LMFDB - Embedded newform 900.6.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,6,Mod(1,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("900.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	900.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:	$144.345437832$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{409}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 102 $ x^2 - x - 102
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 100)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$10.6119$ of defining polynomial
Character	$\chi$	$=$	900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+141.342 q^{7} +O(q^{10})$	$q+141.342 q^{7} -273.356 q^{11} -945.370 q^{13} +1212.32 q^{17} +135.931 q^{19} -2730.88 q^{23} +3077.70 q^{29} -1410.44 q^{31} +3152.03 q^{37} +13499.5 q^{41} -5433.70 q^{43} -3269.53 q^{47} +3170.70 q^{49} +21421.3 q^{53} +41403.1 q^{59} -24373.9 q^{61} +32307.0 q^{67} +38675.0 q^{71} -8083.79 q^{73} -38636.8 q^{77} -15979.3 q^{79} +48494.1 q^{83} -89187.5 q^{89} -133621. q^{91} +56822.2 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 40 q^{7}+O(q^{10}) $ 2 * q + 40 * q^7	$ 2 q + 40 q^{7} + 60 q^{11} - 920 q^{13} + 2910 q^{17} + 2092 q^{19} + 120 q^{23} - 3552 q^{29} - 8888 q^{31} - 12140 q^{37} + 12438 q^{41} - 1160 q^{43} - 1200 q^{47} - 3366 q^{49} + 26340 q^{53} + 36696 q^{59} + 19204 q^{61} + 90460 q^{67} - 2736 q^{71} - 12770 q^{73} - 72420 q^{77} - 16184 q^{79} - 30300 q^{83} + 47322 q^{89} - 136192 q^{91} + 2980 q^{97}+O(q^{100}) $ 2 * q + 40 * q^7 + 60 * q^11 - 920 * q^13 + 2910 * q^17 + 2092 * q^19 + 120 * q^23 - 3552 * q^29 - 8888 * q^31 - 12140 * q^37 + 12438 * q^41 - 1160 * q^43 - 1200 * q^47 - 3366 * q^49 + 26340 * q^53 + 36696 * q^59 + 19204 * q^61 + 90460 * q^67 - 2736 * q^71 - 12770 * q^73 - 72420 * q^77 - 16184 * q^79 - 30300 * q^83 + 47322 * q^89 - 136192 * q^91 + 2980 * 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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	141.342	1.09025	0.545127	−	0.838354i	$-0.316481\pi$
$7$	141.342	1.09025	0.545127	+	0.838354i	$0.316481\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−273.356	−0.681157	−0.340579	−	0.940216i	$-0.610623\pi$
$11$	−273.356	−0.681157	−0.340579	+	0.940216i	$0.610623\pi$
$12$	0	0
$13$	−945.370	−1.55147	−0.775735	−	0.631059i	$-0.782621\pi$
$13$	−945.370	−1.55147	−0.775735	+	0.631059i	$0.782621\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1212.32	1.01740	0.508702	−	0.860943i	$-0.330126\pi$
$17$	1212.32	1.01740	0.508702	+	0.860943i	$0.330126\pi$
$18$	0	0
$19$	135.931	0.0863844	0.0431922	−	0.999067i	$-0.486247\pi$
$19$	135.931	0.0863844	0.0431922	+	0.999067i	$0.486247\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2730.88	−1.07642	−0.538211	−	0.842810i	$-0.680900\pi$
$23$	−2730.88	−1.07642	−0.538211	+	0.842810i	$0.680900\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3077.70	0.679565	0.339783	−	0.940504i	$-0.389646\pi$
$29$	3077.70	0.679565	0.339783	+	0.940504i	$0.389646\pi$
$30$	0	0
$31$	−1410.44	−0.263603	−0.131801	−	0.991276i	$-0.542076\pi$
$31$	−1410.44	−0.263603	−0.131801	+	0.991276i	$0.542076\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3152.03	0.378517	0.189259	−	0.981927i	$-0.439392\pi$
$37$	3152.03	0.378517	0.189259	+	0.981927i	$0.439392\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	13499.5	1.25418	0.627090	−	0.778947i	$-0.284246\pi$
$41$	13499.5	1.25418	0.627090	+	0.778947i	$0.284246\pi$
$42$	0	0
$43$	−5433.70	−0.448151	−0.224076	−	0.974572i	$-0.571936\pi$
$43$	−5433.70	−0.448151	−0.224076	+	0.974572i	$0.571936\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3269.53	−0.215894	−0.107947	−	0.994157i	$-0.534428\pi$
$47$	−3269.53	−0.215894	−0.107947	+	0.994157i	$0.534428\pi$
$48$	0	0
$49$	3170.70	0.188654
$50$	0	0
$51$	0	0
$52$	0	0
$53$	21421.3	1.04750	0.523752	−	0.851871i	$-0.324532\pi$
$53$	21421.3	1.04750	0.523752	+	0.851871i	$0.324532\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	41403.1	1.54847	0.774235	−	0.632899i	$-0.218135\pi$
$59$	41403.1	1.54847	0.774235	+	0.632899i	$0.218135\pi$
$60$	0	0
$61$	−24373.9	−0.838688	−0.419344	−	0.907827i	$-0.637740\pi$
$61$	−24373.9	−0.838688	−0.419344	+	0.907827i	$0.637740\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	32307.0	0.879245	0.439623	−	0.898183i	$-0.355112\pi$
