LMFDB - Embedded newform 225.6.a.t.1.1

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{31}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 31 $ x^2 - 31
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.56776$ 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)

$f(q)$	$=$	$q-2.56776 q^{2} -25.4066 q^{4} +82.6263 q^{7} +147.407 q^{8} +O(q^{10})$	$q-2.56776 q^{2} -25.4066 q^{4} +82.6263 q^{7} +147.407 q^{8} -483.963 q^{11} +7.50538 q^{13} -212.165 q^{14} +434.505 q^{16} -1080.71 q^{17} +3042.38 q^{19} +1242.70 q^{22} +3385.91 q^{23} -19.2720 q^{26} -2099.25 q^{28} -2345.79 q^{29} +3912.43 q^{31} -5832.72 q^{32} +2775.01 q^{34} -12094.3 q^{37} -7812.13 q^{38} -7264.06 q^{41} +3022.36 q^{43} +12295.9 q^{44} -8694.22 q^{46} -17299.5 q^{47} -9979.89 q^{49} -190.686 q^{52} -31151.2 q^{53} +12179.7 q^{56} +6023.45 q^{58} +48709.8 q^{59} -1957.31 q^{61} -10046.2 q^{62} +1072.87 q^{64} -59596.7 q^{67} +27457.2 q^{68} -83672.1 q^{71} -5692.94 q^{73} +31055.4 q^{74} -77296.6 q^{76} -39988.1 q^{77} +31309.9 q^{79} +18652.4 q^{82} -37162.0 q^{83} -7760.72 q^{86} -71339.4 q^{88} -69190.7 q^{89} +620.142 q^{91} -86024.4 q^{92} +44421.1 q^{94} +88249.2 q^{97} +25626.0 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{2} + 16 q^{4} - 102 q^{7} + 228 q^{8}+O(q^{10}) $ 2 * q + 6 * q^2 + 16 * q^4 - 102 * q^7 + 228 * q^8	$ 2 q + 6 q^{2} + 16 q^{4} - 102 q^{7} + 228 q^{8} + 12 q^{11} - 1054 q^{13} - 1794 q^{14} - 200 q^{16} - 1716 q^{17} + 4214 q^{19} + 5492 q^{22} - 444 q^{23} - 9114 q^{26} - 9744 q^{28} - 4068 q^{29} - 2598 q^{31} - 13848 q^{32} - 2668 q^{34} - 4412 q^{37} + 2226 q^{38} - 11232 q^{41} + 8450 q^{43} + 32832 q^{44} - 41508 q^{46} + 2460 q^{47} + 7300 q^{49} - 44144 q^{52} - 65064 q^{53} - 2700 q^{56} - 8732 q^{58} + 63924 q^{59} + 7310 q^{61} - 65826 q^{62} - 47296 q^{64} - 61734 q^{67} + 1152 q^{68} - 98304 q^{71} + 26564 q^{73} + 96876 q^{74} - 28784 q^{76} - 131556 q^{77} + 84000 q^{79} - 15344 q^{82} - 65772 q^{83} + 38742 q^{86} - 31368 q^{88} - 103104 q^{89} + 196602 q^{91} - 244608 q^{92} + 213716 q^{94} - 69374 q^{97} + 173676 q^{98}+O(q^{100}) $ 2 * q + 6 * q^2 + 16 * q^4 - 102 * q^7 + 228 * q^8 + 12 * q^11 - 1054 * q^13 - 1794 * q^14 - 200 * q^16 - 1716 * q^17 + 4214 * q^19 + 5492 * q^22 - 444 * q^23 - 9114 * q^26 - 9744 * q^28 - 4068 * q^29 - 2598 * q^31 - 13848 * q^32 - 2668 * q^34 - 4412 * q^37 + 2226 * q^38 - 11232 * q^41 + 8450 * q^43 + 32832 * q^44 - 41508 * q^46 + 2460 * q^47 + 7300 * q^49 - 44144 * q^52 - 65064 * q^53 - 2700 * q^56 - 8732 * q^58 + 63924 * q^59 + 7310 * q^61 - 65826 * q^62 - 47296 * q^64 - 61734 * q^67 + 1152 * q^68 - 98304 * q^71 + 26564 * q^73 + 96876 * q^74 - 28784 * q^76 - 131556 * q^77 + 84000 * q^79 - 15344 * q^82 - 65772 * q^83 + 38742 * q^86 - 31368 * q^88 - 103104 * q^89 + 196602 * q^91 - 244608 * q^92 + 213716 * q^94 - 69374 * q^97 + 173676 * q^98

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 $

$2$	−2.56776	−0.453921	−0.226960	−	0.973904i	$-0.572879\pi$
$2$	−2.56776	−0.453921	−0.226960	+	0.973904i	$0.572879\pi$
$3$	0	0
$3$	0	0
$4$	−25.4066	−0.793956
$5$	0	0
$5$	0	0
$6$	0	0
$7$	82.6263	0.637343	0.318672	−	0.947865i	$-0.396763\pi$
$7$	82.6263	0.637343	0.318672	+	0.947865i	$0.396763\pi$
$8$	147.407	0.814314
$9$	0	0
$10$	0	0
$11$	−483.963	−1.20595	−0.602977	−	0.797759i	$-0.706019\pi$
$11$	−483.963	−1.20595	−0.602977	+	0.797759i	$0.706019\pi$
$12$	0	0
$13$	7.50538	0.0123173	0.00615863	−	0.999981i	$-0.498040\pi$
$13$	7.50538	0.0123173	0.00615863	+	0.999981i	$0.498040\pi$
$14$	−212.165	−0.289303
$15$	0	0
$16$	434.505	0.424322
$17$	−1080.71	−0.906958	−0.453479	−	0.891267i	$-0.649817\pi$
$17$	−1080.71	−0.906958	−0.453479	+	0.891267i	$0.649817\pi$
$18$	0	0
$19$	3042.38	1.93344	0.966719	−	0.255842i	$-0.0823527\pi$
$19$	3042.38	1.93344	0.966719	+	0.255842i	$0.0823527\pi$
$20$	0	0
$21$	0	0
$22$	1242.70	0.547408
$23$	3385.91	1.33461	0.667307	−	0.744782i	$-0.267447\pi$
$23$	3385.91	1.33461	0.667307	+	0.744782i	$0.267447\pi$
$24$	0	0
$25$	0	0
$26$	−19.2720	−0.00559106
$27$	0	0
$28$	−2099.25	−0.506022
$29$	−2345.79	−0.517959	−0.258979	−	0.965883i	$-0.583386\pi$
$29$	−2345.79	−0.517959	−0.258979	+	0.965883i	$0.583386\pi$
$30$	0	0
$31$	3912.43	0.731210	0.365605	−	0.930770i	$-0.380862\pi$
$31$	3912.43	0.731210	0.365605	+	0.930770i	$0.380862\pi$
$32$	−5832.72	−1.00692
$33$	0	0
$34$	2775.01	0.411687
$35$	0	0
$36$	0	0
$37$	−12094.3	−1.45237	−0.726187	−	0.687498i	$-0.758709\pi$
$37$	−12094.3	−1.45237	−0.726187	+	0.687498i	$0.758709\pi$
$38$	−7812.13	−0.877628
$39$	0	0
$40$	0	0
$41$	−7264.06	−0.674869	−0.337435	−	0.941349i	$-0.609559\pi$
$41$	−7264.06	−0.674869	−0.337435	+	0.941349i	$0.609559\pi$
$42$	0	0
$43$	3022.36	0.249273	0.124637	−	0.992202i	$-0.460224\pi$
$43$	3022.36	0.249273	0.124637	+	0.992202i	$0.460224\pi$
$44$	12295.9	0.957474
$45$	0	0
$46$	−8694.22	−0.605810
$47$	−17299.5	−1.14232	−0.571162	−	0.820837i	$-0.693507\pi$
$47$	−17299.5	−1.14232	−0.571162	+	0.820837i	$0.693507\pi$
$48$	0	0
$49$	−9979.89	−0.593793
$50$	0	0
$51$	0	0
$52$	−190.686	−0.00977936
