LMFDB - Embedded newform 900.6.a.x.1.3

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:	$3$
Coefficient field:	3.3.535753.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 186x - 432 $ x^3 - x^2 - 186*x - 432
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 60)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.43168$ 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)

$f(q)$	$=$	$q+222.092 q^{7} +O(q^{10})$	$q+222.092 q^{7} -682.910 q^{11} -412.376 q^{13} -866.026 q^{17} +668.199 q^{19} +3324.21 q^{23} -1250.97 q^{29} -9701.37 q^{31} +8805.05 q^{37} -6736.49 q^{41} +17084.3 q^{43} +12968.3 q^{47} +32517.9 q^{49} +4152.64 q^{53} +28414.2 q^{59} +10332.4 q^{61} +65968.5 q^{67} +20040.8 q^{71} -47182.3 q^{73} -151669. q^{77} -21037.1 q^{79} -90354.9 q^{83} +41577.9 q^{89} -91585.4 q^{91} +56347.2 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 88 q^{7}+O(q^{10}) $ 3 * q + 88 * q^7	$ 3 q + 88 q^{7} - 148 q^{11} - 868 q^{17} + 3000 q^{19} + 1052 q^{23} - 7962 q^{29} + 132 q^{31} + 11944 q^{37} - 8170 q^{41} + 26188 q^{43} - 37892 q^{47} + 13827 q^{49} - 11076 q^{53} + 46228 q^{59} + 3126 q^{61} + 126332 q^{67} + 80400 q^{71} + 42904 q^{73} - 234608 q^{77} - 64476 q^{79} - 126268 q^{83} + 38030 q^{89} - 49200 q^{91} + 331016 q^{97}+O(q^{100}) $ 3 * q + 88 * q^7 - 148 * q^11 - 868 * q^17 + 3000 * q^19 + 1052 * q^23 - 7962 * q^29 + 132 * q^31 + 11944 * q^37 - 8170 * q^41 + 26188 * q^43 - 37892 * q^47 + 13827 * q^49 - 11076 * q^53 + 46228 * q^59 + 3126 * q^61 + 126332 * q^67 + 80400 * q^71 + 42904 * q^73 - 234608 * q^77 - 64476 * q^79 - 126268 * q^83 + 38030 * q^89 - 49200 * q^91 + 331016 * 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
$7$	222.092	1.71312	0.856561	−	0.516047i	$-0.172597\pi$
$7$	222.092	1.71312	0.856561	+	0.516047i	$0.172597\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−682.910	−1.70169	−0.850847	−	0.525413i	$-0.823911\pi$
$11$	−682.910	−1.70169	−0.850847	+	0.525413i	$0.823911\pi$
$12$	0	0
$13$	−412.376	−0.676760	−0.338380	−	0.941010i	$-0.609879\pi$
$13$	−412.376	−0.676760	−0.338380	+	0.941010i	$0.609879\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−866.026	−0.726790	−0.363395	−	0.931635i	$-0.618382\pi$
$17$	−866.026	−0.726790	−0.363395	+	0.931635i	$0.618382\pi$
$18$	0	0
$19$	668.199	0.424641	0.212320	−	0.977200i	$-0.431898\pi$
$19$	668.199	0.424641	0.212320	+	0.977200i	$0.431898\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3324.21	1.31029	0.655147	−	0.755502i	$-0.272607\pi$
$23$	3324.21	1.31029	0.655147	+	0.755502i	$0.272607\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1250.97	−0.276218	−0.138109	−	0.990417i	$-0.544102\pi$
$29$	−1250.97	−0.276218	−0.138109	+	0.990417i	$0.544102\pi$
$30$	0	0
$31$	−9701.37	−1.81313	−0.906565	−	0.422066i	$-0.861305\pi$
$31$	−9701.37	−1.81313	−0.906565	+	0.422066i	$0.861305\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8805.05	1.05737	0.528686	−	0.848818i	$-0.322685\pi$
$37$	8805.05	1.05737	0.528686	+	0.848818i	$0.322685\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6736.49	−0.625856	−0.312928	−	0.949777i	$-0.601310\pi$
$41$	−6736.49	−0.625856	−0.312928	+	0.949777i	$0.601310\pi$
$42$	0	0
$43$	17084.3	1.40905	0.704524	−	0.709680i	$-0.251161\pi$
$43$	17084.3	1.40905	0.704524	+	0.709680i	$0.251161\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12968.3	0.856325	0.428163	−	0.903702i	$-0.359161\pi$
$47$	12968.3	0.856325	0.428163	+	0.903702i	$0.359161\pi$
