LMFDB - Embedded newform 900.6.a.l.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{94}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 94 $ x^2 - 94
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$9.69536$ 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)

$f(q)$	$=$	$q-71.0000 q^{7} +O(q^{10})$	$q-71.0000 q^{7} +581.722 q^{11} -137.000 q^{13} +581.722 q^{17} -1087.00 q^{19} +4072.05 q^{23} -4072.05 q^{29} -2269.00 q^{31} +6010.00 q^{37} +1745.16 q^{41} -4283.00 q^{43} +1163.44 q^{47} -11766.0 q^{49} -25014.0 q^{53} +29086.1 q^{59} +12719.0 q^{61} -6899.00 q^{67} -40138.8 q^{71} +24010.0 q^{73} -41302.2 q^{77} +12236.0 q^{79} +26759.2 q^{83} -55845.3 q^{89} +9727.00 q^{91} +19651.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 142 q^{7}+O(q^{10}) $ 2 * q - 142 * q^7	$ 2 q - 142 q^{7} - 274 q^{13} - 2174 q^{19} - 4538 q^{31} + 12020 q^{37} - 8566 q^{43} - 23532 q^{49} + 25438 q^{61} - 13798 q^{67} + 48020 q^{73} + 24472 q^{79} + 19454 q^{91} + 39302 q^{97}+O(q^{100}) $ 2 * q - 142 * q^7 - 274 * q^13 - 2174 * q^19 - 4538 * q^31 + 12020 * q^37 - 8566 * q^43 - 23532 * q^49 + 25438 * q^61 - 13798 * q^67 + 48020 * q^73 + 24472 * q^79 + 19454 * q^91 + 39302 * 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
$7$	−71.0000	−0.547663	−0.273831	−	0.961778i	$-0.588291\pi$
$7$	−71.0000	−0.547663	−0.273831	+	0.961778i	$0.588291\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	581.722	1.44955	0.724775	−	0.688985i	$-0.241943\pi$
$11$	581.722	1.44955	0.724775	+	0.688985i	$0.241943\pi$
$12$	0	0
$13$	−137.000	−0.224834	−0.112417	−	0.993661i	$-0.535859\pi$
$13$	−137.000	−0.224834	−0.112417	+	0.993661i	$0.535859\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	581.722	0.488194	0.244097	−	0.969751i	$-0.421508\pi$
$17$	581.722	0.488194	0.244097	+	0.969751i	$0.421508\pi$
$18$	0	0
$19$	−1087.00	−0.690789	−0.345395	−	0.938458i	$-0.612255\pi$
$19$	−1087.00	−0.690789	−0.345395	+	0.938458i	$0.612255\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4072.05	1.60507	0.802534	−	0.596606i	$-0.203484\pi$
$23$	4072.05	1.60507	0.802534	+	0.596606i	$0.203484\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4072.05	−0.899121	−0.449561	−	0.893250i	$-0.648419\pi$
$29$	−4072.05	−0.899121	−0.449561	+	0.893250i	$0.648419\pi$
$30$	0	0
$31$	−2269.00	−0.424063	−0.212031	−	0.977263i	$-0.568008\pi$
$31$	−2269.00	−0.424063	−0.212031	+	0.977263i	$0.568008\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6010.00	0.721722	0.360861	−	0.932620i	$-0.382483\pi$
$37$	6010.00	0.721722	0.360861	+	0.932620i	$0.382483\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1745.16	0.162135	0.0810675	−	0.996709i	$-0.474167\pi$
$41$	1745.16	0.162135	0.0810675	+	0.996709i	$0.474167\pi$
$42$	0	0
$43$	−4283.00	−0.353246	−0.176623	−	0.984279i	$-0.556517\pi$
$43$	−4283.00	−0.353246	−0.176623	+	0.984279i	$0.556517\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1163.44	0.0768246	0.0384123	−	0.999262i	$-0.487770\pi$
$47$	1163.44	0.0768246	0.0384123	+	0.999262i	$0.487770\pi$
$48$	0	0
$49$	−11766.0	−0.700065
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−25014.0	−1.22319	−0.611595	−	0.791171i	$-0.709472\pi$
$53$	−25014.0	−1.22319	−0.611595	+	0.791171i	$0.709472\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	29086.1	1.08782	0.543908	−	0.839145i	$-0.316944\pi$
$59$	29086.1	1.08782	0.543908	+	0.839145i	$0.316944\pi$
$60$	0	0
$61$	12719.0	0.437651	0.218826	−	0.975764i	$-0.429777\pi$
$61$	12719.0	0.437651	0.218826	+	0.975764i	$0.429777\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6899.00	−0.187758	−0.0938791	−	0.995584i	$-0.529927\pi$
$67$	−6899.00	−0.187758	−0.0938791	+	0.995584i	$0.529927\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−40138.8	−0.944971	−0.472485	−	0.881338i	$-0.656643\pi$
$71$	−40138.8	−0.944971	−0.472485	+	0.881338i	$0.656643\pi$
$72$	0	0
$73$	24010.0	0.527333	0.263667	−	0.964614i	$-0.415068\pi$
$73$	24010.0	0.527333	0.263667	+	0.964614i	$0.415068\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−41302.2	−0.793865
$78$	0	0
$79$	12236.0	0.220583	0.110291	−	0.993899i	$-0.464822\pi$
$79$	12236.0	0.220583	0.110291	+	0.993899i	$0.464822\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	26759.2	0.426361	0.213181	−	0.977013i	$-0.431618\pi$