$53$	−31151.2	−1.52330	−0.761649	−	0.647989i	$-0.775610\pi$
$53$	−31151.2	−1.52330	−0.761649	+	0.647989i	$0.775610\pi$
$54$	0	0
$55$	0	0
$56$	12179.7	0.518998
$57$	0	0
$58$	6023.45	0.235112
$59$	48709.8	1.82174	0.910870	−	0.412692i	$-0.135412\pi$
$59$	48709.8	1.82174	0.910870	+	0.412692i	$0.135412\pi$
$60$	0	0
$61$	−1957.31	−0.0673495	−0.0336747	−	0.999433i	$-0.510721\pi$
$61$	−1957.31	−0.0673495	−0.0336747	+	0.999433i	$0.510721\pi$
$62$	−10046.2	−0.331911
$63$	0	0
$64$	1072.87	0.0327415
$65$	0	0
$66$	0	0
$67$	−59596.7	−1.62194	−0.810970	−	0.585087i	$-0.801060\pi$
$67$	−59596.7	−1.62194	−0.810970	+	0.585087i	$0.801060\pi$
$68$	27457.2	0.720084
$69$	0	0
$70$	0	0
$71$	−83672.1	−1.96986	−0.984929	−	0.172958i	$-0.944667\pi$
$71$	−83672.1	−1.96986	−0.984929	+	0.172958i	$0.944667\pi$
$72$	0	0
$73$	−5692.94	−0.125034	−0.0625172	−	0.998044i	$-0.519913\pi$
$73$	−5692.94	−0.125034	−0.0625172	+	0.998044i	$0.519913\pi$
$74$	31055.4	0.659263
$75$	0	0
$76$	−77296.6	−1.53506
$77$	−39988.1	−0.768607
$78$	0	0
$79$	31309.9	0.564435	0.282217	−	0.959350i	$-0.408930\pi$
$79$	31309.9	0.564435	0.282217	+	0.959350i	$0.408930\pi$
$80$	0	0
$81$	0	0
$82$	18652.4	0.306337
$83$	−37162.0	−0.592113	−0.296056	−	0.955170i	$-0.595672\pi$
$83$	−37162.0	−0.592113	−0.296056	+	0.955170i	$0.595672\pi$
$84$	0	0
$85$	0	0
$86$	−7760.72	−0.113150
$87$	0	0
$88$	−71339.4	−0.982025
$89$	−69190.7	−0.925918	−0.462959	−	0.886380i	$-0.653212\pi$
$89$	−69190.7	−0.925918	−0.462959	+	0.886380i	$0.653212\pi$
$90$	0	0
$91$	620.142	0.00785033
$92$	−86024.4	−1.05963
$93$	0	0
$94$	44421.1	0.518525
$95$	0	0
$96$	0	0
$97$	88249.2	0.952317	0.476159	−	0.879359i	$-0.342029\pi$
$97$	88249.2	0.952317	0.476159	+	0.879359i	$0.342029\pi$
$98$	25626.0	0.269535
$99$	0	0
$100$	0	0
$101$	−89217.2	−0.870252	−0.435126	−	0.900370i	$-0.643296\pi$
$101$	−89217.2	−0.870252	−0.435126	+	0.900370i	$0.643296\pi$
$102$	0	0
$103$	−9687.70	−0.0899762	−0.0449881	−	0.998988i	$-0.514325\pi$
$103$	−9687.70	−0.0899762	−0.0449881	+	0.998988i	$0.514325\pi$
$104$	1106.34	0.0100301
$105$	0	0
$106$	79988.9	0.691457
$107$	−59011.6	−0.498285	−0.249143	−	0.968467i	$-0.580149\pi$
$107$	−59011.6	−0.498285	−0.249143	+	0.968467i	$0.580149\pi$
$108$	0	0
$109$	85229.1	0.687103	0.343551	−	0.939134i	$-0.388370\pi$
$109$	85229.1	0.687103	0.343551	+	0.939134i	$0.388370\pi$
$110$	0	0
$111$	0	0
$112$	35901.6	0.270439
$113$	−113652.	−0.837297	−0.418649	−	0.908148i	$-0.637496\pi$
$113$	−113652.	−0.837297	−0.418649	+	0.908148i	$0.637496\pi$
$114$	0	0
$115$	0	0
$116$	59598.6	0.411236
$117$	0	0
$118$	−125075.	−0.826926
$119$	−89295.2	−0.578044
$120$	0	0
$121$	73169.4	0.454325
$122$	5025.90	0.0305713
$123$	0	0
$124$	−99401.4	−0.580548
$125$	0	0
$126$	0	0
$127$	−295771.	−1.62722	−0.813611	−	0.581410i	$-0.802501\pi$
$127$	−295771.	−1.62722	−0.813611	+	0.581410i	$0.802501\pi$
$128$	183892.	0.992060
$129$	0	0
$130$	0	0
$131$	250130.	1.27347	0.636734	−	0.771084i	$-0.280285\pi$
$131$	250130.	1.27347	0.636734	+	0.771084i	$0.280285\pi$
$132$	0	0
$133$	251381.	1.23226
$134$	153030.	0.736233
$135$	0	0
$136$	−159304.	−0.738548
$137$	269811.	1.22817	0.614084	−	0.789241i	$-0.289526\pi$
$137$	269811.	1.22817	0.614084	+	0.789241i	$0.289526\pi$
$138$	0	0
$139$	−262877.	−1.15403	−0.577013	−	0.816735i	$-0.695782\pi$
$139$	−262877.	−1.15403	−0.577013	+	0.816735i	$0.695782\pi$
$140$	0	0
$141$	0	0
$142$	214850.	0.894160
$143$	−3632.33	−0.0148541
$144$	0	0
$145$	0	0
$146$	14618.1	0.0567557
$147$	0	0
$148$	307276.	1.15312
$149$	−131202.	−0.484144	−0.242072	−	0.970258i	$-0.577827\pi$
$149$	−131202.	−0.484144	−0.242072	+	0.970258i	$0.577827\pi$
$150$	0	0
$151$	175144.	0.625104	0.312552	−	0.949901i	$-0.398816\pi$
$151$	175144.	0.625104	0.312552	+	0.949901i	$0.398816\pi$
$152$	448468.	1.57443
$153$	0	0
$154$	102680.	0.348887
$155$	0	0
$156$	0	0
$157$	47706.6	0.154465	0.0772323	−	0.997013i	$-0.475392\pi$
$157$	47706.6	0.154465	0.0772323	+	0.997013i	$0.475392\pi$
$158$	−80396.4	−0.256209
$159$	0	0
$160$	0	0
$161$	279765.	0.850608
$162$	0	0
$163$	−22618.8	−0.0666809	−0.0333404	−	0.999444i	$-0.510615\pi$
$163$	−22618.8	−0.0666809	−0.0333404	+	0.999444i	$0.510615\pi$
$164$	184555.	0.535816
$165$	0	0
$166$	95423.4	0.268772
$167$	339093.	0.940866	0.470433	−	0.882436i	$-0.344098\pi$
$167$	339093.	0.940866	0.470433	+	0.882436i	$0.344098\pi$
$168$	0	0
$169$	−371237.	−0.999848
$170$	0	0
$171$	0	0
$172$	−76787.9	−0.197912
$173$	269075.	0.683530	0.341765	−	0.939785i	$-0.388975\pi$
$173$	269075.	0.683530	0.341765	+	0.939785i	$0.388975\pi$
$174$	0	0
$175$	0	0
$176$	−210285.	−0.511712
$177$	0	0
$178$	177665.	0.420294
$179$	−193640.	−0.451714	−0.225857	−	0.974160i	$-0.572518\pi$
$179$	−193640.	−0.451714	−0.225857	+	0.974160i	$0.572518\pi$
$180$	0	0
$181$	165755.	0.376071	0.188035	−	0.982162i	$-0.439788\pi$
$181$	165755.	0.376071	0.188035	+	0.982162i	$0.439788\pi$
$182$	−1592.38	−0.00356343
$183$	0	0
$184$	499106.	1.08680
$185$	0	0
$186$	0	0
$187$	523024.	1.09375
$188$	439522.	0.906955
$189$	0	0
$190$	0	0
$191$	522342.	1.03603	0.518014	−	0.855372i	$-0.326671\pi$
$191$	522342.	1.03603	0.518014	+	0.855372i	$0.326671\pi$
