828	−0.0904142	+	0.995904i	$0.528819\pi$
$194$	600345.	1.14524
$195$	0	0
$196$	−561716.	−1.04442
$197$	373887.	0.686395	0.343198	−	0.939263i	$-0.388490\pi$
$197$	373887.	0.686395	0.343198	+	0.939263i	$0.388490\pi$
$198$	0	0
$199$	−517469.	−0.926300	−0.463150	−	0.886280i	$-0.653281\pi$
$199$	−517469.	−0.926300	−0.463150	+	0.886280i	$0.653281\pi$
$200$	0	0
$201$	0	0
$202$	781200.	1.34705
$203$	165659.	0.282146
$204$	0	0
$205$	0	0
$206$	282958.	0.464574
$207$	0	0
$208$	831820.	1.33313
$209$	−387876.	−0.614224
$210$	0	0
$211$	22667.0	0.0350500	0.0175250	−	0.999846i	$-0.494421\pi$
$211$	22667.0	0.0350500	0.0175250	+	0.999846i	$0.494421\pi$
$212$	378974.	0.579121
$213$	0	0
$214$	−666960.	−0.995555
$215$	0	0
$216$	0	0
$217$	−163485.	−0.235683
$218$	−953951.	−1.35952
$219$	0	0
$220$	0	0
$221$	1.32895e6	1.83033
$222$	0	0
$223$	−377645.	−0.508536	−0.254268	−	0.967134i	$-0.581835\pi$
$223$	−377645.	−0.508536	−0.254268	+	0.967134i	$0.581835\pi$
$224$	371979.	0.495334
$225$	0	0
$226$	1.67104e6	2.17628
$227$	1.23417e6	1.58969	0.794844	−	0.606814i	$-0.207553\pi$
$227$	1.23417e6	1.58969	0.794844	+	0.606814i	$0.207553\pi$
$228$	0	0
$229$	918669.	1.15763	0.578816	−	0.815458i	$-0.303515\pi$
$229$	918669.	1.15763	0.578816	+	0.815458i	$0.303515\pi$
$230$	0	0
$231$	0	0
$232$	−184800.	−0.225415
$233$	−515048.	−0.621524	−0.310762	−	0.950488i	$-0.600584\pi$
$233$	−515048.	−0.621524	−0.310762	+	0.950488i	$0.600584\pi$
$234$	0	0
$235$	0	0
$236$	1.09368e6	1.27824
$237$	0	0
$238$	478800.	0.547913
$239$	1.41128e6	1.59815	0.799076	−	0.601231i	$-0.205323\pi$
$239$	1.41128e6	1.59815	0.799076	+	0.601231i	$0.205323\pi$
$240$	0	0
$241$	−980783.	−1.08775	−0.543877	−	0.839165i	$-0.683044\pi$
$241$	−980783.	−1.08775	−0.543877	+	0.839165i	$0.683044\pi$
$242$	−410390.	−0.450462
$243$	0	0
$244$	−289294.	−0.311075
$245$	0	0
$246$	0	0
$247$	−1.21115e6	−1.26316
$248$	182375.	0.188294
$249$	0	0
$250$	0	0
$251$	−14055.9	−0.0140823	−0.00704116	−	0.999975i	$-0.502241\pi$
$251$	−14055.9	−0.0140823	−0.00704116	+	0.999975i	$0.502241\pi$
$252$	0	0
$253$	−1.27680e6	−1.25407
$254$	−988765.	−0.961632
$255$	0	0
$256$	552976.	0.527359
$257$	−1.01724e6	−0.960711	−0.480355	−	0.877074i	$-0.659492\pi$
$257$	−1.01724e6	−0.960711	−0.480355	+	0.877074i	$0.659492\pi$
$258$	0	0
$259$	139950.	0.129635
$260$	0	0
$261$	0	0
$262$	−949200.	−0.854288
$263$	−784653.	−0.699501	−0.349751	−	0.936843i	$-0.613734\pi$
$263$	−784653.	−0.699501	−0.349751	+	0.936843i	$0.613734\pi$
$264$	0	0
$265$	0	0
$266$	−436360.	−0.378130
$267$	0	0
$268$	−1.91691e6	−1.63029
$269$	−1.37982e6	−1.16263	−0.581315	−	0.813678i	$-0.697462\pi$
$269$	−1.37982e6	−1.16263	−0.581315	+	0.813678i	$0.697462\pi$
$270$	0	0
$271$	−1.40731e6	−1.16403	−0.582017	−	0.813176i	$-0.697736\pi$
$271$	−1.40731e6	−1.16403	−0.582017	+	0.813176i	$0.697736\pi$
$272$	1.01229e6	0.829628
$273$	0	0
$274$	−2.03280e6	−1.63575
$275$	0	0
$276$	0	0
$277$	−1.21384e6	−0.950526	−0.475263	−	0.879844i	$-0.657647\pi$
$277$	−1.21384e6	−0.950526	−0.475263	+	0.879844i	$0.657647\pi$
$278$	1.77492e6	1.37742
$279$	0	0
$280$	0	0
$281$	−2.38448e6	−1.80147	−0.900737	−	0.434365i	$-0.856973\pi$
$281$	−2.38448e6	−1.80147	−0.900737	+	0.434365i	$0.856973\pi$
$282$	0	0
$283$	−2.30498e6	−1.71080	−0.855402	−	0.517965i	$-0.826690\pi$
$283$	−2.30498e6	−1.71080	−0.855402	+	0.517965i	$0.826690\pi$
$284$	3.06485e6	2.25483
$285$	0	0
$286$	2.92600e6	2.11524
$287$	798174.	0.571996
$288$	0	0
$289$	197423.	0.139044
$290$	0	0
$291$	0	0
$292$	−2.83898e6	−1.94852
$293$	1.15981e6	0.789257	0.394628	−	0.918841i	$-0.370873\pi$
$293$	1.15981e6	0.789257	0.394628	+	0.918841i	$0.370873\pi$
$294$	0	0
$295$	0	0
$296$	−156121.	−0.103569
$297$	0	0
$298$	3.65680e6	2.38540