$67$	32307.0	0.879245	0.439623	+	0.898183i	$0.355112\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	38675.0	0.910510	0.455255	−	0.890361i	$-0.349548\pi$
$71$	38675.0	0.910510	0.455255	+	0.890361i	$0.349548\pi$
$72$	0	0
$73$	−8083.79	−0.177545	−0.0887724	−	0.996052i	$-0.528294\pi$
$73$	−8083.79	−0.177545	−0.0887724	+	0.996052i	$0.528294\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−38636.8	−0.742634
$78$	0	0
$79$	−15979.3	−0.288064	−0.144032	−	0.989573i	$-0.546007\pi$
$79$	−15979.3	−0.288064	−0.144032	+	0.989573i	$0.546007\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	48494.1	0.772670	0.386335	−	0.922359i	$-0.373741\pi$
$83$	48494.1	0.772670	0.386335	+	0.922359i	$0.373741\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−89187.5	−1.19352	−0.596759	−	0.802420i	$-0.703545\pi$
$89$	−89187.5	−1.19352	−0.596759	+	0.802420i	$0.703545\pi$
$90$	0	0
$91$	−133621.	−1.69150
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	56822.2	0.613181	0.306590	−	0.951842i	$-0.400812\pi$
$97$	56822.2	0.613181	0.306590	+	0.951842i	$0.400812\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−67419.2	−0.657627	−0.328814	−	0.944395i	$-0.606649\pi$
$101$	−67419.2	−0.657627	−0.328814	+	0.944395i	$0.606649\pi$
$102$	0	0
$103$	−61779.3	−0.573786	−0.286893	−	0.957963i	$-0.592622\pi$
$103$	−61779.3	−0.573786	−0.286893	+	0.957963i	$0.592622\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−79150.3	−0.668333	−0.334167	−	0.942514i	$-0.608455\pi$
$107$	−79150.3	−0.668333	−0.334167	+	0.942514i	$0.608455\pi$
$108$	0	0
$109$	−177041.	−1.42728	−0.713638	−	0.700515i	$-0.752954\pi$
$109$	−177041.	−1.42728	−0.713638	+	0.700515i	$0.752954\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	197220.	1.45297	0.726483	−	0.687184i	$-0.241153\pi$
$113$	197220.	1.45297	0.726483	+	0.687184i	$0.241153\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	171352.	1.10923
$120$	0	0
$121$	−86327.4	−0.536025
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−75992.4	−0.418081	−0.209041	−	0.977907i	$-0.567034\pi$
$127$	−75992.4	−0.418081	−0.209041	+	0.977907i	$0.567034\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	297561.	1.51495	0.757474	−	0.652866i	$-0.226433\pi$
$131$	297561.	1.51495	0.757474	+	0.652866i	$0.226433\pi$
$132$	0	0
$133$	19212.9	0.0941810
$134$	0	0
$135$	0	0
$136$	0	0
$137$	57190.7	0.260330	0.130165	−	0.991492i	$-0.458449\pi$
$137$	57190.7	0.260330	0.130165	+	0.991492i	$0.458449\pi$
$138$	0	0
$139$	297610.	1.30651	0.653253	−	0.757140i	$-0.273404\pi$
$139$	297610.	1.30651	0.653253	+	0.757140i	$0.273404\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	258423.	1.05679
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	271940.	1.00348	0.501739	−	0.865019i	$-0.332694\pi$
$149$	271940.	1.00348	0.501739	+	0.865019i	$0.332694\pi$
$150$	0	0
$151$	144441.	0.515523	0.257762	−	0.966209i	$-0.417015\pi$
$151$	144441.	0.515523	0.257762	+	0.966209i	$0.417015\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	407289.	1.31872	0.659361	−	0.751826i	$-0.270827\pi$
$157$	407289.	1.31872	0.659361	+	0.751826i	$0.270827\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−385989.	−1.17357
$162$	0	0
$163$	473883.	1.39702	0.698509	−	0.715601i	$-0.253847\pi$
$163$	473883.	1.39702	0.698509	+	0.715601i	$0.253847\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	291047.	0.807555	0.403777	−	0.914857i	$-0.367697\pi$
$167$	291047.	0.807555	0.403777	+	0.914857i	$0.367697\pi$
$168$	0	0
$169$	522431.	1.40706
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−133697.	−0.339631	−0.169815	−	0.985476i	$-0.554317\pi$
$173$	−133697.	−0.339631	−0.169815	+	0.985476i	$0.554317\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	392572.	0.915770	0.457885	−	0.889012i	$-0.348607\pi$
$179$	392572.	0.915770	0.457885	+	0.889012i	$0.348607\pi$
$180$	0	0
$181$	−374040.	−0.848636	−0.424318	−	0.905513i	$-0.639486\pi$
$181$	−374040.	−0.848636	−0.424318	+	0.905513i	$0.639486\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−331394.	−0.693011
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−993045.	−1.96963	−0.984817	−	0.173597i	$-0.944461\pi$
$191$	−993045.	−1.96963	−0.984817	+	0.173597i	$0.944461\pi$
$192$	0	0
$193$	−132234.	−0.255535	−0.127768	−	0.991804i	$-0.540781\pi$
$193$	−132234.	−0.255535	−0.127768	+	0.991804i	$0.540781\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.00791e6	1.85035	0.925177	−	0.379535i	$-0.123916\pi$
$197$	1.00791e6	1.85035	0.925177	+	0.379535i	$0.123916\pi$