$192$	0	0
$193$	−579738.	−1.12031	−0.560156	−	0.828387i	$-0.689259\pi$
$193$	−579738.	−1.12031	−0.560156	+	0.828387i	$0.689259\pi$
$194$	−226603.	−0.432277
$195$	0	0
$196$	253555.	0.471446
$197$	−576819.	−1.05895	−0.529473	−	0.848327i	$-0.677610\pi$
$197$	−576819.	−1.05895	−0.529473	+	0.848327i	$0.677610\pi$
$198$	0	0
$199$	−493665.	−0.883690	−0.441845	−	0.897091i	$-0.645676\pi$
$199$	−493665.	−0.883690	−0.441845	+	0.897091i	$0.645676\pi$
$200$	0	0
$201$	0	0
$202$	229089.	0.395026
$203$	−193824.	−0.330117
$204$	0	0
$205$	0	0
$206$	24875.7	0.0408421
$207$	0	0
$208$	3261.13	0.00522648
$209$	−1.47240e6	−2.33164
$210$	0	0
$211$	−383876.	−0.593588	−0.296794	−	0.954942i	$-0.595917\pi$
$211$	−383876.	−0.593588	−0.296794	+	0.954942i	$0.595917\pi$
$212$	791446.	1.20943
$213$	0	0
$214$	151528.	0.226182
$215$	0	0
$216$	0	0
$217$	323270.	0.466032
$218$	−218848.	−0.311890
$219$	0	0
$220$	0	0
$221$	−8111.14	−0.0111712
$222$	0	0
$223$	−321593.	−0.433056	−0.216528	−	0.976276i	$-0.569473\pi$
$223$	−321593.	−0.433056	−0.216528	+	0.976276i	$0.569473\pi$
$224$	−481936.	−0.641755
$225$	0	0
$226$	291831.	0.380067
$227$	−331402.	−0.426865	−0.213432	−	0.976958i	$-0.568464\pi$
$227$	−331402.	−0.426865	−0.213432	+	0.976958i	$0.568464\pi$
$228$	0	0
$229$	979958.	1.23486	0.617432	−	0.786625i	$-0.288173\pi$
$229$	979958.	1.23486	0.617432	+	0.786625i	$0.288173\pi$
$230$	0	0
$231$	0	0
$232$	−345786.	−0.421781
$233$	−309466.	−0.373442	−0.186721	−	0.982413i	$-0.559786\pi$
$233$	−309466.	−0.373442	−0.186721	+	0.982413i	$0.559786\pi$
$234$	0	0
$235$	0	0
$236$	−1.23755e6	−1.44638
$237$	0	0
$238$	229289.	0.262386
$239$	1.52544e6	1.72743	0.863714	−	0.503983i	$-0.168133\pi$
$239$	1.52544e6	1.72743	0.863714	+	0.503983i	$0.168133\pi$
$240$	0	0
$241$	−772397.	−0.856639	−0.428320	−	0.903627i	$-0.640894\pi$
$241$	−772397.	−0.856639	−0.428320	+	0.903627i	$0.640894\pi$
$242$	−187882.	−0.206227
$243$	0	0
$244$	49728.5	0.0534725
$245$	0	0
$246$	0	0
$247$	22834.2	0.0238147
$248$	576718.	0.595434
$249$	0	0
$250$	0	0
$251$	401992.	0.402747	0.201374	−	0.979514i	$-0.435459\pi$
$251$	401992.	0.402747	0.201374	+	0.979514i	$0.435459\pi$
$252$	0	0
$253$	−1.63866e6	−1.60948
$254$	759471.	0.738630
$255$	0	0
$256$	−506524.	−0.483058
$257$	−484514.	−0.457587	−0.228793	−	0.973475i	$-0.573478\pi$
$257$	−484514.	−0.457587	−0.228793	+	0.973475i	$0.573478\pi$
$258$	0	0
$259$	−999312.	−0.925660
$260$	0	0
$261$	0	0
$262$	−642276.	−0.578054
$263$	−979486.	−0.873190	−0.436595	−	0.899658i	$-0.643816\pi$
$263$	−979486.	−0.873190	−0.436595	+	0.899658i	$0.643816\pi$
$264$	0	0
$265$	0	0
$266$	−645487.	−0.559350
$267$	0	0
$268$	1.51415e6	1.28775
$269$	995409.	0.838727	0.419364	−	0.907818i	$-0.362253\pi$
$269$	995409.	0.838727	0.419364	+	0.907818i	$0.362253\pi$
$270$	0	0
$271$	46175.9	0.0381937	0.0190969	−	0.999818i	$-0.493921\pi$
$271$	46175.9	0.0381937	0.0190969	+	0.999818i	$0.493921\pi$
$272$	−469575.	−0.384842
$273$	0	0
$274$	−692810.	−0.557491
$275$	0	0
$276$	0	0
$277$	−844717.	−0.661473	−0.330736	−	0.943723i	$-0.607297\pi$
$277$	−844717.	−0.661473	−0.330736	+	0.943723i	$0.607297\pi$
$278$	675006.	0.523837
$279$	0	0
$280$	0	0
$281$	453572.	0.342674	0.171337	−	0.985213i	$-0.445191\pi$
$281$	453572.	0.342674	0.171337	+	0.985213i	$0.445191\pi$
$282$	0	0
$283$	2.37359e6	1.76173	0.880867	−	0.473363i	$-0.156960\pi$
$283$	2.37359e6	1.76173	0.880867	+	0.473363i	$0.156960\pi$
$284$	2.12582e6	1.56398
$285$	0	0
$286$	9326.96	0.00674256
$287$	−600203.	−0.430123
$288$	0	0
$289$	−251922.	−0.177427
$290$	0	0
$291$	0	0
$292$	144638.	0.0992718
$293$	−1.71847e6	−1.16943	−0.584714	−	0.811240i	$-0.698793\pi$
$293$	−1.71847e6	−1.16943	−0.584714	+	0.811240i	$0.698793\pi$
$294$	0	0
$295$	0	0
$296$	−1.78279e6	−1.18269
$297$	0	0
$298$	336896.	0.219763
$299$	25412.5	0.0164388
$300$	0	0
$301$	249727.	0.158873
$302$	−449728.	−0.283748
$303$	0	0
$304$	1.32193e6	0.820399
$305$	0	0
$306$	0	0
$307$	−2.89249e6	−1.75156	−0.875781	−	0.482709i	$-0.839653\pi$
$307$	−2.89249e6	−1.75156	−0.875781	+	0.482709i	$0.839653\pi$
$308$	1.01596e6	0.610240
$309$	0	0
$310$	0	0
$311$	1.28455e6	0.753097	0.376548	−	0.926397i	$-0.377111\pi$
$311$	1.28455e6	0.753097	0.376548	+	0.926397i	$0.377111\pi$
$312$	0	0
$313$	−2.32830e6	−1.34332	−0.671659	−	0.740860i	$-0.734418\pi$
$313$	−2.32830e6	−1.34332	−0.671659	+	0.740860i	$0.734418\pi$
$314$	−122499.	−0.0701147
$315$	0	0
$316$	−795477.	−0.448136
$317$	−2.86466e6	−1.60112	−0.800561	−	0.599251i	$-0.795465\pi$
$317$	−2.86466e6	−1.60112	−0.800561	+	0.599251i	$0.795465\pi$
$318$	0	0
$319$	1.13528e6	0.624634
$320$	0	0
$321$	0	0
$322$	−718372.	−0.386109
$323$	−3.28794e6	−1.75355
$324$	0	0
$325$	0	0
$326$	58079.8	0.0302678
$327$	0	0
$328$	−1.07077e6	−0.549556
$329$	−1.42940e6	−0.728053
$330$	0	0
$331$	33623.6	0.0168684	0.00843422	−	0.999964i	$-0.497315\pi$
$331$	33623.6	0.0168684	0.00843422	+	0.999964i	$0.497315\pi$
$332$	944161.	0.470111
$333$	0	0
$334$	−870711.	−0.427079
$335$	0	0
$336$	0	0
$337$	1.18330e6	0.567571	0.283786	−	0.958888i	$-0.408410\pi$
$337$	1.18330e6	0.567571	0.283786	+	0.958888i	$0.408410\pi$