$299$	−3.98685e6	−2.57900
$300$	0	0
$301$	−465975.	−0.296447
$302$	−4.34341e6	−2.74040
$303$	0	0
$304$	−922564.	−0.572549
$305$	0	0
$306$	0	0
$307$	122885.	0.0744137	0.0372069	−	0.999308i	$-0.488154\pi$
$307$	122885.	0.0744137	0.0372069	+	0.999308i	$0.488154\pi$
$308$	572275.	0.343739
$309$	0	0
$310$	0	0
$311$	−1.49796e6	−0.878209	−0.439105	−	0.898436i	$-0.644704\pi$
$311$	−1.49796e6	−0.878209	−0.439105	+	0.898436i	$0.644704\pi$
$312$	0	0
$313$	−858515.	−0.495321	−0.247661	−	0.968847i	$-0.579662\pi$
$313$	−858515.	−0.495321	−0.247661	+	0.968847i	$0.579662\pi$
$314$	−602186.	−0.344672
$315$	0	0
$316$	1.45601e6	0.820250
$317$	−95713.9	−0.0534967	−0.0267483	−	0.999642i	$-0.508515\pi$
$317$	−95713.9	−0.0534967	−0.0267483	+	0.999642i	$0.508515\pi$
$318$	0	0
$319$	1.23200e6	0.677851
$320$	0	0
$321$	0	0
$322$	−1.43640e6	−0.772033
$323$	−1.47393e6	−0.786085
$324$	0	0
$325$	0	0
$326$	−3.64930e6	−1.90180
$327$	0	0
$328$	−890400.	−0.456984
$329$	608419.	0.309894
$330$	0	0
$331$	3.67171e6	1.84204	0.921019	−	0.389517i	$-0.127358\pi$
$331$	3.67171e6	1.84204	0.921019	+	0.389517i	$0.127358\pi$
$332$	−1.35820e6	−0.676268
$333$	0	0
$334$	−5.10552e6	−2.50423
$335$	0	0
$336$	0	0
$337$	1.04775e6	0.502557	0.251278	−	0.967915i	$-0.419149\pi$
$337$	1.04775e6	0.502557	0.251278	+	0.967915i	$0.419149\pi$
$338$	6.03008e6	2.87099
$339$	0	0
$340$	0	0
$341$	−1.21583e6	−0.566224
$342$	0	0
$343$	1.42150e6	0.652399
$344$	519817.	0.236840
$345$	0	0
$346$	−613200.	−0.275367
$347$	−104281.	−0.0464925	−0.0232462	−	0.999730i	$-0.507400\pi$
$347$	−104281.	−0.0464925	−0.0232462	+	0.999730i	$0.507400\pi$
$348$	0	0
$349$	−1.79031e6	−0.786799	−0.393399	−	0.919368i	$-0.628701\pi$
$349$	−1.79031e6	−0.786799	−0.393399	+	0.919368i	$0.628701\pi$
$350$	0	0
$351$	0	0
$352$	2.76640e6	1.19003
$353$	−2.45155e6	−1.04714	−0.523569	−	0.851984i	$-0.675400\pi$
$353$	−2.45155e6	−1.04714	−0.523569	+	0.851984i	$0.675400\pi$
$354$	0	0
$355$	0	0
$356$	4.57820e6	1.91456
$357$	0	0
$358$	1.87040e6	0.771306
$359$	−1.58330e6	−0.648374	−0.324187	−	0.945993i	$-0.605091\pi$
$359$	−1.58330e6	−0.648374	−0.324187	+	0.945993i	$0.605091\pi$
$360$	0	0
$361$	−1.13282e6	−0.457501
$362$	1.18064e6	0.473527
$363$	0	0
$364$	1.78695e6	0.706902
$365$	0	0
$366$	0	0
$367$	−4.58672e6	−1.77761	−0.888805	−	0.458285i	$-0.848464\pi$
$367$	−4.58672e6	−1.77761	−0.888805	+	0.458285i	$0.848464\pi$
$368$	−3.03688e6	−1.16898
$369$	0	0
$370$	0	0
$371$	−448784.	−0.169279
$372$	0	0
$373$	−2.23517e6	−0.831839	−0.415920	−	0.909401i	$-0.636540\pi$
$373$	−2.23517e6	−0.831839	−0.415920	+	0.909401i	$0.636540\pi$
$374$	3.56083e6	1.31635
$375$	0	0
$376$	−678720.	−0.247583
$377$	3.84696e6	1.39401
$378$	0	0
$379$	4.68359e6	1.67487	0.837434	−	0.546538i	$-0.184055\pi$
$379$	4.68359e6	1.67487	0.837434	+	0.546538i	$0.184055\pi$
$380$	0	0
$381$	0	0
$382$	2.25400e6	0.790306
$383$	1.49675e6	0.521378	0.260689	−	0.965423i	$-0.416050\pi$
$383$	1.49675e6	0.521378	0.260689	+	0.965423i	$0.416050\pi$
$384$	0	0
$385$	0	0
$386$	−782905.	−0.267449
$387$	0	0
$388$	2.72669e6	0.919510
$389$	−3.18567e6	−1.06740	−0.533699	−	0.845675i	$-0.679198\pi$
$389$	−3.18567e6	−1.06740	−0.533699	+	0.845675i	$0.679198\pi$
$390$	0	0
$391$	−4.85184e6	−1.60496
$392$	−742051.	−0.243904
$393$	0	0
$394$	3.12816e6	1.01519
$395$	0	0
$396$	0	0
$397$	3.06708e6	0.976674	0.488337	−	0.872655i	$-0.337604\pi$
$397$	3.06708e6	0.976674	0.488337	+	0.872655i	$0.337604\pi$
$398$	−4.32946e6	−1.37002
$399$	0	0
$400$	0	0
$401$	4.40050e6	1.36660	0.683299	−	0.730139i	$-0.260545\pi$
$401$	4.40050e6	1.36660	0.683299	+	0.730139i	$0.260545\pi$
$402$	0	0
$403$	−3.79648e6	−1.16445
$404$	3.54811e6	1.08154
$405$	0	0
$406$	1.38600e6	0.417300
$407$	1.04081e6	0.311446
$408$	0	0