$198$	0	0
$199$	−756589.	−1.35434	−0.677169	−	0.735827i	$-0.736794\pi$
$199$	−756589.	−1.35434	−0.677169	+	0.735827i	$0.736794\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	435010.	0.740899
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−37157.7	−0.0588414
$210$	0	0
$211$	446900.	0.691041	0.345521	−	0.938411i	$-0.387702\pi$
$211$	446900.	0.691041	0.345521	+	0.938411i	$0.387702\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−199355.	−0.287394
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.14609e6	−1.57847
$222$	0	0
$223$	1.28822e6	1.73471	0.867357	−	0.497687i	$-0.165817\pi$
$223$	1.28822e6	1.73471	0.867357	+	0.497687i	$0.165817\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−109761.	−0.141379	−0.0706894	−	0.997498i	$-0.522520\pi$
$227$	−109761.	−0.141379	−0.0706894	+	0.997498i	$0.522520\pi$
$228$	0	0
$229$	1.34217e6	1.69130	0.845648	−	0.533741i	$-0.179214\pi$
$229$	1.34217e6	1.69130	0.845648	+	0.533741i	$0.179214\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	987929.	1.19216	0.596082	−	0.802924i	$-0.296723\pi$
$233$	987929.	1.19216	0.596082	+	0.802924i	$0.296723\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	78478.1	0.0888697	0.0444349	−	0.999012i	$-0.485851\pi$
$239$	78478.1	0.0888697	0.0444349	+	0.999012i	$0.485851\pi$
$240$	0	0
$241$	1.40078e6	1.55356	0.776780	−	0.629772i	$-0.216852\pi$
$241$	1.40078e6	1.55356	0.776780	+	0.629772i	$0.216852\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−128505.	−0.134023
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−181711.	−0.182053	−0.0910264	−	0.995848i	$-0.529015\pi$
$251$	−181711.	−0.182053	−0.0910264	+	0.995848i	$0.529015\pi$
$252$	0	0
$253$	746502.	0.733212
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−267824.	−0.252940	−0.126470	−	0.991970i	$-0.540365\pi$
$257$	−267824.	−0.252940	−0.126470	+	0.991970i	$0.540365\pi$
$258$	0	0
$259$	445516.	0.412680
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.11398e6	0.993090	0.496545	−	0.868011i	$-0.334602\pi$
$263$	1.11398e6	0.993090	0.496545	+	0.868011i	$0.334602\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−124320.	−0.104751	−0.0523757	−	0.998627i	$-0.516679\pi$
$269$	−124320.	−0.104751	−0.0523757	+	0.998627i	$0.516679\pi$
$270$	0	0
$271$	−1.81145e6	−1.49832	−0.749158	−	0.662391i	$-0.769542\pi$
$271$	−1.81145e6	−1.49832	−0.749158	+	0.662391i	$0.769542\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−544092.	−0.426062	−0.213031	−	0.977045i	$-0.568334\pi$
$277$	−544092.	−0.426062	−0.213031	+	0.977045i	$0.568334\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.18452e6	−0.894902	−0.447451	−	0.894308i	$-0.647668\pi$
$281$	−1.18452e6	−0.894902	−0.447451	+	0.894308i	$0.647668\pi$
$282$	0	0
$283$	999650.	0.741963	0.370981	−	0.928640i	$-0.379021\pi$
$283$	999650.	0.741963	0.370981	+	0.928640i	$0.379021\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.90806e6	1.36737
$288$	0	0
$289$	49850.7	0.0351097
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.56682e6	1.74673	0.873365	−	0.487065i	$-0.161933\pi$
$293$	2.56682e6	1.74673	0.873365	+	0.487065i	$0.161933\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.58169e6	1.67004
$300$	0	0
$301$	−768013.	−0.488598
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	925658.	0.560537	0.280269	−	0.959922i	$-0.409576\pi$
$307$	925658.	0.560537	0.280269	+	0.959922i	$0.409576\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.46013e6	−1.44231	−0.721153	−	0.692776i	$-0.756387\pi$
$311$	−2.46013e6	−1.44231	−0.721153	+	0.692776i	$0.756387\pi$
$312$	0	0
$313$	1.43344e6	0.827025	0.413513	−	0.910498i	$-0.364302\pi$
$313$	1.43344e6	0.827025	0.413513	+	0.910498i	$0.364302\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−400932.	−0.224090	−0.112045	−	0.993703i	$-0.535740\pi$
$317$	−400932.	−0.224090	−0.112045	+	0.993703i	$0.535740\pi$
$318$	0	0
$319$	−841308.	−0.462891
$320$	0	0
$321$	0	0
$322$	0	0
$323$	164792.	0.0878878
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−462124.	−0.235380
$330$	0	0
$331$	−337080.	−0.169108	−0.0845539	−	0.996419i	$-0.526947\pi$
$331$	−337080.	−0.169108	−0.0845539	+	0.996419i	$0.526947\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.52668e6	1.21192	0.605962	−	0.795494i	$-0.292788\pi$
$337$	2.52668e6	1.21192	0.605962	+	0.795494i	$0.292788\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	385552.	0.179555
$342$	0	0
$343$	−1.92739e6	−0.884574