$83$	26759.2	0.426361	0.213181	+	0.977013i	$0.431618\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−55845.3	−0.747328	−0.373664	−	0.927564i	$-0.621899\pi$
$89$	−55845.3	−0.747328	−0.373664	+	0.927564i	$0.621899\pi$
$90$	0	0
$91$	9727.00	0.123133
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	19651.0	0.212058	0.106029	−	0.994363i	$-0.466186\pi$
$97$	19651.0	0.212058	0.106029	+	0.994363i	$0.466186\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	136705.	1.33346	0.666729	−	0.745300i	$-0.267694\pi$
$101$	136705.	1.33346	0.666729	+	0.745300i	$0.267694\pi$
$102$	0	0
$103$	152380.	1.41526	0.707628	−	0.706585i	$-0.249765\pi$
$103$	152380.	1.41526	0.707628	+	0.706585i	$0.249765\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	178589.	1.50797	0.753987	−	0.656889i	$-0.228128\pi$
$107$	178589.	1.50797	0.753987	+	0.656889i	$0.228128\pi$
$108$	0	0
$109$	3137.00	0.0252900	0.0126450	−	0.999920i	$-0.495975\pi$
$109$	3137.00	0.0252900	0.0126450	+	0.999920i	$0.495975\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	93075.5	0.685708	0.342854	−	0.939389i	$-0.388606\pi$
$113$	93075.5	0.685708	0.342854	+	0.939389i	$0.388606\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−41302.2	−0.267366
$120$	0	0
$121$	177349.	1.10120
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3620.00	−0.0199159	−0.00995793	−	0.999950i	$-0.503170\pi$
$127$	−3620.00	−0.0199159	−0.00995793	+	0.999950i	$0.503170\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−91330.3	−0.464982	−0.232491	−	0.972598i	$-0.574688\pi$
$131$	−91330.3	−0.464982	−0.232491	+	0.972598i	$0.574688\pi$
$132$	0	0
$133$	77177.0	0.378320
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−268755.	−1.22336	−0.611682	−	0.791104i	$-0.709507\pi$
$137$	−268755.	−1.22336	−0.611682	+	0.791104i	$0.709507\pi$
$138$	0	0
$139$	−311536.	−1.36764	−0.683819	−	0.729652i	$-0.739682\pi$
$139$	−311536.	−1.36764	−0.683819	+	0.729652i	$0.739682\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−79695.9	−0.325908
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	150084.	0.553821	0.276910	−	0.960896i	$-0.410689\pi$
$149$	150084.	0.553821	0.276910	+	0.960896i	$0.410689\pi$
$150$	0	0
$151$	123221.	0.439787	0.219893	−	0.975524i	$-0.429429\pi$
$151$	123221.	0.439787	0.219893	+	0.975524i	$0.429429\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	387061.	1.25323	0.626614	−	0.779330i	$-0.284440\pi$
$157$	387061.	1.25323	0.626614	+	0.779330i	$0.284440\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−289116.	−0.879036
$162$	0	0
$163$	508327.	1.49856	0.749280	−	0.662253i	$-0.230400\pi$
$163$	508327.	1.49856	0.749280	+	0.662253i	$0.230400\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−62244.2	−0.172706	−0.0863531	−	0.996265i	$-0.527521\pi$
$167$	−62244.2	−0.172706	−0.0863531	+	0.996265i	$0.527521\pi$
$168$	0	0
$169$	−352524.	−0.949450
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−478757.	−1.21619	−0.608093	−	0.793866i	$-0.708065\pi$
$173$	−478757.	−1.21619	−0.608093	+	0.793866i	$0.708065\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	276899.	0.645936	0.322968	−	0.946410i	$-0.395319\pi$
$179$	276899.	0.645936	0.322968	+	0.946410i	$0.395319\pi$
$180$	0	0
$181$	317261.	0.719814	0.359907	−	0.932988i	$-0.382808\pi$
$181$	317261.	0.719814	0.359907	+	0.932988i	$0.382808\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	338400.	0.707663
$188$	0	0
$189$	0	0
$190$	0	0
$191$	432219.	0.857276	0.428638	−	0.903476i	$-0.358994\pi$
$191$	432219.	0.857276	0.428638	+	0.903476i	$0.358994\pi$
$192$	0	0
$193$	775273.	1.49817	0.749086	−	0.662473i	$-0.230493\pi$
$193$	775273.	1.49817	0.749086	+	0.662473i	$0.230493\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	671307.	1.23241	0.616205	−	0.787586i	$-0.288669\pi$
$197$	671307.	1.23241	0.616205	+	0.787586i	$0.288669\pi$
$198$	0	0
$199$	−752503.	−1.34702	−0.673512	−	0.739176i	$-0.735215\pi$
$199$	−752503.	−1.34702	−0.673512	+	0.739176i	$0.735215\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	289116.	0.492415
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−632331.	−1.00133
$210$	0	0
$211$	543989.	0.841170	0.420585	−	0.907253i	$-0.361825\pi$