$338$	953248.	0.453852
$339$	0	0
$340$	0	0
$341$	−1.89347e6	−0.881805
$342$	0	0
$343$	−2.21330e6	−1.01579
$344$	445516.	0.202987
$345$	0	0
$346$	−690921.	−0.310269
$347$	−982641.	−0.438098	−0.219049	−	0.975714i	$-0.570295\pi$
$347$	−982641.	−0.438098	−0.219049	+	0.975714i	$0.570295\pi$
$348$	0	0
$349$	3.86491e6	1.69854	0.849270	−	0.527958i	$-0.177042\pi$
$349$	3.86491e6	1.69854	0.849270	+	0.527958i	$0.177042\pi$
$350$	0	0
$351$	0	0
$352$	2.82282e6	1.21430
$353$	3.08371e6	1.31715	0.658577	−	0.752513i	$-0.271159\pi$
$353$	3.08371e6	1.31715	0.658577	+	0.752513i	$0.271159\pi$
$354$	0	0
$355$	0	0
$356$	1.75790e6	0.735138
$357$	0	0
$358$	497223.	0.205042
$359$	−1.39479e6	−0.571179	−0.285589	−	0.958352i	$-0.592189\pi$
$359$	−1.39479e6	−0.571179	−0.285589	+	0.958352i	$0.592189\pi$
$360$	0	0
$361$	6.78000e6	2.73818
$362$	−425619.	−0.170706
$363$	0	0
$364$	−15755.7	−0.00623281
$365$	0	0
$366$	0	0
$367$	−201558.	−0.0781152	−0.0390576	−	0.999237i	$-0.512436\pi$
$367$	−201558.	−0.0781152	−0.0390576	+	0.999237i	$0.512436\pi$
$368$	1.47120e6	0.566306
$369$	0	0
$370$	0	0
$371$	−2.57391e6	−0.970864
$372$	0	0
$373$	3.49422e6	1.30040	0.650202	−	0.759761i	$-0.274684\pi$
$373$	3.49422e6	1.30040	0.650202	+	0.759761i	$0.274684\pi$
$374$	−1.34300e6	−0.496476
$375$	0	0
$376$	−2.55006e6	−0.930211
$377$	−17606.1	−0.00637983
$378$	0	0
$379$	−3.36760e6	−1.20427	−0.602133	−	0.798396i	$-0.705682\pi$
$379$	−3.36760e6	−1.20427	−0.602133	+	0.798396i	$0.705682\pi$
$380$	0	0
$381$	0	0
$382$	−1.34125e6	−0.470275
$383$	−3.53112e6	−1.23003	−0.615015	−	0.788515i	$-0.710850\pi$
$383$	−3.53112e6	−1.23003	−0.615015	+	0.788515i	$0.710850\pi$
$384$	0	0
$385$	0	0
$386$	1.48863e6	0.508533
$387$	0	0
$388$	−2.24211e6	−0.756098
$389$	−3.49889e6	−1.17235	−0.586173	−	0.810186i	$-0.699366\pi$
$389$	−3.49889e6	−1.17235	−0.586173	+	0.810186i	$0.699366\pi$
$390$	0	0
$391$	−3.65919e6	−1.21044
$392$	−1.47110e6	−0.483534
$393$	0	0
$394$	1.48113e6	0.480678
$395$	0	0
$396$	0	0
$397$	4.17547e6	1.32963	0.664813	−	0.747010i	$-0.268511\pi$
$397$	4.17547e6	1.32963	0.664813	+	0.747010i	$0.268511\pi$
$398$	1.26762e6	0.401125
$399$	0	0
$400$	0	0
$401$	893371.	0.277441	0.138721	−	0.990332i	$-0.455701\pi$
$401$	893371.	0.277441	0.138721	+	0.990332i	$0.455701\pi$
$402$	0	0
$403$	29364.3	0.00900651
$404$	2.26670e6	0.690942
$405$	0	0
$406$	497696.	0.149847
$407$	5.85322e6	1.75149
$408$	0	0
$409$	495248.	0.146391	0.0731955	−	0.997318i	$-0.476680\pi$
$409$	495248.	0.146391	0.0731955	+	0.997318i	$0.476680\pi$
$410$	0	0
$411$	0	0
$412$	246131.	0.0714372
$413$	4.02472e6	1.16107
$414$	0	0
$415$	0	0
$416$	−43776.8	−0.0124025
$417$	0	0
$418$	3.78078e6	1.05838
$419$	4.96965e6	1.38290	0.691450	−	0.722425i	$-0.256972\pi$
$419$	4.96965e6	1.38290	0.691450	+	0.722425i	$0.256972\pi$
$420$	0	0
$421$	−932548.	−0.256428	−0.128214	−	0.991747i	$-0.540924\pi$
$421$	−932548.	−0.256428	−0.128214	+	0.991747i	$0.540924\pi$
$422$	985703.	0.269442
$423$	0	0
$424$	−4.59189e6	−1.24044
$425$	0	0
$426$	0	0
$427$	−161725.	−0.0429247
$428$	1.49928e6	0.395617
$429$	0	0
$430$	0	0
$431$	1.81726e6	0.471221	0.235611	−	0.971848i	$-0.424291\pi$
$431$	1.81726e6	0.471221	0.235611	+	0.971848i	$0.424291\pi$
$432$	0	0
$433$	−2.61506e6	−0.670288	−0.335144	−	0.942167i	$-0.608785\pi$
$433$	−2.61506e6	−0.670288	−0.335144	+	0.942167i	$0.608785\pi$
$434$	−830080.	−0.211542
$435$	0	0
$436$	−2.16538e6	−0.545529
$437$	1.03012e7	2.58039
$438$	0	0
$439$	−3.03139e6	−0.750723	−0.375362	−	0.926878i	$-0.622481\pi$
$439$	−3.03139e6	−0.750723	−0.375362	+	0.926878i	$0.622481\pi$
$440$	0	0
$441$	0	0
$442$	20827.5	0.00507086
$443$	−3.47567e6	−0.841452	−0.420726	−	0.907188i	$-0.638225\pi$
$443$	−3.47567e6	−0.841452	−0.420726	+	0.907188i	$0.638225\pi$
$444$	0	0
$445$	0	0
$446$	825775.	0.196573
$447$	0	0
$448$	88647.6	0.0208676
$449$	−5.53025e6	−1.29458	−0.647290	−	0.762244i	$-0.724098\pi$
$449$	−5.53025e6	−1.29458	−0.647290	+	0.762244i	$0.724098\pi$
$450$	0	0
$451$	3.51554e6	0.813861
$452$	2.88750e6	0.664777
$453$	0	0
$454$	850962.	0.193763
$455$	0	0
$456$	0	0
$457$	−1.42548e6	−0.319279	−0.159639	−	0.987175i	$-0.551033\pi$
$457$	−1.42548e6	−0.319279	−0.159639	+	0.987175i	$0.551033\pi$
$458$	−2.51630e6	−0.560530
$459$	0	0
$460$	0	0
$461$	4.21687e6	0.924141	0.462070	−	0.886843i	$-0.347107\pi$
$461$	4.21687e6	0.924141	0.462070	+	0.886843i	$0.347107\pi$
$462$	0	0
$463$	4.18428e6	0.907128	0.453564	−	0.891224i	$-0.350152\pi$
$463$	4.18428e6	0.907128	0.453564	+	0.891224i	$0.350152\pi$
$464$	−1.01926e6	−0.219781
$465$	0	0
$466$	794637.	0.169513
$467$	1.36779e6	0.290219	0.145110	−	0.989416i	$-0.453646\pi$
$467$	1.36779e6	0.290219	0.145110	+	0.989416i	$0.453646\pi$
$468$	0	0
$469$	−4.92425e6	−1.03373
$470$	0	0
$471$	0	0
$472$	7.18015e6	1.48347
$473$	−1.46271e6	−0.300612
$474$	0	0
$475$	0	0
$476$	2.26869e6	0.458941
$477$	0	0
$478$	−3.91697e6	−0.784115
$479$	2.01693e6	0.401654	0.200827	−	0.979627i	$-0.435637\pi$
$479$	2.01693e6	0.401654	0.200827	+	0.979627i	$0.435637\pi$
$480$	0	0
$481$	−90772.7	−0.0178893
$482$	1.98333e6	0.388846
$483$	0	0
$484$	−1.85899e6	−0.360714
$485$	0	0
$486$	0	0