$48$	0	0
$49$	32517.9	1.93478
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4152.64	0.203065	0.101532	−	0.994832i	$-0.467625\pi$
$53$	4152.64	0.203065	0.101532	+	0.994832i	$0.467625\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	28414.2	1.06269	0.531343	−	0.847157i	$-0.321687\pi$
$59$	28414.2	1.06269	0.531343	+	0.847157i	$0.321687\pi$
$60$	0	0
$61$	10332.4	0.355531	0.177766	−	0.984073i	$-0.443113\pi$
$61$	10332.4	0.355531	0.177766	+	0.984073i	$0.443113\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	65968.5	1.79535	0.897677	−	0.440655i	$-0.145254\pi$
$67$	65968.5	1.79535	0.897677	+	0.440655i	$0.145254\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	20040.8	0.471813	0.235907	−	0.971776i	$-0.424194\pi$
$71$	20040.8	0.471813	0.235907	+	0.971776i	$0.424194\pi$
$72$	0	0
$73$	−47182.3	−1.03627	−0.518134	−	0.855299i	$-0.673373\pi$
$73$	−47182.3	−1.03627	−0.518134	+	0.855299i	$0.673373\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−151669.	−2.91521
$78$	0	0
$79$	−21037.1	−0.379243	−0.189621	−	0.981857i	$-0.560726\pi$
$79$	−21037.1	−0.379243	−0.189621	+	0.981857i	$0.560726\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−90354.9	−1.43965	−0.719824	−	0.694156i	$-0.755778\pi$
$83$	−90354.9	−1.43965	−0.719824	+	0.694156i	$0.755778\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	41577.9	0.556401	0.278201	−	0.960523i	$-0.410262\pi$
$89$	41577.9	0.556401	0.278201	+	0.960523i	$0.410262\pi$
$90$	0	0
$91$	−91585.4	−1.15937
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	56347.2	0.608055	0.304028	−	0.952663i	$-0.401668\pi$
$97$	56347.2	0.608055	0.304028	+	0.952663i	$0.401668\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14497.5	0.141413	0.0707065	−	0.997497i	$-0.477475\pi$
$101$	14497.5	0.141413	0.0707065	+	0.997497i	$0.477475\pi$
$102$	0	0
$103$	85813.1	0.797004	0.398502	−	0.917167i	$-0.369530\pi$
$103$	85813.1	0.797004	0.398502	+	0.917167i	$0.369530\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−37465.9	−0.316357	−0.158178	−	0.987411i	$-0.550562\pi$
$107$	−37465.9	−0.316357	−0.158178	+	0.987411i	$0.550562\pi$
$108$	0	0
$109$	−42617.0	−0.343571	−0.171786	−	0.985134i	$-0.554954\pi$
$109$	−42617.0	−0.343571	−0.171786	+	0.985134i	$0.554954\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−181016.	−1.33359	−0.666793	−	0.745243i	$-0.732333\pi$
$113$	−181016.	−1.33359	−0.666793	+	0.745243i	$0.732333\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−192338.	−1.24508
$120$	0	0
$121$	305315.	1.89577
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	51765.9	0.284797	0.142398	−	0.989809i	$-0.454519\pi$
$127$	51765.9	0.284797	0.142398	+	0.989809i	$0.454519\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	170037.	0.865694	0.432847	−	0.901467i	$-0.357509\pi$
$131$	170037.	0.865694	0.432847	+	0.901467i	$0.357509\pi$
$132$	0	0
$133$	148402.	0.727461
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−122685.	−0.558458	−0.279229	−	0.960225i	$-0.590079\pi$
$137$	−122685.	−0.558458	−0.279229	+	0.960225i	$0.590079\pi$
$138$	0	0
$139$	−22126.7	−0.0971359	−0.0485679	−	0.998820i	$-0.515466\pi$
$139$	−22126.7	−0.0971359	−0.0485679	+	0.998820i	$0.515466\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	281616.	1.15164
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	61016.4	0.225155	0.112577	−	0.993643i	$-0.464089\pi$
$149$	61016.4	0.225155	0.112577	+	0.993643i	$0.464089\pi$
$150$	0	0