$211$	543989.	0.841170	0.420585	+	0.907253i	$0.361825\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	161099.	0.232243
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−79695.9	−0.109763
$222$	0	0
$223$	1.16371e6	1.56705	0.783527	−	0.621358i	$-0.213419\pi$
$223$	1.16371e6	1.56705	0.783527	+	0.621358i	$0.213419\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−979037.	−1.26106	−0.630528	−	0.776166i	$-0.717162\pi$
$227$	−979037.	−1.26106	−0.630528	+	0.776166i	$0.717162\pi$
$228$	0	0
$229$	−559927.	−0.705574	−0.352787	−	0.935704i	$-0.614766\pi$
$229$	−559927.	−0.705574	−0.352787	+	0.935704i	$0.614766\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.50608e6	1.81743	0.908714	−	0.417418i	$-0.137065\pi$
$233$	1.50608e6	1.81743	0.908714	+	0.417418i	$0.137065\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−700975.	−0.793793	−0.396897	−	0.917863i	$-0.629913\pi$
$239$	−700975.	−0.793793	−0.396897	+	0.917863i	$0.629913\pi$
$240$	0	0
$241$	1.72941e6	1.91803	0.959014	−	0.283357i	$-0.0914483\pi$
$241$	1.72941e6	1.91803	0.959014	+	0.283357i	$0.0914483\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	148919.	0.155313
$248$	0	0
$249$	0	0
$250$	0	0
$251$	229198.	0.229629	0.114815	−	0.993387i	$-0.463373\pi$
$251$	229198.	0.229629	0.114815	+	0.993387i	$0.463373\pi$
$252$	0	0
$253$	2.36880e6	2.32663
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.01103e6	0.954843	0.477422	−	0.878674i	$-0.341571\pi$
$257$	1.01103e6	0.954843	0.477422	+	0.878674i	$0.341571\pi$
$258$	0	0
$259$	−426710.	−0.395260
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−803358.	−0.716176	−0.358088	−	0.933688i	$-0.616571\pi$
$263$	−803358.	−0.716176	−0.358088	+	0.933688i	$0.616571\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.89932e6	1.60036	0.800180	−	0.599760i	$-0.204737\pi$
$269$	1.89932e6	1.60036	0.800180	+	0.599760i	$0.204737\pi$
$270$	0	0
$271$	1.01205e6	0.837104	0.418552	−	0.908193i	$-0.362538\pi$
$271$	1.01205e6	0.837104	0.418552	+	0.908193i	$0.362538\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.65128e6	1.29307	0.646534	−	0.762886i	$-0.276218\pi$
$277$	1.65128e6	1.29307	0.646534	+	0.762886i	$0.276218\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	164627.	0.124376	0.0621879	−	0.998064i	$-0.480192\pi$
$281$	164627.	0.124376	0.0621879	+	0.998064i	$0.480192\pi$
$282$	0	0
$283$	1.88422e6	1.39851	0.699256	−	0.714871i	$-0.253515\pi$
$283$	1.88422e6	1.39851	0.699256	+	0.714871i	$0.253515\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−123907.	−0.0887953
$288$	0	0
$289$	−1.08146e6	−0.761666
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.08024e6	−1.41561	−0.707805	−	0.706408i	$-0.750314\pi$
$293$	−2.08024e6	−1.41561	−0.707805	+	0.706408i	$0.750314\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−557871.	−0.360874
$300$	0	0
$301$	304093.	0.193460
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	424891.	0.257295	0.128648	−	0.991690i	$-0.458936\pi$
$307$	424891.	0.257295	0.128648	+	0.991690i	$0.458936\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.28677e6	−0.754396	−0.377198	−	0.926133i	$-0.623112\pi$
$311$	−1.28677e6	−0.754396	−0.377198	+	0.926133i	$0.623112\pi$
$312$	0	0
$313$	2.26096e6	1.30446	0.652231	−	0.758020i	$-0.273833\pi$
$313$	2.26096e6	1.30446	0.652231	+	0.758020i	$0.273833\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.71608e6	−0.959156	−0.479578	−	0.877499i	$-0.659210\pi$
$317$	−1.71608e6	−0.959156	−0.479578	+	0.877499i	$0.659210\pi$
$318$	0	0
$319$	−2.36880e6	−1.30332
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−632331.	−0.337239
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−82604.5	−0.0420740
$330$	0	0
$331$	3.39555e6	1.70349	0.851747	−	0.523954i	$-0.175544\pi$
$331$	3.39555e6	1.70349	0.851747	+	0.523954i	$0.175544\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	372979.	0.178900	0.0894499	−	0.995991i	$-0.471489\pi$
$337$	372979.	0.178900	0.0894499	+	0.995991i	$0.471489\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.31993e6	−0.614701
$342$	0	0
$343$	2.02868e6	0.931063
$344$	0	0
$345$	0	0
$346$	0	0
$347$	883053.	0.393698	0.196849	−	0.980434i	$-0.436929\pi$
$347$	883053.	0.393698	0.196849	+	0.980434i	$0.436929\pi$
$348$	0	0
$349$	−2.71799e6	−1.19449	−0.597247	−	0.802058i	$-0.703739\pi$
