LMFDB - Embedded newform 225.6.a.i.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{89}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 22 $ x^2 - x - 22
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.21699$ 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)

$f(q)$	$=$	$q-9.21699 q^{2} +52.9529 q^{4} +167.208 q^{7} -193.123 q^{8} +O(q^{10})$	$q-9.21699 q^{2} +52.9529 q^{4} +167.208 q^{7} -193.123 q^{8} -593.435 q^{11} +251.773 q^{13} -1541.15 q^{14} +85.5179 q^{16} +372.379 q^{17} -1007.62 q^{19} +5469.68 q^{22} -2536.90 q^{23} -2320.59 q^{26} +8854.14 q^{28} +5851.48 q^{29} -6503.25 q^{31} +5391.71 q^{32} -3432.21 q^{34} +5476.07 q^{37} +9287.25 q^{38} +13358.2 q^{41} +2559.64 q^{43} -31424.1 q^{44} +23382.6 q^{46} -7695.48 q^{47} +11151.4 q^{49} +13332.1 q^{52} +1597.41 q^{53} -32291.6 q^{56} -53933.0 q^{58} -12875.5 q^{59} -30013.4 q^{61} +59940.4 q^{62} -52431.9 q^{64} -24101.5 q^{67} +19718.5 q^{68} -23995.0 q^{71} +2007.86 q^{73} -50472.9 q^{74} -53356.6 q^{76} -99226.9 q^{77} -72911.0 q^{79} -123122. q^{82} -34769.5 q^{83} -23592.1 q^{86} +114606. q^{88} -45966.9 q^{89} +42098.4 q^{91} -134337. q^{92} +70929.2 q^{94} -30817.1 q^{97} -102783. q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{2} + 21 q^{4} + 108 q^{7} - 207 q^{8}+O(q^{10}) $ 2 * q - 9 * q^2 + 21 * q^4 + 108 * q^7 - 207 * q^8	$ 2 q - 9 q^{2} + 21 q^{4} + 108 q^{7} - 207 q^{8} - 168 q^{11} + 1296 q^{13} - 1554 q^{14} + 1105 q^{16} - 576 q^{17} - 1336 q^{19} + 5562 q^{22} - 5904 q^{23} - 2094 q^{26} + 10746 q^{28} + 3552 q^{29} - 11648 q^{31} + 6057 q^{32} - 3638 q^{34} + 14688 q^{37} + 9216 q^{38} - 1812 q^{41} - 7560 q^{43} - 45018 q^{44} + 22652 q^{46} - 3240 q^{47} - 2150 q^{49} - 20034 q^{52} + 22176 q^{53} - 31470 q^{56} - 54432 q^{58} - 57336 q^{59} - 30140 q^{61} + 58824 q^{62} - 84911 q^{64} - 5184 q^{67} + 50022 q^{68} - 17424 q^{71} - 3456 q^{73} - 48474 q^{74} - 42864 q^{76} - 124416 q^{77} - 57520 q^{79} - 126414 q^{82} - 133992 q^{83} - 25788 q^{86} + 108702 q^{88} - 136764 q^{89} - 19728 q^{91} - 26748 q^{92} + 71896 q^{94} + 85536 q^{97} - 105669 q^{98}+O(q^{100}) $ 2 * q - 9 * q^2 + 21 * q^4 + 108 * q^7 - 207 * q^8 - 168 * q^11 + 1296 * q^13 - 1554 * q^14 + 1105 * q^16 - 576 * q^17 - 1336 * q^19 + 5562 * q^22 - 5904 * q^23 - 2094 * q^26 + 10746 * q^28 + 3552 * q^29 - 11648 * q^31 + 6057 * q^32 - 3638 * q^34 + 14688 * q^37 + 9216 * q^38 - 1812 * q^41 - 7560 * q^43 - 45018 * q^44 + 22652 * q^46 - 3240 * q^47 - 2150 * q^49 - 20034 * q^52 + 22176 * q^53 - 31470 * q^56 - 54432 * q^58 - 57336 * q^59 - 30140 * q^61 + 58824 * q^62 - 84911 * q^64 - 5184 * q^67 + 50022 * q^68 - 17424 * q^71 - 3456 * q^73 - 48474 * q^74 - 42864 * q^76 - 124416 * q^77 - 57520 * q^79 - 126414 * q^82 - 133992 * q^83 - 25788 * q^86 + 108702 * q^88 - 136764 * q^89 - 19728 * q^91 - 26748 * q^92 + 71896 * q^94 + 85536 * q^97 - 105669 * 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 $

$2$	−9.21699	−1.62935	−0.814675	−	0.579918i	$-0.803084\pi$
$2$	−9.21699	−1.62935	−0.814675	+	0.579918i	$0.803084\pi$
$3$	0	0
$3$	0	0
$4$	52.9529	1.65478
$5$	0	0
$5$	0	0
$6$	0	0
$7$	167.208	1.28977	0.644884	−	0.764281i	$-0.276906\pi$
$7$	167.208	1.28977	0.644884	+	0.764281i	$0.276906\pi$
$8$	−193.123	−1.06686
$9$	0	0
$10$	0	0
$11$	−593.435	−1.47874	−0.739369	−	0.673300i	$-0.764876\pi$
$11$	−593.435	−1.47874	−0.739369	+	0.673300i	$0.764876\pi$
$12$	0	0
$13$	251.773	0.413191	0.206595	−	0.978426i	$-0.433762\pi$
$13$	251.773	0.413191	0.206595	+	0.978426i	$0.433762\pi$
$14$	−1541.15	−2.10148
$15$	0	0
$16$	85.5179	0.0835136
$17$	372.379	0.312509	0.156254	−	0.987717i	$-0.450058\pi$
$17$	372.379	0.312509	0.156254	+	0.987717i	$0.450058\pi$
$18$	0	0
$19$	−1007.62	−0.640345	−0.320173	−	0.947359i	$-0.603741\pi$
$19$	−1007.62	−0.640345	−0.320173	+	0.947359i	$0.603741\pi$
$20$	0	0
$21$	0	0
$22$	5469.68	2.40938
$23$	−2536.90	−0.999965	−0.499982	−	0.866036i	$-0.666660\pi$
$23$	−2536.90	−0.999965	−0.499982	+	0.866036i	$0.666660\pi$
$24$	0	0
$25$	0	0
$26$	−2320.59	−0.673232
$27$	0	0
$28$	8854.14	2.13428
$29$	5851.48	1.29202	0.646012	−	0.763327i	$-0.276436\pi$
$29$	5851.48	1.29202	0.646012	+	0.763327i	$0.276436\pi$
$30$	0	0
$31$	−6503.25	−1.21542	−0.607709	−	0.794159i	$-0.707911\pi$
$31$	−6503.25	−1.21542	−0.607709	+	0.794159i	$0.707911\pi$
$32$	5391.71	0.930790
$33$	0	0
$34$	−3432.21	−0.509186
$35$	0	0
$36$	0	0
$37$	5476.07	0.657605	0.328802	−	0.944399i	$-0.393355\pi$
$37$	5476.07	0.657605	0.328802	+	0.944399i	$0.393355\pi$
$38$	9287.25	1.04335
$39$	0	0
$40$	0	0
$41$	13358.2	1.24105	0.620523	−	0.784189i	$-0.286920\pi$
$41$	13358.2	1.24105	0.620523	+	0.784189i	$0.286920\pi$
$42$	0	0
$43$	2559.64	0.211109	0.105555	−	0.994414i	$-0.466338\pi$
$43$	2559.64	0.211109	0.105555	+	0.994414i	$0.466338\pi$
$44$	−31424.1	−2.44699
$45$	0	0
$46$	23382.6	1.62929
$47$	−7695.48	−0.508149	−0.254075	−	0.967185i	$-0.581771\pi$
$47$	−7695.48	−0.508149	−0.254075	+	0.967185i	$0.581771\pi$
$48$	0	0
$49$	11151.4	0.663500
$50$	0	0
$51$	0	0
$52$	13332.1	0.683739
$53$	1597.41	0.0781139	0.0390569	−	0.999237i	$-0.487565\pi$