$151$	247649.	0.883882	0.441941	−	0.897044i	$-0.354290\pi$
$151$	247649.	0.883882	0.441941	+	0.897044i	$0.354290\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−217742.	−0.705006	−0.352503	−	0.935811i	$-0.614669\pi$
$157$	−217742.	−0.705006	−0.352503	+	0.935811i	$0.614669\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	738280.	2.24469
$162$	0	0
$163$	−101662.	−0.299702	−0.149851	−	0.988709i	$-0.547879\pi$
$163$	−101662.	−0.299702	−0.149851	+	0.988709i	$0.547879\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	309268.	0.858112	0.429056	−	0.903278i	$-0.358846\pi$
$167$	309268.	0.858112	0.429056	+	0.903278i	$0.358846\pi$
$168$	0	0
$169$	−201239.	−0.541996
$170$	0	0
$171$	0	0
$172$	0	0
$173$	679235.	1.72546	0.862730	−	0.505665i	$-0.168753\pi$
$173$	679235.	1.72546	0.862730	+	0.505665i	$0.168753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	692042.	1.61436	0.807180	−	0.590306i	$-0.200993\pi$
$179$	692042.	1.61436	0.807180	+	0.590306i	$0.200993\pi$
$180$	0	0
$181$	613696.	1.39238	0.696189	−	0.717859i	$-0.254878\pi$
$181$	613696.	1.39238	0.696189	+	0.717859i	$0.254878\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	591418.	1.23677
$188$	0	0
$189$	0	0
$190$	0	0
$191$	25798.2	0.0511690	0.0255845	−	0.999673i	$-0.491855\pi$
$191$	25798.2	0.0511690	0.0255845	+	0.999673i	$0.491855\pi$
$192$	0	0
$193$	−8618.50	−0.0166548	−0.00832738	−	0.999965i	$-0.502651\pi$
$193$	−8618.50	−0.0166548	−0.00832738	+	0.999965i	$0.502651\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	307099.	0.563783	0.281892	−	0.959446i	$-0.409038\pi$
$197$	307099.	0.563783	0.281892	+	0.959446i	$0.409038\pi$
$198$	0	0
$199$	657859.	1.17761	0.588803	−	0.808277i	$-0.299599\pi$
$199$	657859.	1.17761	0.588803	+	0.808277i	$0.299599\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−277831.	−0.473196
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−456320.	−0.722609
$210$	0	0
$211$	−858929.	−1.32816	−0.664081	−	0.747660i	$-0.731177\pi$
$211$	−858929.	−1.32816	−0.664081	+	0.747660i	$0.731177\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.15460e6	−3.10611
$218$	0	0
$219$	0	0
$220$	0	0
$221$	357128.	0.491862
$222$	0	0
$223$	76808.9	0.103431	0.0517153	−	0.998662i	$-0.483531\pi$
$223$	76808.9	0.103431	0.0517153	+	0.998662i	$0.483531\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.03732e6	1.33612	0.668062	−	0.744106i	$-0.267124\pi$
$227$	1.03732e6	1.33612	0.668062	+	0.744106i	$0.267124\pi$
$228$	0	0
$229$	−80347.6	−0.101247	−0.0506237	−	0.998718i	$-0.516121\pi$
$229$	−80347.6	−0.101247	−0.0506237	+	0.998718i	$0.516121\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−500585.	−0.604071	−0.302035	−	0.953297i	$-0.597666\pi$
$233$	−500585.	−0.604071	−0.302035	+	0.953297i	$0.597666\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.68223e6	1.90498	0.952491	−	0.304566i	$-0.0985114\pi$
$239$	1.68223e6	1.90498	0.952491	+	0.304566i	$0.0985114\pi$
$240$	0	0
$241$	−782691.	−0.868056	−0.434028	−	0.900899i	$-0.642908\pi$
$241$	−782691.	−0.868056	−0.434028	+	0.900899i	$0.642908\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−275549.	−0.287380
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.55394e6	1.55686	0.778432	−	0.627729i	$-0.216016\pi$
$251$	1.55394e6	1.55686	0.778432	+	0.627729i	$0.216016\pi$
$252$	0	0
$253$	−2.27013e6	−2.22972
$254$	0	0
$255$	0	0
$256$	0	0
$257$	648898.	0.612835	0.306418	−	0.951897i	$-0.400870\pi$
$257$	648898.	0.612835	0.306418	+	0.951897i	$0.400870\pi$