$409$	1.11741e6	0.330296	0.165148	−	0.986269i	$-0.447190\pi$
$409$	1.11741e6	0.330296	0.165148	+	0.986269i	$0.447190\pi$
$410$	0	0
$411$	0	0
$412$	1.28516e6	0.373005
$413$	−1.29515e6	−0.373633
$414$	0	0
$415$	0	0
$416$	8.63818e6	2.44731
$417$	0	0
$418$	−3.24520e6	−0.908449
$419$	2.33361e6	0.649372	0.324686	−	0.945822i	$-0.394741\pi$
$419$	2.33361e6	0.649372	0.324686	+	0.945822i	$0.394741\pi$
$420$	0	0
$421$	−6.29946e6	−1.73220	−0.866100	−	0.499870i	$-0.833381\pi$
$421$	−6.29946e6	−1.73220	−0.866100	+	0.499870i	$0.833381\pi$
$422$	189646.	0.0518396
$423$	0	0
$424$	500640.	0.135242
$425$	0	0
$426$	0	0
$427$	342585.	0.0909282
$428$	−3.02924e6	−0.799328
$429$	0	0
$430$	0	0
$431$	3.68766e6	0.956220	0.478110	−	0.878300i	$-0.341322\pi$
$431$	3.68766e6	0.956220	0.478110	+	0.878300i	$0.341322\pi$
$432$	0	0
$433$	3.35268e6	0.859356	0.429678	−	0.902982i	$-0.358627\pi$
$433$	3.35268e6	0.859356	0.429678	+	0.902982i	$0.358627\pi$
$434$	−1.36781e6	−0.348580
$435$	0	0
$436$	−4.33272e6	−1.09155
$437$	4.42178e6	1.10763
$438$	0	0
$439$	−2.47344e6	−0.612548	−0.306274	−	0.951943i	$-0.599082\pi$
$439$	−2.47344e6	−0.612548	−0.306274	+	0.951943i	$0.599082\pi$
$440$	0	0
$441$	0	0
$442$	1.11188e7	2.70709
$443$	−2.83086e6	−0.685344	−0.342672	−	0.939455i	$-0.611332\pi$
$443$	−2.83086e6	−0.685344	−0.342672	+	0.939455i	$0.611332\pi$
$444$	0	0
$445$	0	0
$446$	−3.15960e6	−0.752135
$447$	0	0
$448$	1.96596e6	0.462785
$449$	2.23355e6	0.522853	0.261426	−	0.965223i	$-0.415807\pi$
$449$	2.23355e6	0.522853	0.261426	+	0.965223i	$0.415807\pi$
$450$	0	0
$451$	5.93600e6	1.37421
$452$	7.58964e6	1.74733
$453$	0	0
$454$	1.03258e7	2.35118
$455$	0	0
$456$	0	0
$457$	1.05809e6	0.236991	0.118496	−	0.992955i	$-0.462193\pi$
$457$	1.05809e6	0.236991	0.118496	+	0.992955i	$0.462193\pi$
$458$	7.68614e6	1.71216
$459$	0	0
$460$	0	0
$461$	−2.35838e6	−0.516846	−0.258423	−	0.966032i	$-0.583203\pi$
$461$	−2.35838e6	−0.516846	−0.258423	+	0.966032i	$0.583203\pi$
$462$	0	0
$463$	733020.	0.158914	0.0794572	−	0.996838i	$-0.474681\pi$
$463$	733020.	0.158914	0.0794572	+	0.996838i	$0.474681\pi$
$464$	2.93032e6	0.631858
$465$	0	0
$466$	−4.30920e6	−0.919246
$467$	−5.47129e6	−1.16091	−0.580453	−	0.814293i	$-0.697125\pi$
$467$	−5.47129e6	−1.16091	−0.580453	+	0.814293i	$0.697125\pi$
$468$	0	0
$469$	2.27002e6	0.476539
$470$	0	0
$471$	0	0
$472$	1.44480e6	0.298506
$473$	−3.46545e6	−0.712207
$474$	0	0
$475$	0	0
$476$	2.17465e6	0.439918
$477$	0	0
$478$	1.18076e7	2.36370
$479$	−596371.	−0.118762	−0.0593811	−	0.998235i	$-0.518913\pi$
$479$	−596371.	−0.118762	−0.0593811	+	0.998235i	$0.518913\pi$
$480$	0	0
$481$	3.24995e6	0.640492
$482$	−8.20582e6	−1.60881
$483$	0	0
$484$	−1.86394e6	−0.361675
$485$	0	0
$486$	0	0
$487$	3.16142e6	0.604033	0.302016	−	0.953303i	$-0.402340\pi$
$487$	3.16142e6	0.604033	0.302016	+	0.953303i	$0.402340\pi$
$488$	−382170.	−0.0726451
$489$	0	0
$490$	0	0
$491$	−2.18736e6	−0.409466	−0.204733	−	0.978818i	$-0.565633\pi$
$491$	−2.18736e6	−0.409466	−0.204733	+	0.978818i	$0.565633\pi$
$492$	0	0
$493$	4.68160e6	0.867515
$494$	−1.01332e7	−1.86823
$495$	0	0
$496$	−2.89187e6	−0.527806
$497$	−3.62943e6	−0.659094
$498$	0	0
$499$	−7.42611e6	−1.33509	−0.667544	−	0.744570i	$-0.732654\pi$
$499$	−7.42611e6	−1.33509	−0.667544	+	0.744570i	$0.732654\pi$
$500$	0	0
$501$	0	0
$502$	−117600.	−0.0208280
$503$	2.14620e6	0.378225	0.189113	−	0.981955i	$-0.439439\pi$
$503$	2.14620e6	0.378225	0.189113	+	0.981955i	$0.439439\pi$
$504$	0	0
$505$	0	0
$506$	−1.06825e7	−1.85479
$507$	0	0
$508$	−4.49084e6	−0.772091
$509$	1.38216e6	0.236464	0.118232	−	0.992986i	$-0.462277\pi$
$509$	1.38216e6	0.236464	0.118232	+	0.992986i	$0.462277\pi$
$510$	0	0
$511$	3.36195e6	0.569559
$512$	7.85858e6	1.32486
$513$	0	0