$53$	1597.41	0.0781139	0.0390569	+	0.999237i	$0.487565\pi$
$54$	0	0
$55$	0	0
$56$	−32291.6	−1.37600
$57$	0	0
$58$	−53933.0	−2.10516
$59$	−12875.5	−0.481542	−0.240771	−	0.970582i	$-0.577400\pi$
$59$	−12875.5	−0.481542	−0.240771	+	0.970582i	$0.577400\pi$
$60$	0	0
$61$	−30013.4	−1.03274	−0.516370	−	0.856366i	$-0.672717\pi$
$61$	−30013.4	−1.03274	−0.516370	+	0.856366i	$0.672717\pi$
$62$	59940.4	1.98034
$63$	0	0
$64$	−52431.9	−1.60010
$65$	0	0
$66$	0	0
$67$	−24101.5	−0.655929	−0.327964	−	0.944690i	$-0.606363\pi$
$67$	−24101.5	−0.655929	−0.327964	+	0.944690i	$0.606363\pi$
$68$	19718.5	0.517133
$69$	0	0
$70$	0	0
$71$	−23995.0	−0.564905	−0.282453	−	0.959281i	$-0.591148\pi$
$71$	−23995.0	−0.564905	−0.282453	+	0.959281i	$0.591148\pi$
$72$	0	0
$73$	2007.86	0.0440987	0.0220493	−	0.999757i	$-0.492981\pi$
$73$	2007.86	0.0440987	0.0220493	+	0.999757i	$0.492981\pi$
$74$	−50472.9	−1.07147
$75$	0	0
$76$	−53356.6	−1.05963
$77$	−99226.9	−1.90723
$78$	0	0
$79$	−72911.0	−1.31439	−0.657197	−	0.753719i	$-0.728258\pi$
$79$	−72911.0	−1.31439	−0.657197	+	0.753719i	$0.728258\pi$
$80$	0	0
$81$	0	0
$82$	−123122.	−2.02210
$83$	−34769.5	−0.553992	−0.276996	−	0.960871i	$-0.589339\pi$
$83$	−34769.5	−0.553992	−0.276996	+	0.960871i	$0.589339\pi$
$84$	0	0
$85$	0	0
$86$	−23592.1	−0.343970
$87$	0	0
$88$	114606.	1.57761
$89$	−45966.9	−0.615134	−0.307567	−	0.951526i	$-0.599515\pi$
$89$	−45966.9	−0.615134	−0.307567	+	0.951526i	$0.599515\pi$
$90$	0	0
$91$	42098.4	0.532920
$92$	−134337.	−1.65472
$93$	0	0
$94$	70929.2	0.827953
$95$	0	0
$96$	0	0
$97$	−30817.1	−0.332554	−0.166277	−	0.986079i	$-0.553175\pi$
$97$	−30817.1	−0.332554	−0.166277	+	0.986079i	$0.553175\pi$
$98$	−102783.	−1.08107
$99$	0	0
$100$	0	0
$101$	−80700.2	−0.787174	−0.393587	−	0.919287i	$-0.628766\pi$
$101$	−80700.2	−0.787174	−0.393587	+	0.919287i	$0.628766\pi$
$102$	0	0
$103$	−120645.	−1.12051	−0.560255	−	0.828320i	$-0.689297\pi$
$103$	−120645.	−1.12051	−0.560255	+	0.828320i	$0.689297\pi$
$104$	−48623.1	−0.440818
$105$	0	0
$106$	−14723.4	−0.127275
$107$	79144.0	0.668280	0.334140	−	0.942523i	$-0.391554\pi$
$107$	79144.0	0.668280	0.334140	+	0.942523i	$0.391554\pi$
$108$	0	0
$109$	223285.	1.80008	0.900042	−	0.435804i	$-0.143536\pi$
$109$	223285.	1.80008	0.900042	+	0.435804i	$0.143536\pi$
$110$	0	0
$111$	0	0
$112$	14299.3	0.107713
$113$	−76664.3	−0.564803	−0.282402	−	0.959296i	$-0.591131\pi$
$113$	−76664.3	−0.564803	−0.282402	+	0.959296i	$0.591131\pi$
$114$	0	0
$115$	0	0
$116$	309853.	2.13801
$117$	0	0
$118$	118674.	0.784601
$119$	62264.6	0.403064
$120$	0	0
$121$	191114.	1.18667
$122$	276633.	1.68269
$123$	0	0
$124$	−344366.	−2.01125
$125$	0	0
$126$	0	0
$127$	357475.	1.96669	0.983345	−	0.181746i	$-0.0581750\pi$
$127$	357475.	1.96669	0.983345	+	0.181746i	$0.0581750\pi$
$128$	310730.	1.67632
$129$	0	0
$130$	0	0
$131$	−13882.8	−0.0706802	−0.0353401	−	0.999375i	$-0.511251\pi$
$131$	−13882.8	−0.0706802	−0.0353401	+	0.999375i	$0.511251\pi$
$132$	0	0
$133$	−168482.	−0.825896
$134$	222143.	1.06874
$135$	0	0
$136$	−71914.8	−0.333404
$137$	−189123.	−0.860880	−0.430440	−	0.902619i	$-0.641642\pi$
$137$	−189123.	−0.860880	−0.430440	+	0.902619i	$0.641642\pi$
$138$	0	0
$139$	−250410.	−1.09930	−0.549648	−	0.835396i	$-0.685238\pi$
$139$	−250410.	−1.09930	−0.549648	+	0.835396i	$0.685238\pi$
$140$	0	0
$141$	0	0
$142$	221162.	0.920428
$143$	−149411.	−0.611001
$144$	0	0
$145$	0	0
$146$	−18506.4	−0.0718521
$147$	0	0
$148$	289974.	1.08819
$149$	43424.1	0.160238	0.0801190	−	0.996785i	$-0.474470\pi$
$149$	43424.1	0.160238	0.0801190	+	0.996785i	$0.474470\pi$
$150$	0	0
$151$	−309623.	−1.10507	−0.552536	−	0.833489i	$-0.686340\pi$
$151$	−309623.	−1.10507	−0.552536	+	0.833489i	$0.686340\pi$
$152$	194595.	0.683161
$153$	0	0
$154$	914574.	3.10754
$155$	0	0
$156$	0	0
$157$	−308759.	−0.999701	−0.499850	−	0.866112i	$-0.666612\pi$
$157$	−308759.	−0.999701	−0.499850	+	0.866112i	$0.666612\pi$
$158$	672020.	2.14161
$159$	0	0
$160$	0	0
$161$	−424190.	−1.28972
$162$	0	0
$163$	−557904.	−1.64471	−0.822357	−	0.568972i	$-0.807341\pi$
$163$	−557904.	−1.64471	−0.822357	+	0.568972i	$0.807341\pi$
$164$	707355.	2.05366
$165$	0	0
$166$	320470.	0.902646
$167$	380471.	1.05567	0.527837	−	0.849346i	$-0.323003\pi$
$167$	380471.	1.05567	0.527837	+	0.849346i	$0.323003\pi$
$168$	0	0
$169$	−307903.	−0.829274
$170$	0	0
$171$	0	0
$172$	135540.	0.349339
$173$	−452767.	−1.15016	−0.575082	−	0.818096i	$-0.695030\pi$
$173$	−452767.	−1.15016	−0.575082	+	0.818096i	$0.695030\pi$
$174$	0	0
$175$	0	0
$176$	−50749.3	−0.123495
$177$	0	0
$178$	423676.	1.00227
$179$	−646150.	−1.50730	−0.753652	−	0.657274i	$-0.771709\pi$
$179$	−646150.	−1.50730	−0.753652	+	0.657274i	$0.771709\pi$
$180$	0	0
$181$	−365967.	−0.830320	−0.415160	−	0.909748i	$-0.636274\pi$
$181$	−365967.	−0.830320	−0.415160	+	0.909748i	$0.636274\pi$
$182$	−388020.	−0.868312
$183$	0	0
$184$	489934.	1.06683
$185$	0	0
$186$	0	0
$187$	−220983.	−0.462119
$188$	−407498.	−0.840874
$189$	0	0
$190$	0	0
$191$	591577.	1.17335	0.586675	−	0.809822i	$-0.300436\pi$
$191$	591577.	1.17335	0.586675	+	0.809822i	$0.300436\pi$
$192$	0	0
$193$	574896.	1.11095	0.555477	−	0.831532i	$-0.312536\pi$