$349$	−2.71799e6	−1.19449	−0.597247	+	0.802058i	$0.703739\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.39262e6	1.02197	0.510984	−	0.859590i	$-0.329281\pi$
$353$	2.39262e6	1.02197	0.510984	+	0.859590i	$0.329281\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−37811.9	−0.0154843	−0.00774216	−	0.999970i	$-0.502464\pi$
$359$	−37811.9	−0.0154843	−0.00774216	+	0.999970i	$0.502464\pi$
$360$	0	0
$361$	−1.29453e6	−0.522810
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.96532e6	0.761672	0.380836	−	0.924643i	$-0.375636\pi$
$367$	1.96532e6	0.761672	0.380836	+	0.924643i	$0.375636\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.77600e6	0.669896
$372$	0	0
$373$	1.23448e6	0.459421	0.229711	−	0.973259i	$-0.426222\pi$
$373$	1.23448e6	0.459421	0.229711	+	0.973259i	$0.426222\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	557871.	0.202153
$378$	0	0
$379$	−2.36800e6	−0.846804	−0.423402	−	0.905942i	$-0.639164\pi$
$379$	−2.36800e6	−0.846804	−0.423402	+	0.905942i	$0.639164\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.22390e6	1.12301	0.561506	−	0.827472i	$-0.310222\pi$
$383$	3.22390e6	1.12301	0.561506	+	0.827472i	$0.310222\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.78414e6	−0.597799	−0.298899	−	0.954285i	$-0.596619\pi$
$389$	−1.78414e6	−0.597799	−0.298899	+	0.954285i	$0.596619\pi$
$390$	0	0
$391$	2.36880e6	0.783586
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.02517e6	0.644889	0.322445	−	0.946588i	$-0.395495\pi$
$397$	2.02517e6	0.644889	0.322445	+	0.946588i	$0.395495\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.52694e6	1.71642	0.858210	−	0.513299i	$-0.171577\pi$
$401$	5.52694e6	1.71642	0.858210	+	0.513299i	$0.171577\pi$
$402$	0	0
$403$	310853.	0.0953438
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.49615e6	1.04617
$408$	0	0
$409$	1.59215e6	0.470627	0.235313	−	0.971920i	$-0.424388\pi$
$409$	1.59215e6	0.470627	0.235313	+	0.971920i	$0.424388\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.06511e6	−0.595756
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.70093e6	0.751586	0.375793	−	0.926704i	$-0.377370\pi$
$419$	2.70093e6	0.751586	0.375793	+	0.926704i	$0.377370\pi$
$420$	0	0
$421$	−92902.0	−0.0255458	−0.0127729	−	0.999918i	$-0.504066\pi$
$421$	−92902.0	−0.0255458	−0.0127729	+	0.999918i	$0.504066\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−903049.	−0.239685
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.43278e6	−0.371524	−0.185762	−	0.982595i	$-0.559475\pi$
$431$	−1.43278e6	−0.371524	−0.185762	+	0.982595i	$0.559475\pi$
$432$	0	0
$433$	3.07045e6	0.787014	0.393507	−	0.919322i	$-0.371262\pi$
$433$	3.07045e6	0.787014	0.393507	+	0.919322i	$0.371262\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.42632e6	−1.10876
$438$	0	0
$439$	1.69979e6	0.420953	0.210476	−	0.977599i	$-0.432499\pi$
$439$	1.69979e6	0.420953	0.210476	+	0.977599i	$0.432499\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	68643.1	0.0166183	0.00830917	−	0.999965i	$-0.497355\pi$
$443$	68643.1	0.0166183	0.00830917	+	0.999965i	$0.497355\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.30470e6	−1.47587	−0.737936	−	0.674871i	$-0.764199\pi$
$449$	−6.30470e6	−1.47587	−0.737936	+	0.674871i	$0.764199\pi$
$450$	0	0
$451$	1.01520e6	0.235023
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−67190.0	−0.0150492	−0.00752461	−	0.999972i	$-0.502395\pi$
$457$	−67190.0	−0.0150492	−0.00752461	+	0.999972i	$0.502395\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8.25288e6	−1.80865	−0.904323	−	0.426849i	$-0.859623\pi$
$461$	−8.25288e6	−1.80865	−0.904323	+	0.426849i	$0.859623\pi$
$462$	0	0
$463$	−1.08602e6	−0.235443	−0.117721	−	0.993047i	$-0.537559\pi$
$463$	−1.08602e6	−0.235443	−0.117721	+	0.993047i	$0.537559\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.65965e6	−0.352148	−0.176074	−	0.984377i	$-0.556340\pi$
$467$	−1.65965e6	−0.352148	−0.176074	+	0.984377i	$0.556340\pi$
$468$	0	0
$469$	489829.	0.102828
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.49151e6	−0.512048
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.12842e6	1.41956	0.709781	−	0.704423i	$-0.248794\pi$
$479$	7.12842e6	1.41956	0.709781	+	0.704423i	$0.248794\pi$
$480$	0	0
$481$	−823370.	−0.162268