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.74945e6	1.22581	0.612904	−	0.790158i	$-0.290001\pi$
$347$	2.74945e6	1.22581	0.612904	+	0.790158i	$0.290001\pi$
$348$	0	0
$349$	2.70732e6	1.18981	0.594903	−	0.803797i	$-0.297190\pi$
$349$	2.70732e6	1.18981	0.594903	+	0.803797i	$0.297190\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.20305e6	1.36813	0.684064	−	0.729422i	$-0.260211\pi$
$353$	3.20305e6	1.36813	0.684064	+	0.729422i	$0.260211\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.39968e6	1.80171	0.900855	−	0.434121i	$-0.142941\pi$
$359$	4.39968e6	1.80171	0.900855	+	0.434121i	$0.142941\pi$
$360$	0	0
$361$	−2.45762e6	−0.992538
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−884314.	−0.342721	−0.171361	−	0.985208i	$-0.554816\pi$
$367$	−884314.	−0.342721	−0.171361	+	0.985208i	$0.554816\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.02774e6	1.14205
$372$	0	0
$373$	−1.02175e6	−0.380251	−0.190126	−	0.981760i	$-0.560890\pi$
$373$	−1.02175e6	−0.380251	−0.190126	+	0.981760i	$0.560890\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.90956e6	−1.05433
$378$	0	0
$379$	623582.	0.222995	0.111498	−	0.993765i	$-0.464435\pi$
$379$	623582.	0.222995	0.111498	+	0.993765i	$0.464435\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.60949e6	−0.908988	−0.454494	−	0.890750i	$-0.650180\pi$
$383$	−2.60949e6	−0.908988	−0.454494	+	0.890750i	$0.650180\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.48187e6	0.496520	0.248260	−	0.968693i	$-0.420141\pi$
$389$	1.48187e6	0.496520	0.248260	+	0.968693i	$0.420141\pi$
$390$	0	0
$391$	−3.31068e6	−1.09516
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.62187e6	1.79021	0.895106	−	0.445853i	$-0.147100\pi$
$397$	5.62187e6	1.79021	0.895106	+	0.445853i	$0.147100\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.64060e6	1.13061	0.565304	−	0.824883i	$-0.308759\pi$
$401$	3.64060e6	1.13061	0.565304	+	0.824883i	$0.308759\pi$
$402$	0	0
$403$	1.33339e6	0.408972
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−861627.	−0.257830
$408$	0	0
$409$	−3.12488e6	−0.923687	−0.461844	−	0.886961i	$-0.652812\pi$
$409$	−3.12488e6	−0.923687	−0.461844	+	0.886961i	$0.652812\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.85201e6	1.68822
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.94722e6	1.09839	0.549195	−	0.835694i	$-0.314935\pi$
$419$	3.94722e6	1.09839	0.549195	+	0.835694i	$0.314935\pi$
$420$	0	0
$421$	3.97485e6	1.09299	0.546494	−	0.837463i	$-0.315962\pi$
$421$	3.97485e6	1.09299	0.546494	+	0.837463i	$0.315962\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.44507e6	−0.914383
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.02193e6	−0.264989	−0.132495	−	0.991184i	$-0.542299\pi$
$431$	−1.02193e6	−0.264989	−0.132495	+	0.991184i	$0.542299\pi$
$432$	0	0
$433$	−4.89035e6	−1.25349	−0.626744	−	0.779225i	$-0.715613\pi$
$433$	−4.89035e6	−1.25349	−0.626744	+	0.779225i	$0.715613\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−371212.	−0.0929861
$438$	0	0
$439$	−2.44464e6	−0.605414	−0.302707	−	0.953084i	$-0.597890\pi$
$439$	−2.44464e6	−0.605414	−0.302707	+	0.953084i	$0.597890\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.84682e6	0.931308	0.465654	−	0.884967i	$-0.345819\pi$
$443$	3.84682e6	0.931308	0.465654	+	0.884967i	$0.345819\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.57119e6	−1.53825	−0.769127	−	0.639096i	$-0.779309\pi$
$449$	−6.57119e6	−1.53825	−0.769127	+	0.639096i	$0.779309\pi$
$450$	0	0
$451$	−3.69019e6	−0.854293
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.79203e6	−0.625359	−0.312679	−	0.949859i	$-0.601227\pi$
$457$	−2.79203e6	−0.625359	−0.312679	+	0.949859i	$0.601227\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	272304.	0.0596762	0.0298381	−	0.999555i	$-0.490501\pi$
$461$	272304.	0.0596762	0.0298381	+	0.999555i	$0.490501\pi$
$462$	0	0
$463$	4.71209e6	1.02155	0.510777	−	0.859713i	$-0.329358\pi$
$463$	4.71209e6	1.02155	0.510777	+	0.859713i	$0.329358\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.42758e6	−1.15163	−0.575816	−	0.817579i	$-0.695316\pi$
$467$	−5.42758e6	−1.15163	−0.575816	+	0.817579i	$0.695316\pi$
$468$	0	0
$469$	4.56636e6	0.958600
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.48534e6	0.305261
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.54900e6	−0.905893	−0.452947	−	0.891538i	$-0.649627\pi$
$479$	−4.54900e6	−0.905893	−0.452947	+	0.891538i	$0.649627\pi$