$514$	−8.51088e6	−1.42091
$515$	0	0
$516$	0	0
$517$	4.52480e6	0.744514
$518$	1.17091e6	0.191733
$519$	0	0
$520$	0	0
$521$	−3.29477e6	−0.531778	−0.265889	−	0.964004i	$-0.585665\pi$
$521$	−3.29477e6	−0.531778	−0.265889	+	0.964004i	$0.585665\pi$
$522$	0	0
$523$	1.66848e6	0.266728	0.133364	−	0.991067i	$-0.457422\pi$
$523$	1.66848e6	0.266728	0.133364	+	0.991067i	$0.457422\pi$
$524$	−4.31114e6	−0.685905
$525$	0	0
$526$	−6.56488e6	−1.03458
$527$	−4.62017e6	−0.724655
$528$	0	0
$529$	8.11918e6	1.26146
$530$	0	0
$531$	0	0
$532$	−1.98189e6	−0.303599
$533$	1.85354e7	2.82607
$534$	0	0
$535$	0	0
$536$	−2.53232e6	−0.380721
$537$	0	0
$538$	−1.15444e7	−1.71955
$539$	4.94700e6	0.733450
$540$	0	0
$541$	1.70322e6	0.250194	0.125097	−	0.992145i	$-0.460076\pi$
$541$	1.70322e6	0.250194	0.125097	+	0.992145i	$0.460076\pi$
$542$	−1.17744e7	−1.72163
$543$	0	0
$544$	1.05123e7	1.52300
$545$	0	0
$546$	0	0
$547$	404360.	0.0577830	0.0288915	−	0.999583i	$-0.490802\pi$
$547$	404360.	0.0577830	0.0288915	+	0.999583i	$0.490802\pi$
$548$	−9.23271e6	−1.31334
$549$	0	0
$550$	0	0
$551$	−4.26663e6	−0.598696
$552$	0	0
$553$	−1.72422e6	−0.239762
$554$	−1.01558e7	−1.40585
$555$	0	0
$556$	8.06147e6	1.10593
$557$	−1.04937e7	−1.43315	−0.716575	−	0.697510i	$-0.754291\pi$
$557$	−1.04937e7	−1.43315	−0.716575	+	0.697510i	$0.754291\pi$
$558$	0	0
$559$	−1.08210e7	−1.46466
$560$	0	0
$561$	0	0
$562$	−1.99500e7	−2.66442
$563$	8.24498e6	1.09627	0.548137	−	0.836389i	$-0.315337\pi$
$563$	8.24498e6	1.09627	0.548137	+	0.836389i	$0.315337\pi$
$564$	0	0
$565$	0	0
$566$	−1.92848e7	−2.53031
$567$	0	0
$568$	4.04880e6	0.526569
$569$	9.83611e6	1.27363	0.636814	−	0.771017i	$-0.280252\pi$
$569$	9.83611e6	1.27363	0.636814	+	0.771017i	$0.280252\pi$
$570$	0	0
$571$	−6.26407e6	−0.804019	−0.402010	−	0.915635i	$-0.631688\pi$
$571$	−6.26407e6	−0.804019	−0.402010	+	0.915635i	$0.631688\pi$
$572$	1.32895e7	1.69832
$573$	0	0
$574$	6.67800e6	0.845993
$575$	0	0
$576$	0	0
$577$	−7.24252e6	−0.905628	−0.452814	−	0.891605i	$-0.649580\pi$
$577$	−7.24252e6	−0.905628	−0.452814	+	0.891605i	$0.649580\pi$
$578$	1.65176e6	0.205649
$579$	0	0
$580$	0	0
$581$	1.60840e6	0.197675
$582$	0	0
$583$	−3.33760e6	−0.406689
$584$	−3.75041e6	−0.455037
$585$	0	0
$586$	9.70368e6	1.16733
$587$	5.34987e6	0.640838	0.320419	−	0.947276i	$-0.396176\pi$
$587$	5.34987e6	0.640838	0.320419	+	0.947276i	$0.396176\pi$
$588$	0	0
$589$	4.21065e6	0.500104
$590$	0	0
$591$	0	0
$592$	2.47556e6	0.290315
$593$	−4.06690e6	−0.474927	−0.237464	−	0.971396i	$-0.576316\pi$
$593$	−4.06690e6	−0.474927	−0.237464	+	0.971396i	$0.576316\pi$
$594$	0	0
$595$	0	0
$596$	1.66087e7	1.91523
$597$	0	0
$598$	−3.33564e7	−3.81440
$599$	−2.66326e6	−0.303281	−0.151641	−	0.988436i	$-0.548456\pi$
$599$	−2.66326e6	−0.303281	−0.151641	+	0.988436i	$0.548456\pi$
$600$	0	0
$601$	−1.76782e6	−0.199642	−0.0998208	−	0.995005i	$-0.531827\pi$
$601$	−1.76782e6	−0.199642	−0.0998208	+	0.995005i	$0.531827\pi$
$602$	−3.89863e6	−0.438450
$603$	0	0
$604$	−1.97272e7	−2.20026
$605$	0	0
$606$	0	0
$607$	−1.12446e7	−1.23872	−0.619358	−	0.785109i	$-0.712607\pi$
$607$	−1.12446e7	−1.23872	−0.619358	+	0.785109i	$0.712607\pi$
$608$	−9.58053e6	−1.05107
$609$	0	0
$610$	0	0
$611$	1.41288e7	1.53110
$612$	0	0
$613$	1.38657e7	1.49036	0.745178	−	0.666865i	$-0.232364\pi$
$613$	1.38657e7	1.49036	0.745178	+	0.666865i	$0.232364\pi$
$614$	1.02813e6	0.110059
$615$	0	0
$616$	756000.	0.0802731
$617$	−6.91690e6	−0.731474	−0.365737	−	0.930718i	$-0.619183\pi$
$617$	−6.91690e6	−0.731474	−0.365737	+	0.930718i	$0.619183\pi$
$618$	0	0
$619$	−8.31857e6	−0.872614	−0.436307	−	0.899798i	$-0.643714\pi$
$619$	−8.31857e6	−0.872614	−0.436307	+	0.899798i	$0.643714\pi$
$620$	0	0
$621$	0	0
$622$	−1.25328e7	−1.29889