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7.39599e6	−1.41310	−0.706552	−	0.707662i	$-0.749750\pi$
$487$	−7.39599e6	−1.41310	−0.706552	+	0.707662i	$0.749750\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.03947e6	−0.756173	−0.378086	−	0.925770i	$-0.623418\pi$
$491$	−4.03947e6	−0.756173	−0.378086	+	0.925770i	$0.623418\pi$
$492$	0	0
$493$	−2.36880e6	−0.438946
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.84985e6	0.517525
$498$	0	0
$499$	427757.	0.0769034	0.0384517	−	0.999260i	$-0.487757\pi$
$499$	427757.	0.0769034	0.0384517	+	0.999260i	$0.487757\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.28727e6	−0.931776	−0.465888	−	0.884844i	$-0.654265\pi$
$503$	−5.28727e6	−0.931776	−0.465888	+	0.884844i	$0.654265\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.82713e6	0.825837	0.412918	−	0.910768i	$-0.364509\pi$
$509$	4.82713e6	0.825837	0.412918	+	0.910768i	$0.364509\pi$
$510$	0	0
$511$	−1.70471e6	−0.288801
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	676800.	0.111361
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.31394e6	1.50328	0.751639	−	0.659575i	$-0.229264\pi$
$521$	9.31394e6	1.50328	0.751639	+	0.659575i	$0.229264\pi$
$522$	0	0
$523$	−7.28609e6	−1.16477	−0.582385	−	0.812913i	$-0.697881\pi$
$523$	−7.28609e6	−1.16477	−0.582385	+	0.812913i	$0.697881\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.31993e6	−0.207025
$528$	0	0
$529$	1.01453e7	1.57625
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−239088.	−0.0364535
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.84454e6	−1.01478
$540$	0	0
$541$	−7.43565e6	−1.09226	−0.546130	−	0.837701i	$-0.683899\pi$
$541$	−7.43565e6	−1.09226	−0.546130	+	0.837701i	$0.683899\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2.89436e6	−0.413604	−0.206802	−	0.978383i	$-0.566306\pi$
$547$	−2.89436e6	−0.413604	−0.206802	+	0.978383i	$0.566306\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.42632e6	0.621103
$552$	0	0
$553$	−868756.	−0.120805
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.77132e6	−0.378485	−0.189243	−	0.981930i	$-0.560603\pi$
$557$	−2.77132e6	−0.378485	−0.189243	+	0.981930i	$0.560603\pi$
$558$	0	0
$559$	586771.	0.0794217
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.19652e6	0.690942	0.345471	−	0.938429i	$-0.387719\pi$
$563$	5.19652e6	0.690942	0.345471	+	0.938429i	$0.387719\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.93763e6	0.768834	0.384417	−	0.923160i	$-0.374402\pi$
$569$	5.93763e6	0.768834	0.384417	+	0.923160i	$0.374402\pi$
$570$	0	0
$571$	−3.73263e6	−0.479098	−0.239549	−	0.970884i	$-0.577000\pi$
$571$	−3.73263e6	−0.479098	−0.239549	+	0.970884i	$0.577000\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.31754e6	0.915010	0.457505	−	0.889207i	$-0.348743\pi$
$577$	7.31754e6	0.915010	0.457505	+	0.889207i	$0.348743\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.89990e6	−0.233502
$582$	0	0
$583$	−1.45512e7	−1.77308
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.36811e6	0.762808	0.381404	−	0.924409i	$-0.375441\pi$
$587$	6.36811e6	0.762808	0.381404	+	0.924409i	$0.375441\pi$
$588$	0	0
$589$	2.46640e6	0.292938
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.24739e7	1.45668	0.728340	−	0.685216i	$-0.240292\pi$
$593$	1.24739e7	1.45668	0.728340	+	0.685216i	$0.240292\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.71410e7	−1.95195	−0.975976	−	0.217876i	$-0.930087\pi$
$599$	−1.71410e7	−1.95195	−0.975976	+	0.217876i	$0.930087\pi$
$600$	0	0
$601$	−8.66405e6	−0.978441	−0.489221	−	0.872160i	$-0.662719\pi$
$601$	−8.66405e6	−0.978441	−0.489221	+	0.872160i	$0.662719\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.07340e7	−1.18247	−0.591236	−	0.806499i	$-0.701360\pi$
$607$	−1.07340e7	−1.18247	−0.591236	+	0.806499i	$0.701360\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−159392.	−0.0172728
$612$	0	0
$613$	−1.66404e7	−1.78860	−0.894298	−	0.447472i	$-0.852325\pi$
$613$	−1.66404e7	−1.78860	−0.894298	+	0.447472i	$0.852325\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.41697e6	0.678605	0.339303	−	0.940677i	$-0.389809\pi$
$617$	6.41697e6	0.678605	0.339303	+	0.940677i	$0.389809\pi$
$618$	0	0
$619$	−1.04602e6	−0.109727	−0.0548636	−	0.998494i	$-0.517472\pi$