$487$	−8.07935e6	−1.54367	−0.771834	−	0.635824i	$-0.780660\pi$
$487$	−8.07935e6	−1.54367	−0.771834	+	0.635824i	$0.780660\pi$
$488$	−288520.	−0.0548436
$489$	0	0
$490$	0	0
$491$	−1.09450e6	−0.204885	−0.102443	−	0.994739i	$-0.532666\pi$
$491$	−1.09450e6	−0.204885	−0.102443	+	0.994739i	$0.532666\pi$
$492$	0	0
$493$	2.53513e6	0.469767
$494$	−58633.0	−0.0108100
$495$	0	0
$496$	1.69997e6	0.310268
$497$	−6.91352e6	−1.25548
$498$	0	0
$499$	3.24830e6	0.583989	0.291994	−	0.956420i	$-0.405681\pi$
$499$	3.24830e6	0.583989	0.291994	+	0.956420i	$0.405681\pi$
$500$	0	0
$501$	0	0
$502$	−1.03222e6	−0.182815
$503$	5.19359e6	0.915267	0.457633	−	0.889141i	$-0.348697\pi$
$503$	5.19359e6	0.915267	0.457633	+	0.889141i	$0.348697\pi$
$504$	0	0
$505$	0	0
$506$	4.20768e6	0.730578
$507$	0	0
$508$	7.51454e6	1.29194
$509$	−4.95131e6	−0.847083	−0.423541	−	0.905877i	$-0.639213\pi$
$509$	−4.95131e6	−0.847083	−0.423541	+	0.905877i	$0.639213\pi$
$510$	0	0
$511$	−470387.	−0.0796898
$512$	−4.58391e6	−0.772790
$513$	0	0
$514$	1.24412e6	0.207708
$515$	0	0
$516$	0	0
$517$	8.37233e6	1.37759
$518$	2.56600e6	0.420177
$519$	0	0
$520$	0	0
$521$	5.43792e6	0.877685	0.438843	−	0.898564i	$-0.355389\pi$
$521$	5.43792e6	0.877685	0.438843	+	0.898564i	$0.355389\pi$
$522$	0	0
$523$	−9.04176e6	−1.44544	−0.722718	−	0.691143i	$-0.757107\pi$
$523$	−9.04176e6	−1.44544	−0.722718	+	0.691143i	$0.757107\pi$
$524$	−6.35496e6	−1.01108
$525$	0	0
$526$	2.51509e6	0.396359
$527$	−4.22820e6	−0.663177
$528$	0	0
$529$	5.02805e6	0.781197
$530$	0	0
$531$	0	0
$532$	−6.38674e6	−0.978363
$533$	−54519.5	−0.00831254
$534$	0	0
$535$	0	0
$536$	−8.78494e6	−1.32077
$537$	0	0
$538$	−2.55598e6	−0.380716
$539$	4.82990e6	0.716088
$540$	0	0
$541$	5.97997e6	0.878427	0.439213	−	0.898383i	$-0.355257\pi$
$541$	5.97997e6	0.878427	0.439213	+	0.898383i	$0.355257\pi$
$542$	−118569.	−0.0173369
$543$	0	0
$544$	6.30348e6	0.913236
$545$	0	0
$546$	0	0
$547$	9.52731e6	1.36145	0.680726	−	0.732538i	$-0.261665\pi$
$547$	9.52731e6	1.36145	0.680726	+	0.732538i	$0.261665\pi$
$548$	−6.85496e6	−0.975110
$549$	0	0
$550$	0	0
$551$	−7.13681e6	−1.00144
$552$	0	0
$553$	2.58702e6	0.359739
$554$	2.16903e6	0.300256
$555$	0	0
$556$	6.67881e6	0.916246
$557$	3.80921e6	0.520232	0.260116	−	0.965577i	$-0.416239\pi$
$557$	3.80921e6	0.520232	0.260116	+	0.965577i	$0.416239\pi$
$558$	0	0
$559$	22684.0	0.00307036
$560$	0	0
$561$	0	0
$562$	−1.16467e6	−0.155547
$563$	1.42604e7	1.89609	0.948047	−	0.318131i	$-0.103055\pi$
$563$	1.42604e7	1.89609	0.948047	+	0.318131i	$0.103055\pi$
$564$	0	0
$565$	0	0
$566$	−6.09483e6	−0.799688
$567$	0	0
$568$	−1.23338e7	−1.60408
$569$	1.07341e6	0.138990	0.0694951	−	0.997582i	$-0.477861\pi$
$569$	1.07341e6	0.138990	0.0694951	+	0.997582i	$0.477861\pi$
$570$	0	0
$571$	−9.35129e6	−1.20028	−0.600138	−	0.799896i	$-0.704888\pi$
$571$	−9.35129e6	−1.20028	−0.600138	+	0.799896i	$0.704888\pi$
$572$	92285.0	0.0117935
$573$	0	0
$574$	1.54118e6	0.195242
$575$	0	0
$576$	0	0
$577$	−3.64162e6	−0.455361	−0.227680	−	0.973736i	$-0.573114\pi$
$577$	−3.64162e6	−0.455361	−0.227680	+	0.973736i	$0.573114\pi$
$578$	646875.	0.0805380
$579$	0	0
$580$	0	0
$581$	−3.07056e6	−0.377379
$582$	0	0
$583$	1.50760e7	1.83703
$584$	−839177.	−0.101817
$585$	0	0
$586$	4.41263e6	0.530828
$587$	−7.66557e6	−0.918225	−0.459113	−	0.888378i	$-0.651833\pi$
$587$	−7.66557e6	−0.918225	−0.459113	+	0.888378i	$0.651833\pi$
$588$	0	0
$589$	1.19031e7	1.41375
$590$	0	0
$591$	0	0
$592$	−5.25506e6	−0.616273
$593$	3.74383e6	0.437199	0.218599	−	0.975815i	$-0.429851\pi$
$593$	3.74383e6	0.437199	0.218599	+	0.975815i	$0.429851\pi$
$594$	0	0
$595$	0	0
$596$	3.33339e6	0.384389
$597$	0	0
$598$	−65253.4	−0.00746192
$599$	3.50682e6	0.399343	0.199672	−	0.979863i	$-0.436012\pi$
$599$	3.50682e6	0.399343	0.199672	+	0.979863i	$0.436012\pi$
$600$	0	0
$601$	−1.63306e7	−1.84423	−0.922116	−	0.386914i	$-0.873541\pi$
$601$	−1.63306e7	−1.84423	−0.922116	+	0.386914i	$0.873541\pi$
$602$	−641240.	−0.0721156
$603$	0	0
$604$	−4.44980e6	−0.496305
$605$	0	0
$606$	0	0
$607$	−1.00001e7	−1.10163	−0.550814	−	0.834628i	$-0.685683\pi$
$607$	−1.00001e7	−1.10163	−0.550814	+	0.834628i	$0.685683\pi$
$608$	−1.77454e7	−1.94682
$609$	0	0
$610$	0	0
$611$	−129839.	−0.0140703
$612$	0	0
$613$	−7.99681e6	−0.859539	−0.429770	−	0.902939i	$-0.641405\pi$
$613$	−7.99681e6	−0.859539	−0.429770	+	0.902939i	$0.641405\pi$
$614$	7.42722e6	0.795070
$615$	0	0
$616$	−5.89451e6	−0.625887
$617$	−1.55620e7	−1.64571	−0.822853	−	0.568255i	$-0.807619\pi$
$617$	−1.55620e7	−1.64571	−0.822853	+	0.568255i	$0.807619\pi$
$618$	0	0
$619$	−9.45241e6	−0.991553	−0.495777	−	0.868450i	$-0.665117\pi$
$619$	−9.45241e6	−0.991553	−0.495777	+	0.868450i	$0.665117\pi$
$620$	0	0
$621$	0	0
$622$	−3.29843e6	−0.341846
$623$	−5.71697e6	−0.590128
$624$	0	0
$625$	0	0
$626$	5.97854e6	0.609760
$627$	0	0
$628$	−1.21206e6	−0.122638
$629$	1.30705e7	1.31724
$630$	0	0
$631$	−3.53523e6	−0.353464	−0.176732	−	0.984259i	$-0.556553\pi$
$631$	−3.53523e6	−0.353464	−0.176732	+	0.984259i	$0.556553\pi$
$632$	4.61528e6	0.459627
$633$	0	0
$634$	7.35577e6	0.726783
$635$	0	0
$636$	0	0
$637$	−74902.8	−0.00731391