$623$	−5.42156e6	−0.559634
$624$	0	0
$625$	0	0
$626$	−7.18285e6	−0.732590
$627$	0	0
$628$	−2.73505e6	−0.276736
$629$	3.95506e6	0.398590
$630$	0	0
$631$	1.65981e7	1.65953	0.829764	−	0.558115i	$-0.188475\pi$
$631$	1.65981e7	1.65953	0.829764	+	0.558115i	$0.188475\pi$
$632$	1.92345e6	0.191552
$633$	0	0
$634$	−800800.	−0.0791227
$635$	0	0
$636$	0	0
$637$	1.54472e7	1.50835
$638$	1.03077e7	1.00255
$639$	0	0
$640$	0	0
$641$	6.98176e6	0.671150	0.335575	−	0.942013i	$-0.391069\pi$
$641$	6.98176e6	0.671150	0.335575	+	0.942013i	$0.391069\pi$
$642$	0	0
$643$	1.05868e7	1.00980	0.504900	−	0.863178i	$-0.331529\pi$
$643$	1.05868e7	1.00980	0.504900	+	0.863178i	$0.331529\pi$
$644$	−6.52394e6	−0.619862
$645$	0	0
$646$	−1.23318e7	−1.16264
$647$	1.39127e7	1.30662	0.653310	−	0.757091i	$-0.273380\pi$
$647$	1.39127e7	1.30662	0.653310	+	0.757091i	$0.273380\pi$
$648$	0	0
$649$	−9.63200e6	−0.897645
$650$	0	0
$651$	0	0
$652$	−1.65747e7	−1.52695
$653$	−9.42842e6	−0.865279	−0.432639	−	0.901567i	$-0.642418\pi$
$653$	−9.42842e6	−0.865279	−0.432639	+	0.901567i	$0.642418\pi$
$654$	0	0
$655$	0	0
$656$	1.41188e7	1.28097
$657$	0	0
$658$	5.09040e6	0.458339
$659$	1.98325e7	1.77895	0.889476	−	0.456981i	$-0.151069\pi$
$659$	1.98325e7	1.77895	0.889476	+	0.456981i	$0.151069\pi$
$660$	0	0
$661$	1.01392e7	0.902606	0.451303	−	0.892371i	$-0.350959\pi$
$661$	1.01392e7	0.902606	0.451303	+	0.892371i	$0.350959\pi$
$662$	3.07197e7	2.72441
$663$	0	0
$664$	−1.79424e6	−0.157928
$665$	0	0
$666$	0	0
$667$	−1.40448e7	−1.22237
$668$	−2.31886e7	−2.01064
$669$	0	0
$670$	0	0
$671$	2.54780e6	0.218453
$672$	0	0
$673$	1.69568e7	1.44313	0.721564	−	0.692348i	$-0.243423\pi$
$673$	1.69568e7	1.44313	0.721564	+	0.692348i	$0.243423\pi$
$674$	8.76615e6	0.743291
$675$	0	0
$676$	2.73878e7	2.30510
$677$	−7.48195e6	−0.627398	−0.313699	−	0.949522i	$-0.601568\pi$
$677$	−7.48195e6	−0.627398	−0.313699	+	0.949522i	$0.601568\pi$
$678$	0	0
$679$	−3.22898e6	−0.268776
$680$	0	0
$681$	0	0
$682$	−1.01724e7	−0.837457
$683$	8.01507e6	0.657439	0.328720	−	0.944428i	$-0.393383\pi$
$683$	8.01507e6	0.657439	0.328720	+	0.944428i	$0.393383\pi$
$684$	0	0
$685$	0	0
$686$	1.18932e7	0.964911
$687$	0	0
$688$	−8.24258e6	−0.663884
$689$	−1.04218e7	−0.836360
$690$	0	0
$691$	1.32056e7	1.05212	0.526058	−	0.850449i	$-0.323670\pi$
$691$	1.32056e7	1.05212	0.526058	+	0.850449i	$0.323670\pi$
$692$	−2.78507e6	−0.221091
$693$	0	0
$694$	−872480.	−0.0687633
$695$	0	0
$696$	0	0
$697$	2.25568e7	1.75872
$698$	−1.49788e7	−1.16369
$699$	0	0
$700$	0	0
$701$	1.68513e7	1.29521	0.647604	−	0.761977i	$-0.275771\pi$
$701$	1.68513e7	1.29521	0.647604	+	0.761977i	$0.275771\pi$
$702$	0	0
$703$	−3.60449e6	−0.275078
$704$	1.46208e7	1.11183
$705$	0	0
$706$	−2.05111e7	−1.54874
$707$	−4.20171e6	−0.316138
$708$	0	0
$709$	−6.32659e6	−0.472666	−0.236333	−	0.971672i	$-0.575946\pi$
$709$	−6.32659e6	−0.472666	−0.236333	+	0.971672i	$0.575946\pi$
$710$	0	0
$711$	0	0
$712$	6.04800e6	0.447107
$713$	1.38605e7	1.02107
$714$	0	0
$715$	0	0
$716$	8.49511e6	0.619279
$717$	0	0
$718$	−1.32468e7	−0.958959
$719$	9.51885e6	0.686692	0.343346	−	0.939209i	$-0.388440\pi$
$719$	9.51885e6	0.686692	0.343346	+	0.939209i	$0.388440\pi$
$720$	0	0
$721$	−1.52190e6	−0.109030
$722$	−9.47784e6	−0.676653
$723$	0	0
$724$	5.36229e6	0.380193
$725$	0	0
$726$	0	0
$727$	−1.31844e6	−0.0925179	−0.0462590	−	0.998929i	$-0.514730\pi$
$727$	−1.31844e6	−0.0925179	−0.0462590	+	0.998929i	$0.514730\pi$
$728$	2.36064e6	0.165082
$729$	0	0
$730$	0	0
$731$	−1.31687e7	−0.911484
$732$	0	0
$733$	−1.03356e7	−0.710523	−0.355261	−	0.934767i	$-0.615608\pi$
$733$	−1.03356e7	−0.710523	−0.355261	+	0.934767i	$0.615608\pi$
$734$	−3.83752e7	−2.62912
$735$	0	0
$736$	−3.15370e7	−2.14598