$193$	574896.	1.11095	0.555477	+	0.831532i	$0.312536\pi$
$194$	284040.	0.541846
$195$	0	0
$196$	590501.	1.09795
$197$	−462197.	−0.848519	−0.424259	−	0.905541i	$-0.639465\pi$
$197$	−462197.	−0.848519	−0.424259	+	0.905541i	$0.639465\pi$
$198$	0	0
$199$	246508.	0.441264	0.220632	−	0.975357i	$-0.429188\pi$
$199$	246508.	0.441264	0.220632	+	0.975357i	$0.429188\pi$
$200$	0	0
$201$	0	0
$202$	743813.	1.28258
$203$	978413.	1.66641
$204$	0	0
$205$	0	0
$206$	1.11198e6	1.82570
$207$	0	0
$208$	21531.1	0.0345070
$209$	597959.	0.946903
$210$	0	0
$211$	352855.	0.545620	0.272810	−	0.962068i	$-0.412047\pi$
$211$	352855.	0.545620	0.272810	+	0.962068i	$0.412047\pi$
$212$	84587.8	0.129261
$213$	0	0
$214$	−729470.	−1.08886
$215$	0	0
$216$	0	0
$217$	−1.08739e6	−1.56761
$218$	−2.05801e6	−2.93296
$219$	0	0
$220$	0	0
$221$	93754.8	0.129126
$222$	0	0
$223$	−1.21399e6	−1.63476	−0.817380	−	0.576099i	$-0.804574\pi$
$223$	−1.21399e6	−1.63476	−0.817380	+	0.576099i	$0.804574\pi$
$224$	901536.	1.20050
$225$	0	0
$226$	706615.	0.920262
$227$	262045.	0.337529	0.168765	−	0.985656i	$-0.446022\pi$
$227$	262045.	0.337529	0.168765	+	0.985656i	$0.446022\pi$
$228$	0	0
$229$	197547.	0.248933	0.124466	−	0.992224i	$-0.460278\pi$
$229$	197547.	0.248933	0.124466	+	0.992224i	$0.460278\pi$
$230$	0	0
$231$	0	0
$232$	−1.13005e6	−1.37841
$233$	−568550.	−0.686086	−0.343043	−	0.939320i	$-0.611458\pi$
$233$	−568550.	−0.686086	−0.343043	+	0.939320i	$0.611458\pi$
$234$	0	0
$235$	0	0
$236$	−681796.	−0.796846
$237$	0	0
$238$	−573892.	−0.656732
$239$	381660.	0.432197	0.216099	−	0.976372i	$-0.430667\pi$
$239$	381660.	0.432197	0.216099	+	0.976372i	$0.430667\pi$
$240$	0	0
$241$	−1.52448e6	−1.69075	−0.845373	−	0.534176i	$-0.820622\pi$
$241$	−1.52448e6	−1.69075	−0.845373	+	0.534176i	$0.820622\pi$
$242$	−1.76150e6	−1.93350
$243$	0	0
$244$	−1.58930e6	−1.70896
$245$	0	0
$246$	0	0
$247$	−253692.	−0.264585
$248$	1.25593e6	1.29669
$249$	0	0
$250$	0	0
$251$	−1.95644e6	−1.96012	−0.980059	−	0.198709i	$-0.936325\pi$
$251$	−1.95644e6	−1.96012	−0.980059	+	0.198709i	$0.936325\pi$
$252$	0	0
$253$	1.50549e6	1.47869
$254$	−3.29484e6	−3.20443
$255$	0	0
$256$	−1.18617e6	−1.13122
$257$	337773.	0.319001	0.159501	−	0.987198i	$-0.449012\pi$
$257$	337773.	0.319001	0.159501	+	0.987198i	$0.449012\pi$
$258$	0	0
$259$	915642.	0.848157
$260$	0	0
$261$	0	0
$262$	127957.	0.115163
$263$	129654.	0.115584	0.0577919	−	0.998329i	$-0.481594\pi$
$263$	129654.	0.115584	0.0577919	+	0.998329i	$0.481594\pi$
$264$	0	0
$265$	0	0
$266$	1.55290e6	1.34567
$267$	0	0
$268$	−1.27624e6	−1.08542
$269$	−993788.	−0.837362	−0.418681	−	0.908133i	$-0.637507\pi$
$269$	−993788.	−0.837362	−0.418681	+	0.908133i	$0.637507\pi$
$270$	0	0
$271$	307758.	0.254558	0.127279	−	0.991867i	$-0.459376\pi$
$271$	307758.	0.254558	0.127279	+	0.991867i	$0.459376\pi$
$272$	31845.1	0.0260988
$273$	0	0
$274$	1.74314e6	1.40267
$275$	0	0
$276$	0	0
$277$	404493.	0.316747	0.158373	−	0.987379i	$-0.449375\pi$
$277$	404493.	0.316747	0.158373	+	0.987379i	$0.449375\pi$
$278$	2.30803e6	1.79114
$279$	0	0
$280$	0	0
$281$	−2.18002e6	−1.64700	−0.823501	−	0.567314i	$-0.807982\pi$
$281$	−2.18002e6	−1.64700	−0.823501	+	0.567314i	$0.807982\pi$
$282$	0	0
$283$	748368.	0.555455	0.277727	−	0.960660i	$-0.410419\pi$
$283$	748368.	0.555455	0.277727	+	0.960660i	$0.410419\pi$
$284$	−1.27061e6	−0.934793
$285$	0	0
$286$	1.37712e6	0.995534
$287$	2.23359e6	1.60066
$288$	0	0
$289$	−1.28119e6	−0.902338
$290$	0	0
$291$	0	0
$292$	106322.	0.0729735
$293$	2.85626e6	1.94370	0.971850	−	0.235600i	$-0.0757056\pi$
$293$	2.85626e6	1.94370	0.971850	+	0.235600i	$0.0757056\pi$
$294$	0	0
$295$	0	0
$296$	−1.05755e6	−0.701574
$297$	0	0
$298$	−400240.	−0.261084
$299$	−638724.	−0.413176
$300$	0	0
$301$	427991.	0.272282
$302$	2.85379e6	1.80055
$303$	0	0
$304$	−86169.9	−0.0534775
$305$	0	0
$306$	0	0
$307$	1.38872e6	0.840946	0.420473	−	0.907305i	$-0.361864\pi$
$307$	1.38872e6	0.840946	0.420473	+	0.907305i	$0.361864\pi$
$308$	−5.25436e6	−3.15604
$309$	0	0
$310$	0	0
$311$	−1.68775e6	−0.989477	−0.494739	−	0.869042i	$-0.664736\pi$
$311$	−1.68775e6	−0.989477	−0.494739	+	0.869042i	$0.664736\pi$
$312$	0	0
$313$	1.74335e6	1.00583	0.502915	−	0.864336i	$-0.332261\pi$
$313$	1.74335e6	1.00583	0.502915	+	0.864336i	$0.332261\pi$
$314$	2.84583e6	1.62886
$315$	0	0
$316$	−3.86085e6	−2.17503
$317$	−1.47408e6	−0.823897	−0.411948	−	0.911207i	$-0.635152\pi$
$317$	−1.47408e6	−0.823897	−0.411948	+	0.911207i	$0.635152\pi$
$318$	0	0
$319$	−3.47247e6	−1.91057
$320$	0	0
$321$	0	0
$322$	3.90976e6	2.10141
$323$	−375217.	−0.200114
$324$	0	0
$325$	0	0
$326$	5.14219e6	2.67981
$327$	0	0
$328$	−2.57977e6	−1.32403
$329$	−1.28674e6	−0.655394
$330$	0	0
$331$	1.26710e6	0.635684	0.317842	−	0.948144i	$-0.397042\pi$
$331$	1.26710e6	0.635684	0.317842	+	0.948144i	$0.397042\pi$
$332$	−1.84115e6	−0.916734
$333$	0	0
$334$	−3.50679e6	−1.72006
$335$	0	0
$336$	0	0
$337$	2.81909e6	1.35218	0.676089	−	0.736820i	$-0.263673\pi$
$337$	2.81909e6	1.35218	0.676089	+	0.736820i	$0.263673\pi$
$338$	2.83794e6	1.35118
$339$	0	0
$340$	0	0
$341$	3.85925e6	1.79729
$342$	0	0
$343$	−945654.	−0.434007