$619$	−1.04602e6	−0.109727	−0.0548636	+	0.998494i	$0.517472\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.96501e6	0.409284
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.49615e6	0.352341
$630$	0	0
$631$	−4.21404e6	−0.421333	−0.210666	−	0.977558i	$-0.567563\pi$
$631$	−4.21404e6	−0.421333	−0.210666	+	0.977558i	$0.567563\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.61194e6	0.157399
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.90383e6	−0.855917	−0.427959	−	0.903798i	$-0.640767\pi$
$641$	−8.90383e6	−0.855917	−0.427959	+	0.903798i	$0.640767\pi$
$642$	0	0
$643$	−1.36992e7	−1.30667	−0.653336	−	0.757068i	$-0.726631\pi$
$643$	−1.36992e7	−1.30667	−0.653336	+	0.757068i	$0.726631\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.46431e7	−1.37522	−0.687610	−	0.726080i	$-0.741340\pi$
$647$	−1.46431e7	−1.37522	−0.687610	+	0.726080i	$0.741340\pi$
$648$	0	0
$649$	1.69200e7	1.57684
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.71130e6	−0.707693	−0.353846	−	0.935304i	$-0.615126\pi$
$653$	−7.71130e6	−0.707693	−0.353846	+	0.935304i	$0.615126\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.92431e6	0.262307	0.131154	−	0.991362i	$-0.458132\pi$
$659$	2.92431e6	0.262307	0.131154	+	0.991362i	$0.458132\pi$
$660$	0	0
$661$	−8.16020e6	−0.726436	−0.363218	−	0.931704i	$-0.618322\pi$
$661$	−8.16020e6	−0.726436	−0.363218	+	0.931704i	$0.618322\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.65816e7	−1.44315
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.39892e6	0.634398
$672$	0	0
$673$	−1.06344e7	−0.905060	−0.452530	−	0.891749i	$-0.649478\pi$
$673$	−1.06344e7	−0.905060	−0.452530	+	0.891749i	$0.649478\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.22865e7	1.03029	0.515143	−	0.857104i	$-0.327739\pi$
$677$	1.22865e7	1.03029	0.515143	+	0.857104i	$0.327739\pi$
$678$	0	0
$679$	−1.39522e6	−0.116136
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.77285e7	−1.45419	−0.727095	−	0.686536i	$-0.759130\pi$
$683$	−1.77285e7	−1.45419	−0.727095	+	0.686536i	$0.759130\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.42692e6	0.275015
$690$	0	0
$691$	−1.10626e7	−0.881374	−0.440687	−	0.897661i	$-0.645265\pi$
$691$	−1.10626e7	−0.881374	−0.440687	+	0.897661i	$0.645265\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.01520e6	0.0791534
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.22910e6	0.401912	0.200956	−	0.979600i	$-0.435595\pi$
$701$	5.22910e6	0.401912	0.200956	+	0.979600i	$0.435595\pi$
$702$	0	0
$703$	−6.53287e6	−0.498558
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.70602e6	−0.730286
$708$	0	0
$709$	1.46271e7	1.09280	0.546402	−	0.837523i	$-0.315997\pi$
$709$	1.46271e7	1.09280	0.546402	+	0.837523i	$0.315997\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.23948e6	−0.680650
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.38791e7	1.72264	0.861322	−	0.508059i	$-0.169637\pi$
$719$	2.38791e7	1.72264	0.861322	+	0.508059i	$0.169637\pi$
$720$	0	0
$721$	−1.08190e7	−0.775083
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.16310e7	−0.816171	−0.408085	−	0.912944i	$-0.633803\pi$
$727$	−1.16310e7	−0.816171	−0.408085	+	0.912944i	$0.633803\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.49151e6	−0.172453
$732$	0	0
$733$	−1.77996e7	−1.22363	−0.611817	−	0.790999i	$-0.709561\pi$
$733$	−1.77996e7	−1.22363	−0.611817	+	0.790999i	$0.709561\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.01330e6	−0.272165
$738$	0	0
$739$	1.03277e7	0.695650	0.347825	−	0.937559i	$-0.386920\pi$
$739$	1.03277e7	0.695650	0.347825	+	0.937559i	$0.386920\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.01741e6	0.134067	0.0670336	−	0.997751i	$-0.478647\pi$
$743$	2.01741e6	0.134067	0.0670336	+	0.997751i	$0.478647\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.26798e7	−0.825862
$750$	0	0
$751$	2.26675e6	0.146657	0.0733286	−	0.997308i	$-0.476638\pi$
$751$	2.26675e6	0.146657	0.0733286	+	0.997308i	$0.476638\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.55369e7	0.985427	0.492714	−	0.870191i	$-0.336005\pi$
$757$	1.55369e7	0.985427	0.492714	+	0.870191i	$0.336005\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.55515e7	−1.59939	−0.799697	−	0.600404i	$-0.795006\pi$
$761$	−2.55515e7	−1.59939	−0.799697	+	0.600404i	$0.795006\pi$
$762$	0	0
$763$	−222727.	−0.0138504