$258$	0	0
$259$	1.95553e6	1.81141
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.55229e6	−1.38383	−0.691917	−	0.721977i	$-0.743233\pi$
$263$	−1.55229e6	−1.38383	−0.691917	+	0.721977i	$0.743233\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	238802.	0.201213	0.100607	−	0.994926i	$-0.467922\pi$
$269$	238802.	0.201213	0.100607	+	0.994926i	$0.467922\pi$
$270$	0	0
$271$	890126.	0.736255	0.368128	−	0.929775i	$-0.379999\pi$
$271$	890126.	0.736255	0.368128	+	0.929775i	$0.379999\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.25908e6	1.76901	0.884507	−	0.466527i	$-0.154495\pi$
$277$	2.25908e6	1.76901	0.884507	+	0.466527i	$0.154495\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	948177.	0.716347	0.358174	−	0.933655i	$-0.383400\pi$
$281$	948177.	0.716347	0.358174	+	0.933655i	$0.383400\pi$
$282$	0	0
$283$	191261.	0.141958	0.0709790	−	0.997478i	$-0.477388\pi$
$283$	191261.	0.141958	0.0709790	+	0.997478i	$0.477388\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.49612e6	−1.07217
$288$	0	0
$289$	−669856.	−0.471777
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.60076e6	1.76983	0.884915	−	0.465752i	$-0.154216\pi$
$293$	2.60076e6	1.76983	0.884915	+	0.465752i	$0.154216\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.37082e6	−0.886754
$300$	0	0
$301$	3.79428e6	2.41387
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.82686e6	−1.10627	−0.553134	−	0.833092i	$-0.686568\pi$
$307$	−1.82686e6	−1.10627	−0.553134	+	0.833092i	$0.686568\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	227191.	0.133196	0.0665978	−	0.997780i	$-0.478786\pi$
$311$	227191.	0.133196	0.0665978	+	0.997780i	$0.478786\pi$
$312$	0	0
$313$	956139.	0.551645	0.275823	−	0.961209i	$-0.411050\pi$
$313$	956139.	0.551645	0.275823	+	0.961209i	$0.411050\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−301774.	−0.168669	−0.0843343	−	0.996438i	$-0.526876\pi$
$317$	−301774.	−0.168669	−0.0843343	+	0.996438i	$0.526876\pi$
$318$	0	0
$319$	854301.	0.470039
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−578678.	−0.308625
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.88016e6	1.46699
$330$	0	0
$331$	2.92496e6	1.46740	0.733702	−	0.679472i	$-0.237791\pi$
$331$	2.92496e6	1.46740	0.733702	+	0.679472i	$0.237791\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−600711.	−0.288132	−0.144066	−	0.989568i	$-0.546018\pi$
$337$	−600711.	−0.288132	−0.144066	+	0.989568i	$0.546018\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.62516e6	3.08539
$342$	0	0
$343$	3.48927e6	1.60140
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−546550.	−0.243672	−0.121836	−	0.992550i	$-0.538878\pi$
$347$	−546550.	−0.243672	−0.121836	+	0.992550i	$0.538878\pi$
$348$	0	0
$349$	−2.06579e6	−0.907868	−0.453934	−	0.891035i	$-0.649980\pi$
$349$	−2.06579e6	−0.907868	−0.453934	+	0.891035i	$0.649980\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.11434e6	0.475971	0.237985	−	0.971269i	$-0.423513\pi$
$353$	1.11434e6	0.475971	0.237985	+	0.971269i	$0.423513\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.21134e6	−0.905564	−0.452782	−	0.891621i	$-0.649568\pi$
$359$	−2.21134e6	−0.905564	−0.452782	+	0.891621i	$0.649568\pi$
$360$	0	0
$361$	−2.02961e6	−0.819680
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.36836e6	1.69298	0.846492	−	0.532401i	$-0.178710\pi$
$367$	4.36836e6	1.69298	0.846492	+	0.532401i	$0.178710\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	922269.	0.347875