$737$	1.68821e7	1.14488
$738$	0	0
$739$	−1.22839e7	−0.827415	−0.413708	−	0.910410i	$-0.635766\pi$
$739$	−1.22839e7	−0.827415	−0.413708	+	0.910410i	$0.635766\pi$
$740$	0	0
$741$	0	0
$742$	−3.75480e6	−0.250367
$743$	−1.67774e7	−1.11494	−0.557471	−	0.830197i	$-0.688228\pi$
$743$	−1.67774e7	−1.11494	−0.557471	+	0.830197i	$0.688228\pi$
$744$	0	0
$745$	0	0
$746$	−1.87008e7	−1.23031
$747$	0	0
$748$	1.61728e7	1.05689
$749$	3.58726e6	0.233646
$750$	0	0
$751$	7.62579e6	0.493384	0.246692	−	0.969094i	$-0.420656\pi$
$751$	7.62579e6	0.493384	0.246692	+	0.969094i	$0.420656\pi$
$752$	1.07623e7	0.693999
$753$	0	0
$754$	3.21860e7	2.06176
$755$	0	0
$756$	0	0
$757$	2.35783e7	1.49545	0.747726	−	0.664007i	$-0.231145\pi$
$757$	2.35783e7	1.49545	0.747726	+	0.664007i	$0.231145\pi$
$758$	3.91857e7	2.47716
$759$	0	0
$760$	0	0
$761$	−1.69434e7	−1.06057	−0.530284	−	0.847820i	$-0.677915\pi$
$761$	−1.69434e7	−1.06057	−0.530284	+	0.847820i	$0.677915\pi$
$762$	0	0
$763$	5.13086e6	0.319064
$764$	1.02374e7	0.634534
$765$	0	0
$766$	1.25227e7	0.771128
$767$	−3.00763e7	−1.84602
$768$	0	0
$769$	1.85270e7	1.12977	0.564884	−	0.825171i	$-0.308921\pi$
$769$	1.85270e7	1.12977	0.564884	+	0.825171i	$0.308921\pi$
$770$	0	0
$771$	0	0
$772$	−3.55585e6	−0.214734
$773$	−2.77156e7	−1.66831	−0.834153	−	0.551534i	$-0.814043\pi$
$773$	−2.77156e7	−1.66831	−0.834153	+	0.551534i	$0.814043\pi$
$774$	0	0
$775$	0	0
$776$	3.60207e6	0.214733
$777$	0	0
$778$	−2.66532e7	−1.57870
$779$	−2.05574e7	−1.21374
$780$	0	0
$781$	−2.69920e7	−1.58346
$782$	−4.05934e7	−2.37377
$783$	0	0
$784$	1.17665e7	0.683685
$785$	0	0
$786$	0	0
$787$	−3.01706e7	−1.73639	−0.868194	−	0.496225i	$-0.834719\pi$
$787$	−3.01706e7	−1.73639	−0.868194	+	0.496225i	$0.834719\pi$
$788$	1.42077e7	0.815095
$789$	0	0
$790$	0	0
$791$	−8.98774e6	−0.510751
$792$	0	0
$793$	7.95558e6	0.449251
$794$	2.56611e7	1.44452
$795$	0	0
$796$	−1.96638e7	−1.09998
$797$	3.21180e7	1.79103	0.895514	−	0.445034i	$-0.146808\pi$
$797$	3.21180e7	1.79103	0.895514	+	0.445034i	$0.146808\pi$
$798$	0	0
$799$	1.71942e7	0.952831
$800$	0	0
$801$	0	0
$802$	3.68172e7	2.02123
$803$	2.50027e7	1.36836
$804$	0	0
$805$	0	0
$806$	−3.17637e7	−1.72224
$807$	0	0
$808$	4.68720e6	0.252572
$809$	−1.81097e7	−0.972835	−0.486418	−	0.873726i	$-0.661697\pi$
$809$	−1.81097e7	−0.972835	−0.486418	+	0.873726i	$0.661697\pi$
$810$	0	0
$811$	−9.66264e6	−0.515874	−0.257937	−	0.966162i	$-0.583043\pi$
$811$	−9.66264e6	−0.515874	−0.257937	+	0.966162i	$0.583043\pi$
$812$	6.29503e6	0.335049
$813$	0	0
$814$	8.70800e6	0.460636
$815$	0	0
$816$	0	0
$817$	1.20014e7	0.629040
$818$	9.34891e6	0.488515
$819$	0	0
$820$	0	0
$821$	−1.86943e7	−0.967948	−0.483974	−	0.875082i	$-0.660807\pi$
$821$	−1.86943e7	−0.967948	−0.483974	+	0.875082i	$0.660807\pi$
$822$	0	0
$823$	1.11328e7	0.572932	0.286466	−	0.958090i	$-0.407519\pi$
$823$	1.11328e7	0.572932	0.286466	+	0.958090i	$0.407519\pi$
$824$	1.69775e6	0.0871076
$825$	0	0
$826$	−1.08360e7	−0.552610
$827$	2.39465e6	0.121753	0.0608764	−	0.998145i	$-0.480610\pi$
$827$	2.39465e6	0.121753	0.0608764	+	0.998145i	$0.480610\pi$
$828$	0	0
$829$	−2.21752e7	−1.12068	−0.560339	−	0.828264i	$-0.689329\pi$
$829$	−2.21752e7	−1.12068	−0.560339	+	0.828264i	$0.689329\pi$
$830$	0	0
$831$	0	0
$832$	4.56540e7	2.28649
$833$	1.87986e7	0.938671
$834$	0	0
$835$	0	0
$836$	−1.47393e7	−0.729391
$837$	0	0
$838$	1.95244e7	0.960434
$839$	−3.58040e7	−1.75601	−0.878005	−	0.478651i	$-0.841126\pi$
$839$	−3.58040e7	−1.75601	−0.878005	+	0.478651i	$0.841126\pi$
$840$	0	0
$841$	−6.95915e6	−0.339286
$842$	−5.27051e7	−2.56196
$843$	0	0
$844$	861346.	0.0416219
$845$	0	0
$846$	0	0
$847$	2.20730e6	0.105719
$848$	−7.93850e6	−0.379096
$849$	0	0
$850$	0	0
$851$	−1.18652e7	−0.561630
$852$	0	0