$344$	−494324.	−0.225224
$345$	0	0
$346$	4.17315e6	1.87402
$347$	−2.27369e6	−1.01369	−0.506847	−	0.862036i	$-0.669189\pi$
$347$	−2.27369e6	−1.01369	−0.506847	+	0.862036i	$0.669189\pi$
$348$	0	0
$349$	4.26326e6	1.87361	0.936803	−	0.349858i	$-0.113770\pi$
$349$	4.26326e6	1.87361	0.936803	+	0.349858i	$0.113770\pi$
$350$	0	0
$351$	0	0
$352$	−3.19963e6	−1.37640
$353$	−4.40011e6	−1.87943	−0.939716	−	0.341957i	$-0.888910\pi$
$353$	−4.40011e6	−1.87943	−0.939716	+	0.341957i	$0.888910\pi$
$354$	0	0
$355$	0	0
$356$	−2.43408e6	−1.01791
$357$	0	0
$358$	5.95556e6	2.45592
$359$	−1.55859e6	−0.638256	−0.319128	−	0.947712i	$-0.603390\pi$
$359$	−1.55859e6	−0.638256	−0.319128	+	0.947712i	$0.603390\pi$
$360$	0	0
$361$	−1.46079e6	−0.589958
$362$	3.37312e6	1.35288
$363$	0	0
$364$	2.22923e6	0.881864
$365$	0	0
$366$	0	0
$367$	−593193.	−0.229896	−0.114948	−	0.993372i	$-0.536670\pi$
$367$	−593193.	−0.229896	−0.114948	+	0.993372i	$0.536670\pi$
$368$	−216951.	−0.0835106
$369$	0	0
$370$	0	0
$371$	267100.	0.100749
$372$	0	0
$373$	2.57747e6	0.959229	0.479614	−	0.877479i	$-0.340777\pi$
$373$	2.57747e6	0.959229	0.479614	+	0.877479i	$0.340777\pi$
$374$	2.03679e6	0.752953
$375$	0	0
$376$	1.48617e6	0.542126
$377$	1.47324e6	0.533852
$378$	0	0
$379$	29117.2	0.0104124	0.00520621	−	0.999986i	$-0.498343\pi$
$379$	29117.2	0.0104124	0.00520621	+	0.999986i	$0.498343\pi$
$380$	0	0
$381$	0	0
$382$	−5.45256e6	−1.91180
$383$	3.70323e6	1.28998	0.644992	−	0.764190i	$-0.276861\pi$
$383$	3.70323e6	1.28998	0.644992	+	0.764190i	$0.276861\pi$
$384$	0	0
$385$	0	0
$386$	−5.29881e6	−1.81013
$387$	0	0
$388$	−1.63185e6	−0.550303
$389$	1.49472e6	0.500824	0.250412	−	0.968139i	$-0.419434\pi$
$389$	1.49472e6	0.500824	0.250412	+	0.968139i	$0.419434\pi$
$390$	0	0
$391$	−944689.	−0.312498
$392$	−2.15360e6	−0.707863
$393$	0	0
$394$	4.26006e6	1.38253
$395$	0	0
$396$	0	0
$397$	5.26993e6	1.67814	0.839070	−	0.544023i	$-0.183100\pi$
$397$	5.26993e6	1.67814	0.839070	+	0.544023i	$0.183100\pi$
$398$	−2.27206e6	−0.718973
$399$	0	0
$400$	0	0
$401$	−4.52062e6	−1.40390	−0.701952	−	0.712224i	$-0.747688\pi$
$401$	−4.52062e6	−1.40390	−0.701952	+	0.712224i	$0.747688\pi$
$402$	0	0
$403$	−1.63734e6	−0.502200
$404$	−4.27331e6	−1.30260
$405$	0	0
$406$	−9.01802e6	−2.71517
$407$	−3.24969e6	−0.972425
$408$	0	0
$409$	−852801.	−0.252081	−0.126040	−	0.992025i	$-0.540227\pi$
$409$	−852801.	−0.252081	−0.126040	+	0.992025i	$0.540227\pi$
$410$	0	0
$411$	0	0
$412$	−6.38850e6	−1.85420
$413$	−2.15289e6	−0.621078
$414$	0	0
$415$	0	0
$416$	1.35749e6	0.384594
$417$	0	0
$418$	−5.51138e6	−1.54284
$419$	919574.	0.255889	0.127944	−	0.991781i	$-0.459162\pi$
$419$	919574.	0.255889	0.127944	+	0.991781i	$0.459162\pi$
$420$	0	0
$421$	1.66012e6	0.456493	0.228247	−	0.973603i	$-0.426701\pi$
$421$	1.66012e6	0.456493	0.228247	+	0.973603i	$0.426701\pi$
$422$	−3.25226e6	−0.889005
$423$	0	0
$424$	−308497.	−0.0833368
$425$	0	0
$426$	0	0
$427$	−5.01848e6	−1.33199
$428$	4.19091e6	1.10586
$429$	0	0
$430$	0	0
$431$	1.58573e6	0.411184	0.205592	−	0.978638i	$-0.434088\pi$
$431$	1.58573e6	0.411184	0.205592	+	0.978638i	$0.434088\pi$
$432$	0	0
$433$	2.67437e6	0.685491	0.342746	−	0.939428i	$-0.388643\pi$
$433$	2.67437e6	0.685491	0.342746	+	0.939428i	$0.388643\pi$
$434$	1.00225e7	2.55418
$435$	0	0
$436$	1.18236e7	2.97874
$437$	2.55624e6	0.640323
$438$	0	0
$439$	−2.49585e6	−0.618097	−0.309049	−	0.951046i	$-0.600011\pi$
$439$	−2.49585e6	−0.618097	−0.309049	+	0.951046i	$0.600011\pi$
$440$	0	0
$441$	0	0
$442$	−864137.	−0.210391
$443$	1.41224e6	0.341899	0.170950	−	0.985280i	$-0.445316\pi$
$443$	1.41224e6	0.341899	0.170950	+	0.985280i	$0.445316\pi$
$444$	0	0
$445$	0	0
$446$	1.11894e7	2.66359
$447$	0	0
$448$	−8.76703e6	−2.06375
$449$	6.31679e6	1.47870	0.739351	−	0.673320i	$-0.235132\pi$
$449$	6.31679e6	1.47870	0.739351	+	0.673320i	$0.235132\pi$
$450$	0	0
$451$	−7.92721e6	−1.83518
$452$	−4.05960e6	−0.934625
$453$	0	0
$454$	−2.41527e6	−0.549953
$455$	0	0
$456$	0	0
$457$	−4.13109e6	−0.925282	−0.462641	−	0.886546i	$-0.653098\pi$
$457$	−4.13109e6	−0.925282	−0.462641	+	0.886546i	$0.653098\pi$
$458$	−1.82079e6	−0.405599
$459$	0	0
$460$	0	0
$461$	518659.	0.113666	0.0568329	−	0.998384i	$-0.481900\pi$
$461$	518659.	0.113666	0.0568329	+	0.998384i	$0.481900\pi$
$462$	0	0
$463$	−2.57886e6	−0.559082	−0.279541	−	0.960134i	$-0.590182\pi$
$463$	−2.57886e6	−0.559082	−0.279541	+	0.960134i	$0.590182\pi$
$464$	500406.	0.107902
$465$	0	0
$466$	5.24032e6	1.11787
$467$	794442.	0.168566	0.0842830	−	0.996442i	$-0.473140\pi$
$467$	794442.	0.168566	0.0842830	+	0.996442i	$0.473140\pi$
$468$	0	0
$469$	−4.02995e6	−0.845995
$470$	0	0
$471$	0	0
$472$	2.48656e6	0.513740
$473$	−1.51898e6	−0.312175
$474$	0	0
$475$	0	0
$476$	3.29709e6	0.666981
$477$	0	0
$478$	−3.51776e6	−0.704201
$479$	−3.86091e6	−0.768867	−0.384433	−	0.923153i	$-0.625603\pi$
$479$	−3.86091e6	−0.768867	−0.384433	+	0.923153i	$0.625603\pi$
$480$	0	0
$481$	1.37873e6	0.271716
$482$	1.40511e7	2.75482
$483$	0	0
$484$	1.01200e7	1.96367
$485$	0	0
$486$	0	0
$487$	3.22447e6	0.616078	0.308039	−	0.951374i	$-0.400327\pi$
$487$	3.22447e6	0.616078	0.308039	+	0.951374i	$0.400327\pi$