$480$	0	0
$481$	−2.97983e6	−0.587259
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.52219e6	0.481897	0.240949	−	0.970538i	$-0.422541\pi$
$487$	2.52219e6	0.481897	0.240949	+	0.970538i	$0.422541\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.33529e6	−0.811548	−0.405774	−	0.913973i	$-0.632998\pi$
$491$	−4.33529e6	−0.811548	−0.405774	+	0.913973i	$0.632998\pi$
$492$	0	0
$493$	3.73114e6	0.691392
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.46642e6	0.992687
$498$	0	0
$499$	−6.31959e6	−1.13615	−0.568077	−	0.822975i	$-0.692313\pi$
$499$	−6.31959e6	−1.13615	−0.568077	+	0.822975i	$0.692313\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−8.09276e6	−1.42619	−0.713094	−	0.701068i	$-0.752707\pi$
$503$	−8.09276e6	−1.42619	−0.713094	+	0.701068i	$0.752707\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.00658e6	−0.343291	−0.171645	−	0.985159i	$-0.554908\pi$
$509$	−2.00658e6	−0.343291	−0.171645	+	0.985159i	$0.554908\pi$
$510$	0	0
$511$	−1.14258e6	−0.193569
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	893748.	0.147058
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.22441e6	−0.359021	−0.179511	−	0.983756i	$-0.557451\pi$
$521$	−2.22441e6	−0.359021	−0.179511	+	0.983756i	$0.557451\pi$
$522$	0	0
$523$	−3.88746e6	−0.621459	−0.310729	−	0.950498i	$-0.600573\pi$
$523$	−3.88746e6	−0.621459	−0.310729	+	0.950498i	$0.600573\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.70989e6	−0.268190
$528$	0	0
$529$	1.02135e6	0.158684
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.27621e7	−1.94582
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−866730.	−0.128503
$540$	0	0
$541$	8.65144e6	1.27085	0.635426	−	0.772162i	$-0.280824\pi$
$541$	8.65144e6	1.27085	0.635426	+	0.772162i	$0.280824\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−5.36845e6	−0.767150	−0.383575	−	0.923510i	$-0.625307\pi$
$547$	−5.36845e6	−0.767150	−0.383575	+	0.923510i	$0.625307\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	418356.	0.0587039
$552$	0	0
$553$	−2.25855e6	−0.314063
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.41362e6	0.875922	0.437961	−	0.898994i	$-0.355701\pi$
$557$	6.41362e6	0.875922	0.437961	+	0.898994i	$0.355701\pi$
$558$	0	0
$559$	5.13686e6	0.695293
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.48424e6	−0.330311	−0.165155	−	0.986268i	$-0.552813\pi$
$563$	−2.48424e6	−0.330311	−0.165155	+	0.986268i	$0.552813\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.11183e6	−0.661905	−0.330953	−	0.943647i	$-0.607370\pi$
$569$	−5.11183e6	−0.661905	−0.330953	+	0.943647i	$0.607370\pi$
$570$	0	0
$571$	1.13467e7	1.45640	0.728200	−	0.685365i	$-0.240357\pi$
$571$	1.13467e7	1.45640	0.728200	+	0.685365i	$0.240357\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.90331e6	−0.237996	−0.118998	−	0.992895i	$-0.537968\pi$
$577$	−1.90331e6	−0.237996	−0.118998	+	0.992895i	$0.537968\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.85428e6	0.842406
$582$	0	0
$583$	−5.85564e6	−0.713515
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.64386e6	−0.316697	−0.158348	−	0.987383i	$-0.550617\pi$
$587$	−2.64386e6	−0.316697	−0.158348	+	0.987383i	$0.550617\pi$
$588$	0	0
$589$	−191723.	−0.0227712
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.02064e6	−0.352746	−0.176373	−	0.984323i	$-0.556437\pi$
$593$	−3.02064e6	−0.352746	−0.176373	+	0.984323i	$0.556437\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	336944.	0.0383699	0.0191849	−	0.999816i	$-0.493893\pi$
$599$	336944.	0.0383699	0.0191849	+	0.999816i	$0.493893\pi$
$600$	0	0
$601$	1.06160e7	1.19887	0.599436	−	0.800422i	$-0.295391\pi$
$601$	1.06160e7	1.19887	0.599436	+	0.800422i	$0.295391\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.71997e6	−0.960602	−0.480301	−	0.877104i	$-0.659472\pi$
$607$	−8.71997e6	−0.960602	−0.480301	+	0.877104i	$0.659472\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.09092e6	0.334954
$612$	0	0
$613$	1.32783e6	0.142722	0.0713611	−	0.997451i	$-0.477266\pi$
$613$	1.32783e6	0.142722	0.0713611	+	0.997451i	$0.477266\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.38634e6	−0.886869	−0.443435	−	0.896307i	$-0.646240\pi$
$617$	−8.38634e6	−0.886869	−0.443435	+	0.896307i	$0.646240\pi$
$618$	0	0
$619$	73772.8	0.00773873	0.00386936	−	0.999993i	$-0.498768\pi$
$619$	73772.8	0.00773873	0.00386936	+	0.999993i	$0.498768\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.26060e7	−1.30124