$853$	1.33935e7	0.630262	0.315131	−	0.949048i	$-0.397952\pi$
$853$	1.33935e7	0.630262	0.315131	+	0.949048i	$0.397952\pi$
$854$	2.86627e6	0.134485
$855$	0	0
$856$	−4.00176e6	−0.186667
$857$	−3.06875e7	−1.42728	−0.713640	−	0.700513i	$-0.752955\pi$
$857$	−3.06875e7	−1.42728	−0.713640	+	0.700513i	$0.752955\pi$
$858$	0	0
$859$	−1.90187e7	−0.879422	−0.439711	−	0.898139i	$-0.644919\pi$
$859$	−1.90187e7	−0.879422	−0.439711	+	0.898139i	$0.644919\pi$
$860$	0	0
$861$	0	0
$862$	3.08532e7	1.41427
$863$	1.25827e7	0.575104	0.287552	−	0.957765i	$-0.407159\pi$
$863$	1.25827e7	0.575104	0.287552	+	0.957765i	$0.407159\pi$
$864$	0	0
$865$	0	0
$866$	2.80506e7	1.27100
$867$	0	0
$868$	−6.21243e6	−0.279874
$869$	−1.28230e7	−0.576023
$870$	0	0
$871$	5.27150e7	2.35445
$872$	−5.72371e6	−0.254910
$873$	0	0
$874$	3.69953e7	1.63820
$875$	0	0
$876$	0	0
$877$	2.61008e7	1.14592	0.572961	−	0.819582i	$-0.305795\pi$
$877$	2.61008e7	1.14592	0.572961	+	0.819582i	$0.305795\pi$
$878$	−2.06943e7	−0.905970
$879$	0	0
$880$	0	0
$881$	4.74353e6	0.205903	0.102951	−	0.994686i	$-0.467171\pi$
$881$	4.74353e6	0.205903	0.102951	+	0.994686i	$0.467171\pi$
$882$	0	0
$883$	2.34369e7	1.01158	0.505788	−	0.862658i	$-0.331202\pi$
$883$	2.34369e7	1.01158	0.505788	+	0.862658i	$0.331202\pi$
$884$	5.05001e7	2.17351
$885$	0	0
$886$	−2.36846e7	−1.01364
$887$	4.18677e7	1.78677	0.893387	−	0.449287i	$-0.148322\pi$
$887$	4.18677e7	1.78677	0.893387	+	0.449287i	$0.148322\pi$
$888$	0	0
$889$	5.31810e6	0.225685
$890$	0	0
$891$	0	0
$892$	−1.43505e7	−0.603886
$893$	−1.56702e7	−0.657575
$894$	0	0
$895$	0	0
$896$	4.54507e6	0.189134
$897$	0	0
$898$	1.86872e7	0.773310
$899$	−1.33742e7	−0.551909
$900$	0	0
$901$	−1.26829e7	−0.520482
$902$	4.96641e7	2.03248
$903$	0	0
$904$	1.00262e7	0.408053
$905$	0	0
$906$	0	0
$907$	−2.07476e7	−0.837434	−0.418717	−	0.908117i	$-0.637520\pi$
$907$	−2.07476e7	−0.837434	−0.418717	+	0.908117i	$0.637520\pi$
$908$	4.68986e7	1.88775
$909$	0	0
$910$	0	0
$911$	−2.34064e6	−0.0934413	−0.0467206	−	0.998908i	$-0.514877\pi$
$911$	−2.34064e6	−0.0934413	−0.0467206	+	0.998908i	$0.514877\pi$
$912$	0	0
$913$	1.19616e7	0.474911
$914$	8.85262e6	0.350515
$915$	0	0
$916$	3.49094e7	1.37469
$917$	5.10530e6	0.200492
$918$	0	0
$919$	−1.00899e7	−0.394092	−0.197046	−	0.980394i	$-0.563135\pi$
$919$	−1.00899e7	−0.394092	−0.197046	+	0.980394i	$0.563135\pi$
$920$	0	0
$921$	0	0
$922$	−1.97316e7	−0.764425
$923$	−8.42835e7	−3.25640
$924$	0	0
$925$	0	0
$926$	6.13289e6	0.235038
$927$	0	0
$928$	3.04304e7	1.15995
$929$	6.20501e6	0.235886	0.117943	−	0.993020i	$-0.462370\pi$
$929$	6.20501e6	0.235886	0.117943	+	0.993020i	$0.462370\pi$
$930$	0	0
$931$	−1.71323e7	−0.647802
$932$	−1.95718e7	−0.738060
$933$	0	0
$934$	−4.57761e7	−1.71700
$935$	0	0
$936$	0	0
$937$	3.80727e7	1.41666	0.708328	−	0.705883i	$-0.249450\pi$
$937$	3.80727e7	1.41666	0.708328	+	0.705883i	$0.249450\pi$
$938$	1.89924e7	0.704811
$939$	0	0
$940$	0	0
$941$	7.35056e6	0.270612	0.135306	−	0.990804i	$-0.456798\pi$
$941$	7.35056e6	0.270612	0.135306	+	0.990804i	$0.456798\pi$
$942$	0	0
$943$	−6.76704e7	−2.47810
$944$	−2.29098e7	−0.836740
$945$	0	0
$946$	−2.89940e7	−1.05337
$947$	3.11699e6	0.112943	0.0564717	−	0.998404i	$-0.482015\pi$
$947$	3.11699e6	0.112943	0.0564717	+	0.998404i	$0.482015\pi$
$948$	0	0
$949$	7.80720e7	2.81404
$950$	0	0
$951$	0	0
$952$	2.87280e6	0.102734
$953$	−1.27084e7	−0.453272	−0.226636	−	0.973980i	$-0.572773\pi$
$953$	−1.27084e7	−0.453272	−0.226636	+	0.973980i	$0.572773\pi$
$954$	0	0
$955$	0	0
$956$	5.36286e7	1.89780
$957$	0	0
$958$	−4.98960e6	−0.175652
$959$	1.09335e7	0.383894
$960$	0	0
$961$	−1.54305e7	−0.538977
$962$	2.71910e7	0.947301
$963$	0	0
$964$	−3.72698e7	−1.29171
$965$	0	0
$966$	0	0