$488$	5.79628e6	1.10179
$489$	0	0
$490$	0	0
$491$	1.21994e6	0.228368	0.114184	−	0.993460i	$-0.463575\pi$
$491$	1.21994e6	0.228368	0.114184	+	0.993460i	$0.463575\pi$
$492$	0	0
$493$	2.17897e6	0.403769
$494$	2.33828e6	0.431101
$495$	0	0
$496$	−556144.	−0.101504
$497$	−4.01216e6	−0.728597
$498$	0	0
$499$	9.42042e6	1.69363	0.846816	−	0.531886i	$-0.178517\pi$
$499$	9.42042e6	1.69363	0.846816	+	0.531886i	$0.178517\pi$
$500$	0	0
$501$	0	0
$502$	1.80325e7	3.19372
$503$	2.05215e6	0.361651	0.180825	−	0.983515i	$-0.442123\pi$
$503$	2.05215e6	0.361651	0.180825	+	0.983515i	$0.442123\pi$
$504$	0	0
$505$	0	0
$506$	−1.38761e7	−2.40930
$507$	0	0
$508$	1.89293e7	3.25444
$509$	2.38747e6	0.408455	0.204227	−	0.978923i	$-0.434532\pi$
$509$	2.38747e6	0.408455	0.204227	+	0.978923i	$0.434532\pi$
$510$	0	0
$511$	335729.	0.0568770
$512$	989583.	0.166831
$513$	0	0
$514$	−3.11325e6	−0.519765
$515$	0	0
$516$	0	0
$517$	4.56677e6	0.751420
$518$	−8.43946e6	−1.38194
$519$	0	0
$520$	0	0
$521$	8.23648e6	1.32937	0.664687	−	0.747122i	$-0.268565\pi$
$521$	8.23648e6	1.32937	0.664687	+	0.747122i	$0.268565\pi$
$522$	0	0
$523$	3.68989e6	0.589874	0.294937	−	0.955517i	$-0.404701\pi$
$523$	3.68989e6	0.589874	0.294937	+	0.955517i	$0.404701\pi$
$524$	−735133.	−0.116960
$525$	0	0
$526$	−1.19502e6	−0.188326
$527$	−2.42167e6	−0.379829
$528$	0	0
$529$	−456.883	−7.09848e−5 0
$530$	0	0
$531$	0	0
$532$	−8.92164e6	−1.36668
$533$	3.36323e6	0.512788
$534$	0	0
$535$	0	0
$536$	4.65455e6	0.699786
$537$	0	0
$538$	9.15974e6	1.36435
$539$	−6.61765e6	−0.981143
$540$	0	0
$541$	1.13420e7	1.66608	0.833042	−	0.553210i	$-0.186597\pi$
$541$	1.13420e7	1.66608	0.833042	+	0.553210i	$0.186597\pi$
$542$	−2.83661e6	−0.414764
$543$	0	0
$544$	2.00776e6	0.290880
$545$	0	0
$546$	0	0
$547$	−8.24361e6	−1.17801	−0.589005	−	0.808129i	$-0.700480\pi$
$547$	−8.24361e6	−1.17801	−0.589005	+	0.808129i	$0.700480\pi$
$548$	−1.00146e7	−1.42457
$549$	0	0
$550$	0	0
$551$	−5.89609e6	−0.827342
$552$	0	0
$553$	−1.21913e7	−1.69526
$554$	−3.72821e6	−0.516091
$555$	0	0
$556$	−1.32599e7	−1.81909
$557$	6.50585e6	0.888518	0.444259	−	0.895898i	$-0.353467\pi$
$557$	6.50585e6	0.888518	0.444259	+	0.895898i	$0.353467\pi$
$558$	0	0
$559$	644447.	0.0872283
$560$	0	0
$561$	0	0
$562$	2.00932e7	2.68354
$563$	5.91671e6	0.786700	0.393350	−	0.919389i	$-0.371316\pi$
$563$	5.91671e6	0.786700	0.393350	+	0.919389i	$0.371316\pi$
$564$	0	0
$565$	0	0
$566$	−6.89770e6	−0.905030
$567$	0	0
$568$	4.63399e6	0.602677
$569$	−4.67027e6	−0.604730	−0.302365	−	0.953192i	$-0.597776\pi$
$569$	−4.67027e6	−0.604730	−0.302365	+	0.953192i	$0.597776\pi$
$570$	0	0
$571$	−1.50901e6	−0.193687	−0.0968437	−	0.995300i	$-0.530875\pi$
$571$	−1.50901e6	−0.193687	−0.0968437	+	0.995300i	$0.530875\pi$
$572$	−7.91174e6	−1.01107
$573$	0	0
$574$	−2.05870e7	−2.60803
$575$	0	0
$576$	0	0
$577$	−7.24904e6	−0.906444	−0.453222	−	0.891398i	$-0.649725\pi$
$577$	−7.24904e6	−0.906444	−0.453222	+	0.891398i	$0.649725\pi$
$578$	1.18087e7	1.47022
$579$	0	0
$580$	0	0
$581$	−5.81373e6	−0.714521
$582$	0	0
$583$	−947962.	−0.115510
$584$	−387763.	−0.0470472
$585$	0	0
$586$	−2.63262e7	−3.16697
$587$	4.58091e6	0.548727	0.274364	−	0.961626i	$-0.411533\pi$
$587$	4.58091e6	0.548727	0.274364	+	0.961626i	$0.411533\pi$
$588$	0	0
$589$	6.55282e6	0.778288
$590$	0	0
$591$	0	0
$592$	468302.	0.0549189
$593$	−1.54683e7	−1.80636	−0.903182	−	0.429257i	$-0.858775\pi$
$593$	−1.54683e7	−1.80636	−0.903182	+	0.429257i	$0.858775\pi$
$594$	0	0
$595$	0	0
$596$	2.29943e6	0.265158
$597$	0	0
$598$	5.88711e6	0.673208
$599$	8.04034e6	0.915603	0.457801	−	0.889054i	$-0.348637\pi$
$599$	8.04034e6	0.915603	0.457801	+	0.889054i	$0.348637\pi$
$600$	0	0
$601$	6.64916e6	0.750897	0.375449	−	0.926843i	$-0.377489\pi$
$601$	6.64916e6	0.750897	0.375449	+	0.926843i	$0.377489\pi$
$602$	−3.94479e6	−0.443642
$603$	0	0
$604$	−1.63954e7	−1.82865
$605$	0	0
$606$	0	0
$607$	−3.15754e6	−0.347838	−0.173919	−	0.984760i	$-0.555643\pi$
$607$	−3.15754e6	−0.347838	−0.173919	+	0.984760i	$0.555643\pi$
$608$	−5.43282e6	−0.596027
$609$	0	0
$610$	0	0
$611$	−1.93751e6	−0.209962
$612$	0	0
$613$	−9.30156e6	−0.999780	−0.499890	−	0.866089i	$-0.666626\pi$
$613$	−9.30156e6	−0.999780	−0.499890	+	0.866089i	$0.666626\pi$
$614$	−1.27998e7	−1.37019
$615$	0	0
$616$	1.91630e7	2.03475
$617$	−4.33733e6	−0.458680	−0.229340	−	0.973346i	$-0.573657\pi$
$617$	−4.33733e6	−0.458680	−0.229340	+	0.973346i	$0.573657\pi$
$618$	0	0
$619$	−1.08485e7	−1.13800	−0.569002	−	0.822336i	$-0.692670\pi$
$619$	−1.08485e7	−1.13800	−0.569002	+	0.822336i	$0.692670\pi$
$620$	0	0
$621$	0	0
$622$	1.55559e7	1.61220
$623$	−7.68602e6	−0.793380
$624$	0	0
$625$	0	0
$626$	−1.60685e7	−1.63885
$627$	0	0
$628$	−1.63497e7	−1.65428
$629$	2.03917e6	0.205507
$630$	0	0
$631$	1.06268e7	1.06250	0.531252	−	0.847214i	$-0.321722\pi$
$631$	1.06268e7	1.06250	0.531252	+	0.847214i	$0.321722\pi$
$632$	1.40808e7	1.40228
$633$	0	0
$634$	1.35866e7	1.34242
$635$	0	0
$636$	0	0
$637$	2.80763e6	0.274152
$638$	3.20057e7	3.11298
$639$	0	0
$640$	0	0
$641$	1.13915e7	1.09505	0.547526	−	0.836789i	$-0.315570\pi$
$641$	1.13915e7	1.09505	0.547526	+	0.836789i	$0.315570\pi$