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.82125e6	0.385105
$630$	0	0
$631$	1.03680e7	1.03663	0.518313	−	0.855191i	$-0.326560\pi$
$631$	1.03680e7	1.03663	0.518313	+	0.855191i	$0.326560\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.99748e6	−0.292690
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.77488e6	0.266746	0.133373	−	0.991066i	$-0.457419\pi$
$641$	2.77488e6	0.266746	0.133373	+	0.991066i	$0.457419\pi$
$642$	0	0
$643$	3.99310e6	0.380875	0.190438	−	0.981699i	$-0.439009\pi$
$643$	3.99310e6	0.380875	0.190438	+	0.981699i	$0.439009\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.30149e6	−0.216147	−0.108073	−	0.994143i	$-0.534468\pi$
$647$	−2.30149e6	−0.216147	−0.108073	+	0.994143i	$0.534468\pi$
$648$	0	0
$649$	−1.13178e7	−1.05475
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.94361e7	−1.78371	−0.891857	−	0.452317i	$-0.850598\pi$
$653$	−1.94361e7	−1.78371	−0.891857	+	0.452317i	$0.850598\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.05797e7	−0.948982	−0.474491	−	0.880260i	$-0.657368\pi$
$659$	−1.05797e7	−0.948982	−0.474491	+	0.880260i	$0.657368\pi$
$660$	0	0
$661$	1.01568e6	0.0904180	0.0452090	−	0.998978i	$-0.485605\pi$
$661$	1.01568e6	0.0904180	0.0452090	+	0.998978i	$0.485605\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.40482e6	−0.731499
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.66276e6	0.571278
$672$	0	0
$673$	−5.87188e6	−0.499735	−0.249867	−	0.968280i	$-0.580387\pi$
$673$	−5.87188e6	−0.499735	−0.249867	+	0.968280i	$0.580387\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.10597e7	1.76596	0.882978	−	0.469415i	$-0.155535\pi$
$677$	2.10597e7	1.76596	0.882978	+	0.469415i	$0.155535\pi$
$678$	0	0
$679$	8.03139e6	0.668523
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2.05169e7	−1.68290	−0.841452	−	0.540331i	$-0.818299\pi$
$683$	−2.05169e7	−1.68290	−0.841452	+	0.540331i	$0.818299\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.02510e7	−1.62517
$690$	0	0
$691$	−1.61554e7	−1.28713	−0.643565	−	0.765392i	$-0.722545\pi$
$691$	−1.61554e7	−1.28713	−0.643565	+	0.765392i	$0.722545\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.63657e7	1.27601
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.60014e6	−0.507292	−0.253646	−	0.967297i	$-0.581630\pi$
$701$	−6.60014e6	−0.507292	−0.253646	+	0.967297i	$0.581630\pi$
$702$	0	0
$703$	428460.	0.0326980
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.52919e6	−0.716981
$708$	0	0
$709$	−2.10409e6	−0.157199	−0.0785995	−	0.996906i	$-0.525045\pi$
$709$	−2.10409e6	−0.157199	−0.0785995	+	0.996906i	$0.525045\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.85173e6	0.283748
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.86112e6	0.134262	0.0671309	−	0.997744i	$-0.478615\pi$
$719$	1.86112e6	0.134262	0.0671309	+	0.997744i	$0.478615\pi$
$720$	0	0
$721$	−8.73203e6	−0.625572
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.05408e7	0.739671	0.369836	−	0.929097i	$-0.379414\pi$
$727$	1.05408e7	0.739671	0.369836	+	0.929097i	$0.379414\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6.58736e6	−0.455950
$732$	0	0
$733$	−1.47763e7	−1.01579	−0.507897	−	0.861418i	$-0.669577\pi$
$733$	−1.47763e7	−1.01579	−0.507897	+	0.861418i	$0.669577\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.83133e6	−0.598904
$738$	0	0
$739$	−3.19319e6	−0.215087	−0.107543	−	0.994200i	$-0.534298\pi$
$739$	−3.19319e6	−0.215087	−0.107543	+	0.994200i	$0.534298\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.58587e7	1.05389	0.526945	−	0.849900i	$-0.323337\pi$
$743$	1.58587e7	1.05389	0.526945	+	0.849900i	$0.323337\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.11873e7	−0.728653
$750$	0	0
$751$	2.14341e7	1.38677	0.693387	−	0.720566i	$-0.256118\pi$
$751$	2.14341e7	1.38677	0.693387	+	0.720566i	$0.256118\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.29941e7	−1.45840	−0.729201	−	0.684299i	$-0.760108\pi$
$757$	−2.29941e7	−1.45840	−0.729201	+	0.684299i	$0.760108\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.59525e7	0.998547	0.499273	−	0.866445i	$-0.333600\pi$
$761$	1.59525e7	0.998547	0.499273	+	0.866445i	$0.333600\pi$
$762$	0	0
$763$	−2.50234e7	−1.55609
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.91412e7	−2.40240
$768$	0	0
$769$	−1.41126e7	−0.860577	−0.430288	−	0.902691i	$-0.641588\pi$