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.98479e6	−0.244578
$768$	0	0
$769$	2.63952e7	1.60957	0.804785	−	0.593566i	$-0.202281\pi$
$769$	2.63952e7	1.60957	0.804785	+	0.593566i	$0.202281\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.82515e7	1.09863	0.549313	−	0.835616i	$-0.314889\pi$
$773$	1.82515e7	1.09863	0.549313	+	0.835616i	$0.314889\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.89699e6	−0.112001
$780$	0	0
$781$	−2.33496e7	−1.36978
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−3.35204e7	−1.92918	−0.964588	−	0.263761i	$-0.915037\pi$
$787$	−3.35204e7	−1.92918	−0.964588	+	0.263761i	$0.915037\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.60836e6	−0.375537
$792$	0	0
$793$	−1.74250e6	−0.0983990
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.09881e6	0.228566	0.114283	−	0.993448i	$-0.463543\pi$
$797$	4.09881e6	0.228566	0.114283	+	0.993448i	$0.463543\pi$
$798$	0	0
$799$	676800.	0.0375054
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.39671e7	0.764396
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.35198e7	−0.726271	−0.363135	−	0.931736i	$-0.618294\pi$
$809$	−1.35198e7	−0.726271	−0.363135	+	0.931736i	$0.618294\pi$
$810$	0	0
$811$	1.99657e7	1.06594	0.532970	−	0.846134i	$-0.321076\pi$
$811$	1.99657e7	1.06594	0.532970	+	0.846134i	$0.321076\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.65562e6	0.244018
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.63018e7	1.87962	0.939809	−	0.341700i	$-0.111003\pi$
$821$	3.63018e7	1.87962	0.939809	+	0.341700i	$0.111003\pi$
$822$	0	0
$823$	−3.57994e7	−1.84237	−0.921183	−	0.389129i	$-0.872776\pi$
$823$	−3.57994e7	−1.84237	−0.921183	+	0.389129i	$0.872776\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.86969e7	1.45905	0.729527	−	0.683952i	$-0.239740\pi$
$827$	2.86969e7	1.45905	0.729527	+	0.683952i	$0.239740\pi$
$828$	0	0
$829$	5.02531e6	0.253967	0.126983	−	0.991905i	$-0.459471\pi$
$829$	5.02531e6	0.253967	0.126983	+	0.991905i	$0.459471\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.84454e6	−0.341768
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.05528e7	−1.00801	−0.504007	−	0.863700i	$-0.668141\pi$
$839$	−2.05528e7	−1.00801	−0.504007	+	0.863700i	$0.668141\pi$
$840$	0	0
$841$	−3.92955e6	−0.191581
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.25918e7	−0.603085
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.44730e7	1.15841
$852$	0	0
$853$	2.24442e7	1.05616	0.528081	−	0.849194i	$-0.322912\pi$
$853$	2.24442e7	1.05616	0.528081	+	0.849194i	$0.322912\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.13909e7	−1.45999	−0.729997	−	0.683450i	$-0.760479\pi$
$857$	−3.13909e7	−1.45999	−0.729997	+	0.683450i	$0.760479\pi$
$858$	0	0
$859$	−2.28763e7	−1.05780	−0.528898	−	0.848685i	$-0.677395\pi$
$859$	−2.28763e7	−1.05780	−0.528898	+	0.848685i	$0.677395\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.61827e6	0.256788	0.128394	−	0.991723i	$-0.459018\pi$
$863$	5.61827e6	0.256788	0.128394	+	0.991723i	$0.459018\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.11795e6	0.319746
$870$	0	0
$871$	945163.	0.0422145
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.71980e7	1.19409	0.597046	−	0.802207i	$-0.296341\pi$
$877$	2.71980e7	1.19409	0.597046	+	0.802207i	$0.296341\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.26867e6	−0.185290	−0.0926452	−	0.995699i	$-0.529532\pi$
$881$	−4.26867e6	−0.185290	−0.0926452	+	0.995699i	$0.529532\pi$
$882$	0	0
$883$	−1.81011e7	−0.781272	−0.390636	−	0.920545i	$-0.627745\pi$
$883$	−1.81011e7	−0.781272	−0.390636	+	0.920545i	$0.627745\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.16282e7	1.34979	0.674894	−	0.737915i	$-0.264189\pi$
$887$	3.16282e7	1.34979	0.674894	+	0.737915i	$0.264189\pi$
$888$	0	0
$889$	257020.	0.0109072
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.26466e6	−0.0530696
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.23948e6	0.381284
$900$	0	0
$901$	−1.45512e7	−0.597155
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.91596e6	−0.117696	−0.0588482	−	0.998267i	$-0.518743\pi$
$907$	−2.91596e6	−0.117696	−0.0588482	+	0.998267i	$0.518743\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.31514e7	−0.924231	−0.462115	−	0.886820i	$-0.652909\pi$