$642$	0	0
$643$	−6.15160e6	−0.586760	−0.293380	−	0.955996i	$-0.594780\pi$
$643$	−6.15160e6	−0.586760	−0.293380	+	0.955996i	$0.594780\pi$
$644$	−2.24621e7	−2.13420
$645$	0	0
$646$	3.45838e6	0.326055
$647$	1.75958e7	1.65252	0.826262	−	0.563286i	$-0.190463\pi$
$647$	1.75958e7	1.65252	0.826262	+	0.563286i	$0.190463\pi$
$648$	0	0
$649$	7.64078e6	0.712075
$650$	0	0
$651$	0	0
$652$	−2.95426e7	−2.72164
$653$	−8.66136e6	−0.794883	−0.397441	−	0.917628i	$-0.630102\pi$
$653$	−8.66136e6	−0.794883	−0.397441	+	0.917628i	$0.630102\pi$
$654$	0	0
$655$	0	0
$656$	1.14236e6	0.103644
$657$	0	0
$658$	1.18599e7	1.06787
$659$	5.96764e6	0.535290	0.267645	−	0.963518i	$-0.413755\pi$
$659$	5.96764e6	0.535290	0.267645	+	0.963518i	$0.413755\pi$
$660$	0	0
$661$	−8.33723e6	−0.742195	−0.371098	−	0.928594i	$-0.621019\pi$
$661$	−8.33723e6	−0.742195	−0.371098	+	0.928594i	$0.621019\pi$
$662$	−1.16789e7	−1.03575
$663$	0	0
$664$	6.71479e6	0.591034
$665$	0	0
$666$	0	0
$667$	−1.48446e7	−1.29198
$668$	2.01470e7	1.74691
$669$	0	0
$670$	0	0
$671$	1.78110e7	1.52715
$672$	0	0
$673$	−2.17866e7	−1.85418	−0.927088	−	0.374843i	$-0.877697\pi$
$673$	−2.17866e7	−1.85418	−0.927088	+	0.374843i	$0.877697\pi$
$674$	−2.59835e7	−2.20317
$675$	0	0
$676$	−1.63044e7	−1.37226
$677$	−1.43012e7	−1.19923	−0.599614	−	0.800289i	$-0.704679\pi$
$677$	−1.43012e7	−1.19923	−0.599614	+	0.800289i	$0.704679\pi$
$678$	0	0
$679$	−5.15285e6	−0.428917
$680$	0	0
$681$	0	0
$682$	−3.55707e7	−2.92841
$683$	4.11287e6	0.337360	0.168680	−	0.985671i	$-0.446050\pi$
$683$	4.11287e6	0.337360	0.168680	+	0.985671i	$0.446050\pi$
$684$	0	0
$685$	0	0
$686$	8.71608e6	0.707149
$687$	0	0
$688$	218895.	0.0176305
$689$	402186.	0.0322759
$690$	0	0
$691$	−5.53375e6	−0.440884	−0.220442	−	0.975400i	$-0.570750\pi$
$691$	−5.53375e6	−0.440884	−0.220442	+	0.975400i	$0.570750\pi$
$692$	−2.39753e7	−1.90327
$693$	0	0
$694$	2.09565e7	1.65166
$695$	0	0
$696$	0	0
$697$	4.97430e6	0.387838
$698$	−3.92944e7	−3.05276
$699$	0	0
$700$	0	0
$701$	7.95064e6	0.611093	0.305546	−	0.952177i	$-0.401161\pi$
$701$	7.95064e6	0.611093	0.305546	+	0.952177i	$0.401161\pi$
$702$	0	0
$703$	−5.51782e6	−0.421094
$704$	3.11149e7	2.36612
$705$	0	0
$706$	4.05557e7	3.06225
$707$	−1.34937e7	−1.01527
$708$	0	0
$709$	−1.67069e7	−1.24819	−0.624093	−	0.781350i	$-0.714532\pi$
$709$	−1.67069e7	−1.24819	−0.624093	+	0.781350i	$0.714532\pi$
$710$	0	0
$711$	0	0
$712$	8.87725e6	0.656264
$713$	1.64981e7	1.21538
$714$	0	0
$715$	0	0
$716$	−3.42155e7	−2.49425
$717$	0	0
$718$	1.43655e7	1.03994
$719$	−4.19041e6	−0.302297	−0.151149	−	0.988511i	$-0.548297\pi$
$719$	−4.19041e6	−0.302297	−0.151149	+	0.988511i	$0.548297\pi$
$720$	0	0
$721$	−2.01728e7	−1.44520
$722$	1.34641e7	0.961247
$723$	0	0
$724$	−1.93790e7	−1.37400
$725$	0	0
$726$	0	0
$727$	2.00481e7	1.40682	0.703408	−	0.710787i	$-0.251661\pi$
$727$	2.00481e7	1.40682	0.703408	+	0.710787i	$0.251661\pi$
$728$	−8.13016e6	−0.568552
$729$	0	0
$730$	0	0
$731$	953154.	0.0659735
$732$	0	0
$733$	2.13286e7	1.46623	0.733115	−	0.680105i	$-0.238066\pi$
$733$	2.13286e7	1.46623	0.733115	+	0.680105i	$0.238066\pi$
$734$	5.46745e6	0.374580
$735$	0	0
$736$	−1.36783e7	−0.930757
$737$	1.43027e7	0.969947
$738$	0	0
$739$	1.36132e7	0.916959	0.458480	−	0.888705i	$-0.348394\pi$
$739$	1.36132e7	0.916959	0.458480	+	0.888705i	$0.348394\pi$
$740$	0	0
$741$	0	0
$742$	−2.46186e6	−0.164155
$743$	−1.75261e7	−1.16470	−0.582350	−	0.812938i	$-0.697866\pi$
$743$	−1.75261e7	−1.16470	−0.582350	+	0.812938i	$0.697866\pi$
$744$	0	0
$745$	0	0
$746$	−2.37566e7	−1.56292
$747$	0	0
$748$	−1.17017e7	−0.764705
$749$	1.32335e7	0.861926
$750$	0	0
$751$	7.19421e6	0.465461	0.232731	−	0.972541i	$-0.425234\pi$
$751$	7.19421e6	0.465461	0.232731	+	0.972541i	$0.425234\pi$
$752$	−658102.	−0.0424374
$753$	0	0
$754$	−1.35789e7	−0.869832
$755$	0	0
$756$	0	0
$757$	−1.13265e7	−0.718385	−0.359192	−	0.933264i	$-0.616948\pi$
$757$	−1.13265e7	−0.718385	−0.359192	+	0.933264i	$0.616948\pi$
$758$	−268373.	−0.0169655
$759$	0	0
$760$	0	0
$761$	2.05113e7	1.28390	0.641951	−	0.766746i	$-0.278125\pi$
$761$	2.05113e7	1.28390	0.641951	+	0.766746i	$0.278125\pi$
$762$	0	0
$763$	3.73349e7	2.32169
$764$	3.13257e7	1.94164
$765$	0	0
$766$	−3.41327e7	−2.10183
$767$	−3.24170e6	−0.198969
$768$	0	0
$769$	1.47042e6	0.0896655	0.0448328	−	0.998995i	$-0.485725\pi$
$769$	1.47042e6	0.0896655	0.0448328	+	0.998995i	$0.485725\pi$
$770$	0	0
$771$	0	0
$772$	3.04424e7	1.83838
$773$	2.88444e7	1.73625	0.868125	−	0.496345i	$-0.165325\pi$
$773$	2.88444e7	1.73625	0.868125	+	0.496345i	$0.165325\pi$
$774$	0	0
$775$	0	0
$776$	5.95148e6	0.354789
$777$	0	0
$778$	−1.37768e7	−0.816017
$779$	−1.34600e7	−0.794697
$780$	0	0
$781$	1.42395e7	0.835348
$782$	8.70719e6	0.509168
$783$	0	0
$784$	953648.	0.0554113
$785$	0	0
$786$	0	0
$787$	1.76526e7	1.01595	0.507974	−	0.861372i	$-0.330394\pi$
$787$	1.76526e7	1.01595	0.507974	+	0.861372i	$0.330394\pi$
$788$	−2.44747e7	−1.40411
$789$	0	0
$790$	0	0
$791$	−1.28189e7	−0.728465
$792$	0	0
$793$	−7.55656e6	−0.426718
$794$	−4.85729e7	−2.73428
$795$	0	0
$796$	1.30533e7	0.730194
$797$	−1.58295e6	−0.0882718	−0.0441359	−	0.999026i	$-0.514053\pi$