$638$	−2.91513e6	−0.283534
$639$	0	0
$640$	0	0
$641$	−1.19103e7	−1.14493	−0.572465	−	0.819929i	$-0.694013\pi$
$641$	−1.19103e7	−1.14493	−0.572465	+	0.819929i	$0.694013\pi$
$642$	0	0
$643$	8.67256e6	0.827218	0.413609	−	0.910455i	$-0.364268\pi$
$643$	8.67256e6	0.827218	0.413609	+	0.910455i	$0.364268\pi$
$644$	−7.10789e6	−0.675345
$645$	0	0
$646$	8.44265e6	0.795971
$647$	−1.21563e7	−1.14167	−0.570837	−	0.821063i	$-0.693381\pi$
$647$	−1.21563e7	−1.14167	−0.570837	+	0.821063i	$0.693381\pi$
$648$	0	0
$649$	−2.35738e7	−2.19694
$650$	0	0
$651$	0	0
$652$	574667.	0.0529417
$653$	475958.	0.0436803	0.0218401	−	0.999761i	$-0.493048\pi$
$653$	475958.	0.0436803	0.0218401	+	0.999761i	$0.493048\pi$
$654$	0	0
$655$	0	0
$656$	−3.15627e6	−0.286362
$657$	0	0
$658$	3.67035e6	0.330478
$659$	1.95724e7	1.75562	0.877810	−	0.479008i	$-0.159004\pi$
$659$	1.95724e7	1.75562	0.877810	+	0.479008i	$0.159004\pi$
$660$	0	0
$661$	−1.23639e7	−1.10066	−0.550328	−	0.834948i	$-0.685497\pi$
$661$	−1.23639e7	−1.10066	−0.550328	+	0.834948i	$0.685497\pi$
$662$	−86337.6	−0.00765693
$663$	0	0
$664$	−5.47793e6	−0.482166
$665$	0	0
$666$	0	0
$667$	−7.94265e6	−0.691275
$668$	−8.61520e6	−0.747006
$669$	0	0
$670$	0	0
$671$	947264.	0.0812204
$672$	0	0
$673$	−1.85745e6	−0.158081	−0.0790403	−	0.996871i	$-0.525186\pi$
$673$	−1.85745e6	−0.158081	−0.0790403	+	0.996871i	$0.525186\pi$
$674$	−3.03844e6	−0.257632
$675$	0	0
$676$	9.43186e6	0.793835
$677$	1.96586e7	1.64847	0.824236	−	0.566246i	$-0.191605\pi$
$677$	1.96586e7	1.64847	0.824236	+	0.566246i	$0.191605\pi$
$678$	0	0
$679$	7.29171e6	0.606953
$680$	0	0
$681$	0	0
$682$	4.86199e6	0.400270
$683$	−1.00629e7	−0.825411	−0.412706	−	0.910864i	$-0.635416\pi$
$683$	−1.00629e7	−0.825411	−0.412706	+	0.910864i	$0.635416\pi$
$684$	0	0
$685$	0	0
$686$	5.68324e6	0.461090
$687$	0	0
$688$	1.31323e6	0.105772
$689$	−233802.	−0.0187629
$690$	0	0
$691$	5.94278e6	0.473472	0.236736	−	0.971574i	$-0.423922\pi$
$691$	5.94278e6	0.473472	0.236736	+	0.971574i	$0.423922\pi$
$692$	−6.83627e6	−0.542693
$693$	0	0
$694$	2.52319e6	0.198862
$695$	0	0
$696$	0	0
$697$	7.85034e6	0.612078
$698$	−9.92418e6	−0.771003
$699$	0	0
$700$	0	0
$701$	−9.79494e6	−0.752847	−0.376423	−	0.926448i	$-0.622846\pi$
$701$	−9.79494e6	−0.752847	−0.376423	+	0.926448i	$0.622846\pi$
$702$	0	0
$703$	−3.67957e7	−2.80807
$704$	−519231.	−0.0394847
$705$	0	0
$706$	−7.91824e6	−0.597884
$707$	−7.37169e6	−0.554649
$708$	0	0
$709$	3.95802e6	0.295707	0.147854	−	0.989009i	$-0.452764\pi$
$709$	3.95802e6	0.295707	0.147854	+	0.989009i	$0.452764\pi$
$710$	0	0
$711$	0	0
$712$	−1.01992e7	−0.753988
$713$	1.32471e7	0.975884
$714$	0	0
$715$	0	0
$716$	4.91974e6	0.358641
$717$	0	0
$718$	3.58149e6	0.259270
$719$	−1.16721e7	−0.842029	−0.421014	−	0.907054i	$-0.638326\pi$
$719$	−1.16721e7	−0.842029	−0.421014	+	0.907054i	$0.638326\pi$
$720$	0	0
$721$	−800460.	−0.0573458
$722$	−1.74095e7	−1.24292
$723$	0	0
$724$	−4.21127e6	−0.298584
$725$	0	0
$726$	0	0
$727$	1.05723e7	0.741879	0.370940	−	0.928657i	$-0.379036\pi$
$727$	1.05723e7	0.741879	0.370940	+	0.928657i	$0.379036\pi$
$728$	91413.0	0.00639263
$729$	0	0
$730$	0	0
$731$	−3.26630e6	−0.226080
$732$	0	0
$733$	2.80641e7	1.92927	0.964633	−	0.263598i	$-0.0849093\pi$
$733$	2.80641e7	1.92927	0.964633	+	0.263598i	$0.0849093\pi$
$734$	517554.	0.0354581
$735$	0	0
$736$	−1.97491e7	−1.34385
$737$	2.88426e7	1.95599
$738$	0	0
$739$	2.17470e7	1.46483	0.732417	−	0.680857i	$-0.238392\pi$
$739$	2.17470e7	1.46483	0.732417	+	0.680857i	$0.238392\pi$
$740$	0	0
$741$	0	0
$742$	6.60919e6	0.440696
$743$	1.68996e7	1.12306	0.561531	−	0.827456i	$-0.310213\pi$
$743$	1.68996e7	1.12306	0.561531	+	0.827456i	$0.310213\pi$
$744$	0	0
$745$	0	0
$746$	−8.97234e6	−0.590281
$747$	0	0
$748$	−1.32883e7	−0.868389
$749$	−4.87591e6	−0.317579
$750$	0	0
$751$	−3.68515e6	−0.238427	−0.119213	−	0.992869i	$-0.538037\pi$
$751$	−3.68515e6	−0.238427	−0.119213	+	0.992869i	$0.538037\pi$
$752$	−7.51673e6	−0.484713
$753$	0	0
$754$	45208.3	0.00289594
$755$	0	0
$756$	0	0
$757$	−1.58441e7	−1.00491	−0.502455	−	0.864603i	$-0.667570\pi$
$757$	−1.58441e7	−1.00491	−0.502455	+	0.864603i	$0.667570\pi$
$758$	8.64720e6	0.546641
$759$	0	0
$760$	0	0
$761$	1.08409e7	0.678584	0.339292	−	0.940681i	$-0.389813\pi$
$761$	1.08409e7	0.678584	0.339292	+	0.940681i	$0.389813\pi$
$762$	0	0
$763$	7.04217e6	0.437920
$764$	−1.32709e7	−0.822561
$765$	0	0
$766$	9.06709e6	0.558337
$767$	365586.	0.0224389
$768$	0	0
$769$	4.56486e6	0.278363	0.139182	−	0.990267i	$-0.455553\pi$
$769$	4.56486e6	0.278363	0.139182	+	0.990267i	$0.455553\pi$
$770$	0	0
$771$	0	0
$772$	1.47292e7	0.889478
$773$	1.39415e7	0.839189	0.419595	−	0.907712i	$-0.362172\pi$
$773$	1.39415e7	0.839189	0.419595	+	0.907712i	$0.362172\pi$
$774$	0	0
$775$	0	0
$776$	1.30085e7	0.775485
$777$	0	0
$778$	8.98432e6	0.532152
$779$	−2.21001e7	−1.30482
$780$	0	0
$781$	4.04942e7	2.37556
$782$	9.39594e6	0.549444
$783$	0	0
$784$	−4.33631e6	−0.251959
$785$	0	0
$786$	0	0
$787$	1.64075e7	0.944290	0.472145	−	0.881521i	$-0.343480\pi$
$787$	1.64075e7	0.944290	0.472145	+	0.881521i	$0.343480\pi$
$788$	1.46550e7	0.840756
$789$	0	0