$769$	−1.41126e7	−0.860577	−0.430288	+	0.902691i	$0.641588\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.05372e6	−0.544977	−0.272489	−	0.962159i	$-0.587847\pi$
$773$	−9.05372e6	−0.544977	−0.272489	+	0.962159i	$0.587847\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.83501e6	0.108342
$780$	0	0
$781$	−1.05721e7	−0.620200
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.33625e6	0.192009	0.0960046	−	0.995381i	$-0.469394\pi$
$787$	3.33625e6	0.192009	0.0960046	+	0.995381i	$0.469394\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.78756e7	1.58410
$792$	0	0
$793$	2.30424e7	1.30120
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.58206e7	0.882219	0.441110	−	0.897453i	$-0.354585\pi$
$797$	1.58206e7	0.882219	0.441110	+	0.897453i	$0.354585\pi$
$798$	0	0
$799$	−3.96371e6	−0.219652
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.20976e6	0.120936
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.69505e7	−0.910564	−0.455282	−	0.890347i	$-0.650462\pi$
$809$	−1.69505e7	−0.910564	−0.455282	+	0.890347i	$0.650462\pi$
$810$	0	0
$811$	−2.93880e6	−0.156898	−0.0784491	−	0.996918i	$-0.524997\pi$
$811$	−2.93880e6	−0.156898	−0.0784491	+	0.996918i	$0.524997\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−738610.	−0.0387133
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.13058e7	1.62094	0.810469	−	0.585781i	$-0.199212\pi$
$821$	3.13058e7	1.62094	0.810469	+	0.585781i	$0.199212\pi$
$822$	0	0
$823$	−1.19264e7	−0.613774	−0.306887	−	0.951746i	$-0.599287\pi$
$823$	−1.19264e7	−0.613774	−0.306887	+	0.951746i	$0.599287\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.96149e7	−0.997292	−0.498646	−	0.866806i	$-0.666169\pi$
$827$	−1.96149e7	−0.997292	−0.498646	+	0.866806i	$0.666169\pi$
$828$	0	0
$829$	−3.33477e7	−1.68531	−0.842656	−	0.538453i	$-0.819009\pi$
$829$	−3.33477e7	−1.68531	−0.842656	+	0.538453i	$0.819009\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.84389e6	0.191937
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.78255e7	0.874254	0.437127	−	0.899400i	$-0.355996\pi$
$839$	1.78255e7	0.874254	0.437127	+	0.899400i	$0.355996\pi$
$840$	0	0
$841$	−1.10389e7	−0.538191
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.22017e7	−0.584403
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.60781e6	−0.407445
$852$	0	0
$853$	−2.37711e6	−0.111860	−0.0559302	−	0.998435i	$-0.517812\pi$
$853$	−2.37711e6	−0.111860	−0.0559302	+	0.998435i	$0.517812\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.68298e6	−0.264316	−0.132158	−	0.991229i	$-0.542191\pi$
$857$	−5.68298e6	−0.264316	−0.132158	+	0.991229i	$0.542191\pi$
$858$	0	0
$859$	2.02691e7	0.937240	0.468620	−	0.883400i	$-0.344751\pi$
$859$	2.02691e7	0.937240	0.468620	+	0.883400i	$0.344751\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.83084e6	0.266504	0.133252	−	0.991082i	$-0.457458\pi$
$863$	5.83084e6	0.266504	0.133252	+	0.991082i	$0.457458\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.36803e6	0.196217
$870$	0	0
$871$	−3.05421e7	−1.36412
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.35496e7	1.91199	0.955995	−	0.293383i	$-0.0947812\pi$
$877$	4.35496e7	1.91199	0.955995	+	0.293383i	$0.0947812\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.38261e7	1.46829	0.734146	−	0.678992i	$-0.237583\pi$
$881$	3.38261e7	1.46829	0.734146	+	0.678992i	$0.237583\pi$
$882$	0	0
$883$	−1.15279e7	−0.497562	−0.248781	−	0.968560i	$-0.580030\pi$
$883$	−1.15279e7	−0.497562	−0.248781	+	0.968560i	$0.580030\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.62136e7	−1.11871	−0.559354	−	0.828929i	$-0.688951\pi$
$887$	−2.62136e7	−1.11871	−0.559354	+	0.828929i	$0.688951\pi$
$888$	0	0
$889$	−1.07409e7	−0.455815
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−444432.	−0.0186499
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.34090e6	−0.179135
$900$	0	0
$901$	2.59694e7	1.06573
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.88118e7	0.759296	0.379648	−	0.925131i	$-0.376045\pi$
$907$	1.88118e7	0.759296	0.379648	+	0.925131i	$0.376045\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.51622e7	−1.00451	−0.502253	−	0.864721i	$-0.667495\pi$
$911$	−2.51622e7	−1.00451	−0.502253	+	0.864721i	$0.667495\pi$
$912$	0	0
$913$	−1.32562e7	−0.526310
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.20580e7	1.65168
$918$	0	0
$919$	2.96599e7	1.15846	0.579230	−	0.815164i	$-0.303353\pi$
$919$	2.96599e7	1.15846	0.579230	+	0.815164i	$0.303353\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.65622e7	−1.41263