$797$	−1.58295e6	−0.0882718	−0.0441359	+	0.999026i	$0.514053\pi$
$798$	0	0
$799$	−2.86563e6	−0.158801
$800$	0	0
$801$	0	0
$802$	4.16665e7	2.28745
$803$	−1.19153e6	−0.0652104
$804$	0	0
$805$	0	0
$806$	1.50914e7	0.818259
$807$	0	0
$808$	1.55850e7	0.839807
$809$	2.06307e7	1.10826	0.554132	−	0.832429i	$-0.313050\pi$
$809$	2.06307e7	1.10826	0.554132	+	0.832429i	$0.313050\pi$
$810$	0	0
$811$	1.25838e7	0.671833	0.335916	−	0.941892i	$-0.390954\pi$
$811$	1.25838e7	0.671833	0.335916	+	0.941892i	$0.390954\pi$
$812$	5.18098e7	2.75754
$813$	0	0
$814$	2.99524e7	1.58442
$815$	0	0
$816$	0	0
$817$	−2.57915e6	−0.135183
$818$	7.86026e6	0.410728
$819$	0	0
$820$	0	0
$821$	5.41423e6	0.280336	0.140168	−	0.990128i	$-0.455236\pi$
$821$	5.41423e6	0.280336	0.140168	+	0.990128i	$0.455236\pi$
$822$	0	0
$823$	−6.31437e6	−0.324960	−0.162480	−	0.986712i	$-0.551949\pi$
$823$	−6.31437e6	−0.324960	−0.162480	+	0.986712i	$0.551949\pi$
$824$	2.32993e7	1.19543
$825$	0	0
$826$	1.98431e7	1.01195
$827$	−1.78088e6	−0.0905465	−0.0452732	−	0.998975i	$-0.514416\pi$
$827$	−1.78088e6	−0.0905465	−0.0452732	+	0.998975i	$0.514416\pi$
$828$	0	0
$829$	1.01553e7	0.513223	0.256611	−	0.966515i	$-0.417394\pi$
$829$	1.01553e7	0.513223	0.256611	+	0.966515i	$0.417394\pi$
$830$	0	0
$831$	0	0
$832$	−1.32009e7	−0.661144
$833$	4.15256e6	0.207350
$834$	0	0
$835$	0	0
$836$	3.16637e7	1.56692
$837$	0	0
$838$	−8.47570e6	−0.416933
$839$	−3.03356e7	−1.48781	−0.743904	−	0.668286i	$-0.767028\pi$
$839$	−3.03356e7	−1.48781	−0.743904	+	0.668286i	$0.767028\pi$
$840$	0	0
$841$	1.37287e7	0.669327
$842$	−1.53013e7	−0.743786
$843$	0	0
$844$	1.86847e7	0.902880
$845$	0	0
$846$	0	0
$847$	3.19558e7	1.53053
$848$	136608.	0.00652357
$849$	0	0
$850$	0	0
$851$	−1.38923e7	−0.657581
$852$	0	0
$853$	3.11502e7	1.46585	0.732924	−	0.680311i	$-0.238155\pi$
$853$	3.11502e7	1.46585	0.732924	+	0.680311i	$0.238155\pi$
$854$	4.62553e7	2.17028
$855$	0	0
$856$	−1.52845e7	−0.712964
$857$	−1.28434e7	−0.597350	−0.298675	−	0.954355i	$-0.596545\pi$
$857$	−1.28434e7	−0.597350	−0.298675	+	0.954355i	$0.596545\pi$
$858$	0	0
$859$	−7.65787e6	−0.354099	−0.177050	−	0.984202i	$-0.556655\pi$
$859$	−7.65787e6	−0.354099	−0.177050	+	0.984202i	$0.556655\pi$
$860$	0	0
$861$	0	0
$862$	−1.46157e7	−0.669962
$863$	−7.94715e6	−0.363232	−0.181616	−	0.983370i	$-0.558133\pi$
$863$	−7.94715e6	−0.363232	−0.181616	+	0.983370i	$0.558133\pi$
$864$	0	0
$865$	0	0
$866$	−2.46497e7	−1.11690
$867$	0	0
$868$	−5.75806e7	−2.59404
$869$	4.32680e7	1.94365
$870$	0	0
$871$	−6.06810e6	−0.271024
$872$	−4.31214e7	−1.92044
$873$	0	0
$874$	−2.35609e7	−1.04331
$875$	0	0
$876$	0	0
$877$	−3.27048e6	−0.143586	−0.0717930	−	0.997420i	$-0.522872\pi$
$877$	−3.27048e6	−0.143586	−0.0717930	+	0.997420i	$0.522872\pi$
$878$	2.30042e7	1.00710
$879$	0	0
$880$	0	0
$881$	−2.67525e6	−0.116125	−0.0580623	−	0.998313i	$-0.518492\pi$
$881$	−2.67525e6	−0.116125	−0.0580623	+	0.998313i	$0.518492\pi$
$882$	0	0
$883$	4.00213e7	1.72739	0.863693	−	0.504018i	$-0.168145\pi$
$883$	4.00213e7	1.72739	0.863693	+	0.504018i	$0.168145\pi$
$884$	4.96459e6	0.213675
$885$	0	0
$886$	−1.30166e7	−0.557073
$887$	7.68156e6	0.327824	0.163912	−	0.986475i	$-0.447589\pi$
$887$	7.68156e6	0.327824	0.163912	+	0.986475i	$0.447589\pi$
$888$	0	0
$889$	5.97726e7	2.53657
$890$	0	0
$891$	0	0
$892$	−6.42845e7	−2.70517
$893$	7.75415e6	0.325391
$894$	0	0
$895$	0	0
$896$	5.19564e7	2.16207
$897$	0	0
$898$	−5.82218e7	−2.40932
$899$	−3.80536e7	−1.57035
$900$	0	0
$901$	594843.	0.0244113
$902$	7.30650e7	2.99015
$903$	0	0
$904$	1.48056e7	0.602568
$905$	0	0
$906$	0	0
$907$	8.53681e6	0.344570	0.172285	−	0.985047i	$-0.444885\pi$
$907$	8.53681e6	0.344570	0.172285	+	0.985047i	$0.444885\pi$
$908$	1.38761e7	0.558536
$909$	0	0
$910$	0	0
$911$	8.02303e6	0.320289	0.160145	−	0.987094i	$-0.448804\pi$
$911$	8.02303e6	0.320289	0.160145	+	0.987094i	$0.448804\pi$
$912$	0	0
$913$	2.06334e7	0.819209
$914$	3.80762e7	1.50761
$915$	0	0
$916$	1.04607e7	0.411929
$917$	−2.32131e6	−0.0911610
$918$	0	0
$919$	−4.86848e7	−1.90154	−0.950768	−	0.309903i	$-0.899703\pi$
$919$	−4.86848e7	−1.90154	−0.950768	+	0.309903i	$0.899703\pi$
$920$	0	0
$921$	0	0
$922$	−4.78048e6	−0.185201
$923$	−6.04130e6	−0.233414
$924$	0	0
$925$	0	0
$926$	2.37693e7	0.910939
$927$	0	0
$928$	3.15495e7	1.20260
$929$	3.45089e7	1.31187	0.655937	−	0.754816i	$-0.272274\pi$
$929$	3.45089e7	1.31187	0.655937	+	0.754816i	$0.272274\pi$
$930$	0	0
$931$	−1.12365e7	−0.424869
$932$	−3.01064e7	−1.13532
$933$	0	0
$934$	−7.32237e6	−0.274653
$935$	0	0
$936$	0	0
$937$	−2.19715e7	−0.817543	−0.408772	−	0.912637i	$-0.634043\pi$
$937$	−2.19715e7	−0.817543	−0.408772	+	0.912637i	$0.634043\pi$
$938$	3.71441e7	1.37842
$939$	0	0
$940$	0	0
$941$	−2.56834e7	−0.945537	−0.472768	−	0.881187i	$-0.656745\pi$
$941$	−2.56834e7	−0.945537	−0.472768	+	0.881187i	$0.656745\pi$
$942$	0	0
$943$	−3.38884e7	−1.24100
$944$	−1.10109e6	−0.0402153
$945$	0	0
$946$	1.40004e7	0.508642
$947$	−4.01332e7	−1.45422	−0.727108	−	0.686523i	$-0.759136\pi$
$947$	−4.01332e7	−1.45422	−0.727108	+	0.686523i	$0.759136\pi$
$948$	0	0
$949$	505524.	0.0182212
$950$	0	0
$951$	0	0
$952$	−1.20247e7	−0.430014