$790$	0	0
$791$	−9.39062e6	−0.533646
$792$	0	0
$793$	−14690.3	−0.000829561 0
$794$	−1.07216e7	−0.603545
$795$	0	0
$796$	1.25424e7	0.701611
$797$	1.72151e7	0.959981	0.479990	−	0.877274i	$-0.340640\pi$
$797$	1.72151e7	0.959981	0.479990	+	0.877274i	$0.340640\pi$
$798$	0	0
$799$	1.86958e7	1.03604
$800$	0	0
$801$	0	0
$802$	−2.29397e6	−0.125936
$803$	2.75517e6	0.150786
$804$	0	0
$805$	0	0
$806$	−75400.5	−0.00408824
$807$	0	0
$808$	−1.31512e7	−0.708658
$809$	1.64138e7	0.881736	0.440868	−	0.897572i	$-0.354671\pi$
$809$	1.64138e7	0.881736	0.440868	+	0.897572i	$0.354671\pi$
$810$	0	0
$811$	1.74140e7	0.929706	0.464853	−	0.885388i	$-0.346107\pi$
$811$	1.74140e7	0.929706	0.464853	+	0.885388i	$0.346107\pi$
$812$	4.92442e6	0.262099
$813$	0	0
$814$	−1.50297e7	−0.795040
$815$	0	0
$816$	0	0
$817$	9.19519e6	0.481954
$818$	−1.27168e6	−0.0664500
$819$	0	0
$820$	0	0
$821$	1.90041e7	0.983986	0.491993	−	0.870599i	$-0.336269\pi$
$821$	1.90041e7	0.983986	0.491993	+	0.870599i	$0.336269\pi$
$822$	0	0
$823$	2.26008e7	1.16312	0.581559	−	0.813505i	$-0.302443\pi$
$823$	2.26008e7	1.16312	0.581559	+	0.813505i	$0.302443\pi$
$824$	−1.42803e6	−0.0732689
$825$	0	0
$826$	−1.03345e7	−0.527036
$827$	−1.71443e7	−0.871679	−0.435840	−	0.900024i	$-0.643548\pi$
$827$	−1.71443e7	−0.871679	−0.435840	+	0.900024i	$0.643548\pi$
$828$	0	0
$829$	−2.99057e7	−1.51136	−0.755680	−	0.654941i	$-0.772693\pi$
$829$	−2.99057e7	−1.51136	−0.755680	+	0.654941i	$0.772693\pi$
$830$	0	0
$831$	0	0
$832$	8052.32	0.000403286 0
$833$	1.07854e7	0.538546
$834$	0	0
$835$	0	0
$836$	3.74087e7	1.85122
$837$	0	0
$838$	−1.27609e7	−0.627727
$839$	1.35881e7	0.666431	0.333215	−	0.942851i	$-0.391866\pi$
$839$	1.35881e7	0.666431	0.333215	+	0.942851i	$0.391866\pi$
$840$	0	0
$841$	−1.50084e7	−0.731719
$842$	2.39456e6	0.116398
$843$	0	0
$844$	9.75298e6	0.471282
$845$	0	0
$846$	0	0
$847$	6.04572e6	0.289561
$848$	−1.35354e7	−0.646369
$849$	0	0
$850$	0	0
$851$	−4.09504e7	−1.93836
$852$	0	0
$853$	9.45702e6	0.445022	0.222511	−	0.974930i	$-0.428575\pi$
$853$	9.45702e6	0.445022	0.222511	+	0.974930i	$0.428575\pi$
$854$	415272.	0.0194844
$855$	0	0
$856$	−8.69870e6	−0.405761
$857$	4.10899e7	1.91110	0.955549	−	0.294832i	$-0.0952638\pi$
$857$	4.10899e7	1.91110	0.955549	+	0.294832i	$0.0952638\pi$
$858$	0	0
$859$	2.17103e7	1.00388	0.501941	−	0.864902i	$-0.332620\pi$
$859$	2.17103e7	1.00388	0.501941	+	0.864902i	$0.332620\pi$
$860$	0	0
$861$	0	0
$862$	−4.66631e6	−0.213897
$863$	−1.59005e7	−0.726750	−0.363375	−	0.931643i	$-0.618376\pi$
$863$	−1.59005e7	−0.726750	−0.363375	+	0.931643i	$0.618376\pi$
$864$	0	0
$865$	0	0
$866$	6.71485e6	0.304258
$867$	0	0
$868$	−8.21318e6	−0.370009
$869$	−1.51528e7	−0.680682
$870$	0	0
$871$	−447296.	−0.0199779
$872$	1.25633e7	0.559517
$873$	0	0
$874$	−2.64512e7	−1.17129
$875$	0	0
$876$	0	0
$877$	−1.04121e7	−0.457129	−0.228564	−	0.973529i	$-0.573403\pi$
$877$	−1.04121e7	−0.457129	−0.228564	+	0.973529i	$0.573403\pi$
$878$	7.78388e6	0.340769
$879$	0	0
$880$	0	0
$881$	4.04001e7	1.75365	0.876824	−	0.480811i	$-0.159658\pi$
$881$	4.04001e7	1.75365	0.876824	+	0.480811i	$0.159658\pi$
$882$	0	0
$883$	6.22610e6	0.268729	0.134365	−	0.990932i	$-0.457101\pi$
$883$	6.22610e6	0.268729	0.134365	+	0.990932i	$0.457101\pi$
$884$	206076.	0.00886947
$885$	0	0
$886$	8.92470e6	0.381953
$887$	8.02230e6	0.342365	0.171183	−	0.985239i	$-0.445241\pi$
$887$	8.02230e6	0.342365	0.171183	+	0.985239i	$0.445241\pi$
$888$	0	0
$889$	−2.44385e7	−1.03710
$890$	0	0
$891$	0	0
$892$	8.17058e6	0.343828
$893$	−5.26318e7	−2.20861
$894$	0	0
$895$	0	0
$896$	1.51943e7	0.632283
$897$	0	0
$898$	1.42004e7	0.587637
$899$	−9.17775e6	−0.378736
$900$	0	0
$901$	3.36654e7	1.38157
$902$	−9.02707e6	−0.369429
$903$	0	0
$904$	−1.67530e7	−0.681823
$905$	0	0
$906$	0	0
$907$	−3.16239e7	−1.27643	−0.638215	−	0.769858i	$-0.720327\pi$
$907$	−3.16239e7	−1.27643	−0.638215	+	0.769858i	$0.720327\pi$
$908$	8.41979e6	0.338912
$909$	0	0
$910$	0	0
$911$	−469497.	−0.0187429	−0.00937146	−	0.999956i	$-0.502983\pi$
$911$	−469497.	−0.0187429	−0.00937146	+	0.999956i	$0.502983\pi$
$912$	0	0
$913$	1.79851e7	0.714061
$914$	3.66029e6	0.144927
$915$	0	0
$916$	−2.48974e7	−0.980427
$917$	2.06674e7	0.811636
$918$	0	0
$919$	−1.27281e7	−0.497137	−0.248568	−	0.968614i	$-0.579960\pi$
$919$	−1.27281e7	−0.497137	−0.248568	+	0.968614i	$0.579960\pi$
$920$	0	0
$921$	0	0
$922$	−1.08279e7	−0.419487
$923$	−627991.	−0.0242633
$924$	0	0
$925$	0	0
$926$	−1.07443e7	−0.411764
$927$	0	0
$928$	1.36824e7	0.521544
$929$	−2.23196e7	−0.848492	−0.424246	−	0.905547i	$-0.639461\pi$
$929$	−2.23196e7	−0.848492	−0.424246	+	0.905547i	$0.639461\pi$
$930$	0	0
$931$	−3.03627e7	−1.14806
$932$	7.86248e6	0.296497
$933$	0	0
$934$	−3.51216e6	−0.131737
$935$	0	0
$936$	0	0
$937$	4.56779e7	1.69964	0.849820	−	0.527072i	$-0.176710\pi$
$937$	4.56779e7	1.69964	0.849820	+	0.527072i	$0.176710\pi$
$938$	1.26443e7	0.469233
$939$	0	0
$940$	0	0
$941$	6.91771e6	0.254676	0.127338	−	0.991859i	$-0.459357\pi$
$941$	6.91771e6	0.254676	0.127338	+	0.991859i	$0.459357\pi$
$942$	0	0
$943$	−2.45955e7	−0.900691
$944$	2.11647e7	0.773004
$945$	0	0
$946$	3.75590e6	0.136454