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.31730e7	1.64124	0.820622	−	0.571471i	$-0.193627\pi$
$929$	4.31730e7	1.64124	0.820622	+	0.571471i	$0.193627\pi$
$930$	0	0
$931$	430997.	0.0162967
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.49826e6	−0.167377	−0.0836885	−	0.996492i	$-0.526670\pi$
$937$	−4.49826e6	−0.167377	−0.0836885	+	0.996492i	$0.526670\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.01968e7	−0.375395	−0.187697	−	0.982227i	$-0.560102\pi$
$941$	−1.01968e7	−0.375395	−0.187697	+	0.982227i	$0.560102\pi$
$942$	0	0
$943$	−3.68656e7	−1.35003
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.08376e7	0.392696	0.196348	−	0.980534i	$-0.437092\pi$
$947$	1.08376e7	0.392696	0.196348	+	0.980534i	$0.437092\pi$
$948$	0	0
$949$	7.64218e6	0.275456
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.39870e7	1.56889	0.784444	−	0.620199i	$-0.212948\pi$
$953$	4.39870e7	1.56889	0.784444	+	0.620199i	$0.212948\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.08347e6	0.283826
$960$	0	0
$961$	−2.66398e7	−0.930514
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.72431e7	−0.592992	−0.296496	−	0.955034i	$-0.595818\pi$
$967$	−1.72431e7	−0.592992	−0.296496	+	0.955034i	$0.595818\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.11158e7	−1.73983	−0.869915	−	0.493201i	$-0.835827\pi$
$971$	−5.11158e7	−1.73983	−0.869915	+	0.493201i	$0.835827\pi$
$972$	0	0
$973$	4.20650e7	1.42442
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.28599e7	1.77170	0.885849	−	0.463974i	$-0.153577\pi$
$977$	5.28599e7	1.77170	0.885849	+	0.463974i	$0.153577\pi$
$978$	0	0
$979$	2.43800e7	0.812973
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.40562e6	0.0463965	0.0231983	−	0.999731i	$-0.492615\pi$
$983$	1.40562e6	0.0463965	0.0231983	+	0.999731i	$0.492615\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.48388e7	0.482400
$990$	0	0
$991$	−3.12457e6	−0.101066	−0.0505331	−	0.998722i	$-0.516092\pi$
$991$	−3.12457e6	−0.101066	−0.0505331	+	0.998722i	$0.516092\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.07225e7	−0.341630	−0.170815	−	0.985303i	$-0.554640\pi$
$997$	−1.07225e7	−0.341630	−0.170815	+	0.985303i	$0.554640\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.a.s.1.2		2
3.2	odd	2		100.6.a.c.1.1	✓	2
5.2	odd	4		900.6.d.m.649.4		4
5.3	odd	4		900.6.d.m.649.1		4
5.4	even	2		900.6.a.m.1.1		2
12.11	even	2		400.6.a.v.1.2		2
15.2	even	4		100.6.c.c.49.4		4
15.8	even	4		100.6.c.c.49.1		4
15.14	odd	2		100.6.a.e.1.2	yes	2
60.23	odd	4		400.6.c.m.49.4		4
60.47	odd	4		400.6.c.m.49.1		4
60.59	even	2		400.6.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.6.a.c.1.1	✓	2	3.2	odd	2
100.6.a.e.1.2	yes	2	15.14	odd	2
100.6.c.c.49.1		4	15.8	even	4
100.6.c.c.49.4		4	15.2	even	4
400.6.a.p.1.1		2	60.59	even	2
400.6.a.v.1.2		2	12.11	even	2
400.6.c.m.49.1		4	60.47	odd	4
400.6.c.m.49.4		4	60.23	odd	4
900.6.a.m.1.1		2	5.4	even	2
900.6.a.s.1.2		2	1.1	even	1	trivial
900.6.d.m.649.1		4	5.3	odd	4
900.6.d.m.649.4		4	5.2	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)

\(f(q)\)	\(=\)	\(q+141.342 q^{7} +O(q^{10})\)	\(q+141.342 q^{7} -273.356 q^{11} -945.370 q^{13} +1212.32 q^{17} +135.931 q^{19} -2730.88 q^{23} +3077.70 q^{29} -1410.44 q^{31} +3152.03 q^{37} +13499.5 q^{41} -5433.70 q^{43} -3269.53 q^{47} +3170.70 q^{49} +21421.3 q^{53} +41403.1 q^{59} -24373.9 q^{61} +32307.0 q^{67} +38675.0 q^{71} -8083.79 q^{73} -38636.8 q^{77} -15979.3 q^{79} +48494.1 q^{83} -89187.5 q^{89} -133621. q^{91} +56822.2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 40 q^{7}+O(q^{10}) \) 2 * q + 40 * q^7	\( 2 q + 40 q^{7} + 60 q^{11} - 920 q^{13} + 2910 q^{17} + 2092 q^{19} + 120 q^{23} - 3552 q^{29} - 8888 q^{31} - 12140 q^{37} + 12438 q^{41} - 1160 q^{43} - 1200 q^{47} - 3366 q^{49} + 26340 q^{53} + 36696 q^{59} + 19204 q^{61} + 90460 q^{67} - 2736 q^{71} - 12770 q^{73} - 72420 q^{77} - 16184 q^{79} - 30300 q^{83} + 47322 q^{89} - 136192 q^{91} + 2980 q^{97}+O(q^{100}) \) 2 * q + 40 * q^7 + 60 * q^11 - 920 * q^13 + 2910 * q^17 + 2092 * q^19 + 120 * q^23 - 3552 * q^29 - 8888 * q^31 - 12140 * q^37 + 12438 * q^41 - 1160 * q^43 - 1200 * q^47 - 3366 * q^49 + 26340 * q^53 + 36696 * q^59 + 19204 * q^61 + 90460 * q^67 - 2736 * q^71 - 12770 * q^73 - 72420 * q^77 - 16184 * q^79 - 30300 * q^83 + 47322 * q^89 - 136192 * q^91 + 2980 * q^97

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