$967$	−8.52938e6	−0.293326	−0.146663	−	0.989186i	$-0.546853\pi$
$967$	−8.52938e6	−0.293326	−0.146663	+	0.989186i	$0.546853\pi$
$968$	−2.46234e6	−0.0844617
$969$	0	0
$970$	0	0
$971$	4.64085e7	1.57961	0.789805	−	0.613358i	$-0.210182\pi$
$971$	4.64085e7	1.57961	0.789805	+	0.613358i	$0.210182\pi$
$972$	0	0
$973$	−9.54648e6	−0.323267
$974$	2.64504e7	0.893376
$975$	0	0
$976$	6.05995e6	0.203631
$977$	3.62619e7	1.21539	0.607693	−	0.794172i	$-0.292095\pi$
$977$	3.62619e7	1.21539	0.607693	+	0.794172i	$0.292095\pi$
$978$	0	0
$979$	−4.03200e7	−1.34451
$980$	0	0
$981$	0	0
$982$	−1.83008e7	−0.605608
$983$	8.74437e6	0.288632	0.144316	−	0.989532i	$-0.453902\pi$
$983$	8.74437e6	0.288632	0.144316	+	0.989532i	$0.453902\pi$
$984$	0	0
$985$	0	0
$986$	3.91691e7	1.28307
$987$	0	0
$988$	−4.60239e7	−1.50000
$989$	3.95061e7	1.28432
$990$	0	0
$991$	2.48761e7	0.804635	0.402317	−	0.915500i	$-0.368205\pi$
$991$	2.48761e7	0.804635	0.402317	+	0.915500i	$0.368205\pi$
$992$	−3.00311e7	−0.968930
$993$	0	0
$994$	−3.03660e7	−0.974814
$995$	0	0
$996$	0	0
$997$	−5.63519e7	−1.79544	−0.897720	−	0.440567i	$-0.854777\pi$
$997$	−5.63519e7	−1.79544	−0.897720	+	0.440567i	$0.854777\pi$
$998$	−6.21313e7	−1.97462
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.a.o.1.2	yes	2
3.2	odd	2	inner	225.6.a.o.1.1	✓	2
5.2	odd	4		225.6.b.h.199.3		4
5.3	odd	4		225.6.b.h.199.2		4
5.4	even	2		225.6.a.p.1.1	yes	2
15.2	even	4		225.6.b.h.199.1		4
15.8	even	4		225.6.b.h.199.4		4
15.14	odd	2		225.6.a.p.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.6.a.o.1.1	✓	2	3.2	odd	2	inner
225.6.a.o.1.2	yes	2	1.1	even	1	trivial
225.6.a.p.1.1	yes	2	5.4	even	2
225.6.a.p.1.2	yes	2	15.14	odd	2
225.6.b.h.199.1		4	15.2	even	4
225.6.b.h.199.2		4	5.3	odd	4
225.6.b.h.199.3		4	5.2	odd	4
225.6.b.h.199.4		4	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+8.36660 q^{2} +38.0000 q^{4} -45.0000 q^{7} +50.1996 q^{8} +O(q^{10})\)	\(q+8.36660 q^{2} +38.0000 q^{4} -45.0000 q^{7} +50.1996 q^{8} -334.664 q^{11} -1045.00 q^{13} -376.497 q^{14} -796.000 q^{16} -1271.72 q^{17} +1159.00 q^{19} -2800.00 q^{22} +3815.17 q^{23} -8743.10 q^{26} -1710.00 q^{28} -3681.30 q^{29} +3633.00 q^{31} -8266.20 q^{32} -10640.0 q^{34} -3110.00 q^{37} +9696.89 q^{38} -17737.2 q^{41} +10355.0 q^{43} -12717.2 q^{44} +31920.0 q^{46} -13520.4 q^{47} -14782.0 q^{49} -39710.0 q^{52} +9972.99 q^{53} -2258.98 q^{56} -30800.0 q^{58} +28781.1 q^{59} -7613.00 q^{61} +30395.9 q^{62} -43688.0 q^{64} -50445.0 q^{67} -48325.5 q^{68} +80654.0 q^{71} -74710.0 q^{73} -26020.1 q^{74} +44042.0 q^{76} +15059.9 q^{77} +38316.0 q^{79} -148400. q^{82} -35742.1 q^{83} +86636.1 q^{86} -16800.0 q^{88} +120479. q^{89} +47025.0 q^{91} +144976. q^{92} -113120. q^{94} +71755.0 q^{97} -123675. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 76 q^{4} - 90 q^{7}+O(q^{10}) \) 2 * q + 76 * q^4 - 90 * q^7	\( 2 q + 76 q^{4} - 90 q^{7} - 2090 q^{13} - 1592 q^{16} + 2318 q^{19} - 5600 q^{22} - 3420 q^{28} + 7266 q^{31} - 21280 q^{34} - 6220 q^{37} + 20710 q^{43} + 63840 q^{46} - 29564 q^{49} - 79420 q^{52} - 61600 q^{58} - 15226 q^{61} - 87376 q^{64} - 100890 q^{67} - 149420 q^{73} + 88084 q^{76} + 76632 q^{79} - 296800 q^{82} - 33600 q^{88} + 94050 q^{91} - 226240 q^{94} + 143510 q^{97}+O(q^{100}) \) 2 * q + 76 * q^4 - 90 * q^7 - 2090 * q^13 - 1592 * q^16 + 2318 * q^19 - 5600 * q^22 - 3420 * q^28 + 7266 * q^31 - 21280 * q^34 - 6220 * q^37 + 20710 * q^43 + 63840 * q^46 - 29564 * q^49 - 79420 * q^52 - 61600 * q^58 - 15226 * q^61 - 87376 * q^64 - 100890 * q^67 - 149420 * q^73 + 88084 * q^76 + 76632 * q^79 - 296800 * q^82 - 33600 * q^88 + 94050 * q^91 - 226240 * q^94 + 143510 * q^97

Introduction

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