$947$	−7.12916e6	−0.258323	−0.129162	−	0.991624i	$-0.541229\pi$
$947$	−7.12916e6	−0.258323	−0.129162	+	0.991624i	$0.541229\pi$
$948$	0	0
$949$	−42727.7	−0.00154008
$950$	0	0
$951$	0	0
$952$	−1.31627e7	−0.470709
$953$	2.68417e7	0.957365	0.478683	−	0.877988i	$-0.341114\pi$
$953$	2.68417e7	0.957365	0.478683	+	0.877988i	$0.341114\pi$
$954$	0	0
$955$	0	0
$956$	−3.87562e7	−1.37150
$957$	0	0
$958$	−5.17900e6	−0.182319
$959$	2.22935e7	0.782764
$960$	0	0
$961$	−1.33221e7	−0.465332
$962$	233083.	0.00812031
$963$	0	0
$964$	1.96240e7	0.680134
$965$	0	0
$966$	0	0
$967$	−1.01997e7	−0.350770	−0.175385	−	0.984500i	$-0.556117\pi$
$967$	−1.01997e7	−0.350770	−0.175385	+	0.984500i	$0.556117\pi$
$968$	1.07857e7	0.369963
$969$	0	0
$970$	0	0
$971$	3.54899e6	0.120797	0.0603986	−	0.998174i	$-0.480763\pi$
$971$	3.54899e6	0.120797	0.0603986	+	0.998174i	$0.480763\pi$
$972$	0	0
$973$	−2.17206e7	−0.735511
$974$	2.07459e7	0.700703
$975$	0	0
$976$	−850460.	−0.0285778
$977$	−94269.8	−0.00315963	−0.00157981	−	0.999999i	$-0.500503\pi$
$977$	−94269.8	−0.00315963	−0.00157981	+	0.999999i	$0.500503\pi$
$978$	0	0
$979$	3.34857e7	1.11661
$980$	0	0
$981$	0	0
$982$	2.81041e6	0.0930018
$983$	2.17884e7	0.719185	0.359593	−	0.933109i	$-0.382916\pi$
$983$	2.17884e7	0.719185	0.359593	+	0.933109i	$0.382916\pi$
$984$	0	0
$985$	0	0
$986$	−6.50960e6	−0.213237
$987$	0	0
$988$	−580140.	−0.0189078
$989$	1.02335e7	0.332684
$990$	0	0
$991$	−672678.	−0.0217582	−0.0108791	−	0.999941i	$-0.503463\pi$
$991$	−672678.	−0.0217582	−0.0108791	+	0.999941i	$0.503463\pi$
$992$	−2.28201e7	−0.736272
$993$	0	0
$994$	1.77523e7	0.569887
$995$	0	0
$996$	0	0
$997$	391424.	0.0124712	0.00623562	−	0.999981i	$-0.498015\pi$
$997$	391424.	0.0124712	0.00623562	+	0.999981i	$0.498015\pi$
$998$	−8.34086e6	−0.265085
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.a.t.1.1		2
3.2	odd	2		75.6.a.g.1.2	✓	2
5.2	odd	4		225.6.b.l.199.2		4
5.3	odd	4		225.6.b.l.199.3		4
5.4	even	2		225.6.a.j.1.2		2
15.2	even	4		75.6.b.f.49.3		4
15.8	even	4		75.6.b.f.49.2		4
15.14	odd	2		75.6.a.i.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.a.g.1.2	✓	2	3.2	odd	2
75.6.a.i.1.1	yes	2	15.14	odd	2
75.6.b.f.49.2		4	15.8	even	4
75.6.b.f.49.3		4	15.2	even	4
225.6.a.j.1.2		2	5.4	even	2
225.6.a.t.1.1		2	1.1	even	1	trivial
225.6.b.l.199.2		4	5.2	odd	4
225.6.b.l.199.3		4	5.3	odd	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)

\(f(q)\)	\(=\)	\(q-2.56776 q^{2} -25.4066 q^{4} +82.6263 q^{7} +147.407 q^{8} +O(q^{10})\)	\(q-2.56776 q^{2} -25.4066 q^{4} +82.6263 q^{7} +147.407 q^{8} -483.963 q^{11} +7.50538 q^{13} -212.165 q^{14} +434.505 q^{16} -1080.71 q^{17} +3042.38 q^{19} +1242.70 q^{22} +3385.91 q^{23} -19.2720 q^{26} -2099.25 q^{28} -2345.79 q^{29} +3912.43 q^{31} -5832.72 q^{32} +2775.01 q^{34} -12094.3 q^{37} -7812.13 q^{38} -7264.06 q^{41} +3022.36 q^{43} +12295.9 q^{44} -8694.22 q^{46} -17299.5 q^{47} -9979.89 q^{49} -190.686 q^{52} -31151.2 q^{53} +12179.7 q^{56} +6023.45 q^{58} +48709.8 q^{59} -1957.31 q^{61} -10046.2 q^{62} +1072.87 q^{64} -59596.7 q^{67} +27457.2 q^{68} -83672.1 q^{71} -5692.94 q^{73} +31055.4 q^{74} -77296.6 q^{76} -39988.1 q^{77} +31309.9 q^{79} +18652.4 q^{82} -37162.0 q^{83} -7760.72 q^{86} -71339.4 q^{88} -69190.7 q^{89} +620.142 q^{91} -86024.4 q^{92} +44421.1 q^{94} +88249.2 q^{97} +25626.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{2} + 16 q^{4} - 102 q^{7} + 228 q^{8}+O(q^{10}) \) 2 * q + 6 * q^2 + 16 * q^4 - 102 * q^7 + 228 * q^8	\( 2 q + 6 q^{2} + 16 q^{4} - 102 q^{7} + 228 q^{8} + 12 q^{11} - 1054 q^{13} - 1794 q^{14} - 200 q^{16} - 1716 q^{17} + 4214 q^{19} + 5492 q^{22} - 444 q^{23} - 9114 q^{26} - 9744 q^{28} - 4068 q^{29} - 2598 q^{31} - 13848 q^{32} - 2668 q^{34} - 4412 q^{37} + 2226 q^{38} - 11232 q^{41} + 8450 q^{43} + 32832 q^{44} - 41508 q^{46} + 2460 q^{47} + 7300 q^{49} - 44144 q^{52} - 65064 q^{53} - 2700 q^{56} - 8732 q^{58} + 63924 q^{59} + 7310 q^{61} - 65826 q^{62} - 47296 q^{64} - 61734 q^{67} + 1152 q^{68} - 98304 q^{71} + 26564 q^{73} + 96876 q^{74} - 28784 q^{76} - 131556 q^{77} + 84000 q^{79} - 15344 q^{82} - 65772 q^{83} + 38742 q^{86} - 31368 q^{88} - 103104 q^{89} + 196602 q^{91} - 244608 q^{92} + 213716 q^{94} - 69374 q^{97} + 173676 q^{98}+O(q^{100}) \) 2 * q + 6 * q^2 + 16 * q^4 - 102 * q^7 + 228 * q^8 + 12 * q^11 - 1054 * q^13 - 1794 * q^14 - 200 * q^16 - 1716 * q^17 + 4214 * q^19 + 5492 * q^22 - 444 * q^23 - 9114 * q^26 - 9744 * q^28 - 4068 * q^29 - 2598 * q^31 - 13848 * q^32 - 2668 * q^34 - 4412 * q^37 + 2226 * q^38 - 11232 * q^41 + 8450 * q^43 + 32832 * q^44 - 41508 * q^46 + 2460 * q^47 + 7300 * q^49 - 44144 * q^52 - 65064 * q^53 - 2700 * q^56 - 8732 * q^58 + 63924 * q^59 + 7310 * q^61 - 65826 * q^62 - 47296 * q^64 - 61734 * q^67 + 1152 * q^68 - 98304 * q^71 + 26564 * q^73 + 96876 * q^74 - 28784 * q^76 - 131556 * q^77 + 84000 * q^79 - 15344 * q^82 - 65772 * q^83 + 38742 * q^86 - 31368 * q^88 - 103104 * q^89 + 196602 * q^91 - 244608 * q^92 + 213716 * q^94 - 69374 * q^97 + 173676 * q^98

Introduction

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