$953$	2.22515e7	0.793647	0.396823	−	0.917895i	$-0.370113\pi$
$953$	2.22515e7	0.793647	0.396823	+	0.917895i	$0.370113\pi$
$954$	0	0
$955$	0	0
$956$	2.02100e7	0.715191
$957$	0	0
$958$	3.55860e7	1.25275
$959$	−3.16228e7	−1.11034
$960$	0	0
$961$	1.36631e7	0.477243
$962$	−1.27077e7	−0.442720
$963$	0	0
$964$	−8.07256e7	−2.79781
$965$	0	0
$966$	0	0
$967$	−1.65096e7	−0.567767	−0.283884	−	0.958859i	$-0.591623\pi$
$967$	−1.65096e7	−0.567767	−0.283884	+	0.958859i	$0.591623\pi$
$968$	−3.69085e7	−1.26601
$969$	0	0
$970$	0	0
$971$	−3.76485e7	−1.28144	−0.640722	−	0.767773i	$-0.721365\pi$
$971$	−3.76485e7	−1.28144	−0.640722	+	0.767773i	$0.721365\pi$
$972$	0	0
$973$	−4.18705e7	−1.41784
$974$	−2.97199e7	−1.00381
$975$	0	0
$976$	−2.56669e6	−0.0862478
$977$	4.35118e7	1.45838	0.729190	−	0.684312i	$-0.239897\pi$
$977$	4.35118e7	1.45838	0.729190	+	0.684312i	$0.239897\pi$
$978$	0	0
$979$	2.72783e7	0.909623
$980$	0	0
$981$	0	0
$982$	−1.12442e7	−0.372091
$983$	−3.33555e7	−1.10099	−0.550496	−	0.834838i	$-0.685561\pi$
$983$	−3.33555e7	−1.10099	−0.550496	+	0.834838i	$0.685561\pi$
$984$	0	0
$985$	0	0
$986$	−2.00835e7	−0.657881
$987$	0	0
$988$	−1.34337e7	−0.437829
$989$	−6.49355e6	−0.211102
$990$	0	0
$991$	−3.71091e7	−1.20032	−0.600159	−	0.799881i	$-0.704896\pi$
$991$	−3.71091e7	−1.20032	−0.600159	+	0.799881i	$0.704896\pi$
$992$	−3.50636e7	−1.13130
$993$	0	0
$994$	3.69800e7	1.18714
$995$	0	0
$996$	0	0
$997$	2.17259e7	0.692212	0.346106	−	0.938195i	$-0.387504\pi$
$997$	2.17259e7	0.692212	0.346106	+	0.938195i	$0.387504\pi$
$998$	−8.68279e7	−2.75952
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.a.i.1.1		2
3.2	odd	2		75.6.a.j.1.2		2
5.2	odd	4		45.6.b.c.19.1		4
5.3	odd	4		45.6.b.c.19.4		4
5.4	even	2		225.6.a.u.1.2		2
15.2	even	4		15.6.b.a.4.4	yes	4
15.8	even	4		15.6.b.a.4.1	✓	4
15.14	odd	2		75.6.a.f.1.1		2
20.3	even	4		720.6.f.h.289.2		4
20.7	even	4		720.6.f.h.289.3		4
60.23	odd	4		240.6.f.c.49.4		4
60.47	odd	4		240.6.f.c.49.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.6.b.a.4.1	✓	4	15.8	even	4
15.6.b.a.4.4	yes	4	15.2	even	4
45.6.b.c.19.1		4	5.2	odd	4
45.6.b.c.19.4		4	5.3	odd	4
75.6.a.f.1.1		2	15.14	odd	2
75.6.a.j.1.2		2	3.2	odd	2
225.6.a.i.1.1		2	1.1	even	1	trivial
225.6.a.u.1.2		2	5.4	even	2
240.6.f.c.49.1		4	60.47	odd	4
240.6.f.c.49.4		4	60.23	odd	4
720.6.f.h.289.2		4	20.3	even	4
720.6.f.h.289.3		4	20.7	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)

\(f(q)\)	\(=\)	\(q-9.21699 q^{2} +52.9529 q^{4} +167.208 q^{7} -193.123 q^{8} +O(q^{10})\)	\(q-9.21699 q^{2} +52.9529 q^{4} +167.208 q^{7} -193.123 q^{8} -593.435 q^{11} +251.773 q^{13} -1541.15 q^{14} +85.5179 q^{16} +372.379 q^{17} -1007.62 q^{19} +5469.68 q^{22} -2536.90 q^{23} -2320.59 q^{26} +8854.14 q^{28} +5851.48 q^{29} -6503.25 q^{31} +5391.71 q^{32} -3432.21 q^{34} +5476.07 q^{37} +9287.25 q^{38} +13358.2 q^{41} +2559.64 q^{43} -31424.1 q^{44} +23382.6 q^{46} -7695.48 q^{47} +11151.4 q^{49} +13332.1 q^{52} +1597.41 q^{53} -32291.6 q^{56} -53933.0 q^{58} -12875.5 q^{59} -30013.4 q^{61} +59940.4 q^{62} -52431.9 q^{64} -24101.5 q^{67} +19718.5 q^{68} -23995.0 q^{71} +2007.86 q^{73} -50472.9 q^{74} -53356.6 q^{76} -99226.9 q^{77} -72911.0 q^{79} -123122. q^{82} -34769.5 q^{83} -23592.1 q^{86} +114606. q^{88} -45966.9 q^{89} +42098.4 q^{91} -134337. q^{92} +70929.2 q^{94} -30817.1 q^{97} -102783. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{2} + 21 q^{4} + 108 q^{7} - 207 q^{8}+O(q^{10}) \) 2 * q - 9 * q^2 + 21 * q^4 + 108 * q^7 - 207 * q^8	\( 2 q - 9 q^{2} + 21 q^{4} + 108 q^{7} - 207 q^{8} - 168 q^{11} + 1296 q^{13} - 1554 q^{14} + 1105 q^{16} - 576 q^{17} - 1336 q^{19} + 5562 q^{22} - 5904 q^{23} - 2094 q^{26} + 10746 q^{28} + 3552 q^{29} - 11648 q^{31} + 6057 q^{32} - 3638 q^{34} + 14688 q^{37} + 9216 q^{38} - 1812 q^{41} - 7560 q^{43} - 45018 q^{44} + 22652 q^{46} - 3240 q^{47} - 2150 q^{49} - 20034 q^{52} + 22176 q^{53} - 31470 q^{56} - 54432 q^{58} - 57336 q^{59} - 30140 q^{61} + 58824 q^{62} - 84911 q^{64} - 5184 q^{67} + 50022 q^{68} - 17424 q^{71} - 3456 q^{73} - 48474 q^{74} - 42864 q^{76} - 124416 q^{77} - 57520 q^{79} - 126414 q^{82} - 133992 q^{83} - 25788 q^{86} + 108702 q^{88} - 136764 q^{89} - 19728 q^{91} - 26748 q^{92} + 71896 q^{94} + 85536 q^{97} - 105669 q^{98}+O(q^{100}) \) 2 * q - 9 * q^2 + 21 * q^4 + 108 * q^7 - 207 * q^8 - 168 * q^11 + 1296 * q^13 - 1554 * q^14 + 1105 * q^16 - 576 * q^17 - 1336 * q^19 + 5562 * q^22 - 5904 * q^23 - 2094 * q^26 + 10746 * q^28 + 3552 * q^29 - 11648 * q^31 + 6057 * q^32 - 3638 * q^34 + 14688 * q^37 + 9216 * q^38 - 1812 * q^41 - 7560 * q^43 - 45018 * q^44 + 22652 * q^46 - 3240 * q^47 - 2150 * q^49 - 20034 * q^52 + 22176 * q^53 - 31470 * q^56 - 54432 * q^58 - 57336 * q^59 - 30140 * q^61 + 58824 * q^62 - 84911 * q^64 - 5184 * q^67 + 50022 * q^68 - 17424 * q^71 - 3456 * q^73 - 48474 * q^74 - 42864 * q^76 - 124416 * q^77 - 57520 * q^79 - 126414 * q^82 - 133992 * q^83 - 25788 * q^86 + 108702 * q^88 - 136764 * q^89 - 19728 * q^91 - 26748 * q^92 + 71896 * q^94 + 85536 * q^97 - 105669 * q^98

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