$911$	−2.31514e7	−0.924231	−0.462115	+	0.886820i	$0.652909\pi$
$912$	0	0
$913$	1.55664e7	0.618032
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.48445e6	0.254654
$918$	0	0
$919$	−5.63575e6	−0.220122	−0.110061	−	0.993925i	$-0.535105\pi$
$919$	−5.63575e6	−0.220122	−0.110061	+	0.993925i	$0.535105\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.49901e6	0.212462
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.95906e7	0.744748	0.372374	−	0.928083i	$-0.378544\pi$
$929$	1.95906e7	0.744748	0.372374	+	0.928083i	$0.378544\pi$
$930$	0	0
$931$	1.27896e7	0.483598
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.41955e7	0.900295	0.450148	−	0.892954i	$-0.351371\pi$
$937$	2.41955e7	0.900295	0.450148	+	0.892954i	$0.351371\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.88245e6	0.0693026	0.0346513	−	0.999399i	$-0.488968\pi$
$941$	1.88245e6	0.0693026	0.0346513	+	0.999399i	$0.488968\pi$
$942$	0	0
$943$	7.10640e6	0.260238
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.59300e7	−1.30192	−0.650958	−	0.759114i	$-0.725632\pi$
$947$	−3.59300e7	−1.30192	−0.650958	+	0.759114i	$0.725632\pi$
$948$	0	0
$949$	−3.28937e6	−0.118562
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.11351e7	−0.753828	−0.376914	−	0.926248i	$-0.623015\pi$
$953$	−2.11351e7	−0.753828	−0.376914	+	0.926248i	$0.623015\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.90816e7	0.669991
$960$	0	0
$961$	−2.34808e7	−0.820171
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−2.59260e7	−0.891599	−0.445800	−	0.895133i	$-0.647081\pi$
$967$	−2.59260e7	−0.891599	−0.445800	+	0.895133i	$0.647081\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.34951e7	−1.82082	−0.910408	−	0.413712i	$-0.864232\pi$
$971$	−5.34951e7	−1.82082	−0.910408	+	0.413712i	$0.864232\pi$
$972$	0	0
$973$	2.21191e7	0.749005
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.12243e7	−1.38171	−0.690855	−	0.722994i	$-0.742766\pi$
$977$	−4.12243e7	−1.38171	−0.690855	+	0.722994i	$0.742766\pi$
$978$	0	0
$979$	−3.24864e7	−1.08329
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.53235e7	−0.835872	−0.417936	−	0.908476i	$-0.637246\pi$
$983$	−2.53235e7	−0.835872	−0.417936	+	0.908476i	$0.637246\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.74406e7	−0.566984
$990$	0	0
$991$	3.96797e7	1.28347	0.641733	−	0.766928i	$-0.278216\pi$
$991$	3.96797e7	1.28347	0.641733	+	0.766928i	$0.278216\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.03652e7	1.60470	0.802348	−	0.596857i	$-0.203584\pi$
$997$	5.03652e7	1.60470	0.802348	+	0.596857i	$0.203584\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.a.l.1.2	yes	2
3.2	odd	2	inner	900.6.a.l.1.1	✓	2
5.2	odd	4		900.6.d.l.649.2		4
5.3	odd	4		900.6.d.l.649.4		4
5.4	even	2		900.6.a.v.1.2	yes	2
15.2	even	4		900.6.d.l.649.1		4
15.8	even	4		900.6.d.l.649.3		4
15.14	odd	2		900.6.a.v.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.6.a.l.1.1	✓	2	3.2	odd	2	inner
900.6.a.l.1.2	yes	2	1.1	even	1	trivial
900.6.a.v.1.1	yes	2	15.14	odd	2
900.6.a.v.1.2	yes	2	5.4	even	2
900.6.d.l.649.1		4	15.2	even	4
900.6.d.l.649.2		4	5.2	odd	4
900.6.d.l.649.3		4	15.8	even	4
900.6.d.l.649.4		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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)

\(f(q)\)	\(=\)	\(q-71.0000 q^{7} +O(q^{10})\)	\(q-71.0000 q^{7} +581.722 q^{11} -137.000 q^{13} +581.722 q^{17} -1087.00 q^{19} +4072.05 q^{23} -4072.05 q^{29} -2269.00 q^{31} +6010.00 q^{37} +1745.16 q^{41} -4283.00 q^{43} +1163.44 q^{47} -11766.0 q^{49} -25014.0 q^{53} +29086.1 q^{59} +12719.0 q^{61} -6899.00 q^{67} -40138.8 q^{71} +24010.0 q^{73} -41302.2 q^{77} +12236.0 q^{79} +26759.2 q^{83} -55845.3 q^{89} +9727.00 q^{91} +19651.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 142 q^{7}+O(q^{10}) \) 2 * q - 142 * q^7	\( 2 q - 142 q^{7} - 274 q^{13} - 2174 q^{19} - 4538 q^{31} + 12020 q^{37} - 8566 q^{43} - 23532 q^{49} + 25438 q^{61} - 13798 q^{67} + 48020 q^{73} + 24472 q^{79} + 19454 q^{91} + 39302 q^{97}+O(q^{100}) \) 2 * q - 142 * q^7 - 274 * q^13 - 2174 * q^19 - 4538 * q^31 + 12020 * q^37 - 8566 * q^43 - 23532 * q^49 + 25438 * q^61 - 13798 * q^67 + 48020 * q^73 + 24472 * q^79 + 19454 * q^91 + 39302 * q^97

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