$372$	0	0
$373$	−32942.5	−0.0122598	−0.00612992	−	0.999981i	$-0.501951\pi$
$373$	−32942.5	−0.0122598	−0.00612992	+	0.999981i	$0.501951\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	515871.	0.186934
$378$	0	0
$379$	−4.13275e6	−1.47789	−0.738943	−	0.673768i	$-0.764675\pi$
$379$	−4.13275e6	−1.47789	−0.738943	+	0.673768i	$0.764675\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.14084e6	−1.09408	−0.547040	−	0.837107i	$-0.684245\pi$
$383$	−3.14084e6	−1.09408	−0.547040	+	0.837107i	$0.684245\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.63511e6	0.882928	0.441464	−	0.897279i	$-0.354459\pi$
$389$	2.63511e6	0.882928	0.441464	+	0.897279i	$0.354459\pi$
$390$	0	0
$391$	−2.87885e6	−0.952307
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	681886.	0.217138	0.108569	−	0.994089i	$-0.465373\pi$
$397$	681886.	0.217138	0.108569	+	0.994089i	$0.465373\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.16584e6	−0.983169	−0.491585	−	0.870830i	$-0.663582\pi$
$401$	−3.16584e6	−0.983169	−0.491585	+	0.870830i	$0.663582\pi$
$402$	0	0
$403$	4.00061e6	1.22705
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.01306e6	−1.79932
$408$	0	0
$409$	2.06335e6	0.609908	0.304954	−	0.952367i	$-0.401359\pi$
$409$	2.06335e6	0.609908	0.304954	+	0.952367i	$0.401359\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.31057e6	1.82051
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−83142.7	−0.0231360	−0.0115680	−	0.999933i	$-0.503682\pi$
$419$	−83142.7	−0.0231360	−0.0115680	+	0.999933i	$0.503682\pi$
$420$	0	0
$421$	3.63077e6	0.998375	0.499187	−	0.866494i	$-0.333632\pi$
$421$	3.63077e6	0.998375	0.499187	+	0.866494i	$0.333632\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.29475e6	0.609068
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.97283e6	0.511560	0.255780	−	0.966735i	$-0.417668\pi$
$431$	1.97283e6	0.511560	0.255780	+	0.966735i	$0.417668\pi$
$432$	0	0
$433$	−5.03684e6	−1.29104	−0.645518	−	0.763745i	$-0.723358\pi$
$433$	−5.03684e6	−1.29104	−0.645518	+	0.763745i	$0.723358\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.22123e6	0.556404
$438$	0	0
$439$	−3.03585e6	−0.751829	−0.375915	−	0.926654i	$-0.622671\pi$
$439$	−3.03585e6	−0.751829	−0.375915	+	0.926654i	$0.622671\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.77723e6	−0.672361	−0.336180	−	0.941798i	$-0.609135\pi$
$443$	−2.77723e6	−0.672361	−0.336180	+	0.941798i	$0.609135\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.51254e6	−1.52453	−0.762263	−	0.647268i	$-0.775912\pi$
$449$	−6.51254e6	−1.52453	−0.762263	+	0.647268i	$0.775912\pi$
$450$	0	0
$451$	4.60042e6	1.06502
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.00632e6	1.34530	0.672648	−	0.739962i	$-0.265157\pi$
$457$	6.00632e6	1.34530	0.672648	+	0.739962i	$0.265157\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.92568e6	−0.422020	−0.211010	−	0.977484i	$-0.567675\pi$
$461$	−1.92568e6	−0.422020	−0.211010	+	0.977484i	$0.567675\pi$
$462$	0	0
$463$	−3.87061e6	−0.839125	−0.419562	−	0.907727i	$-0.637816\pi$
$463$	−3.87061e6	−0.839125	−0.419562	+	0.907727i	$0.637816\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.77952e6	−1.86285	−0.931426	−	0.363929i	$-0.881435\pi$
$467$	−8.77952e6	−1.86285	−0.931426	+	0.363929i	$0.881435\pi$
$468$	0	0
$469$	1.46511e7	3.07566
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.16670e7	−2.39777
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.98716e6	1.59057	0.795286	−	0.606234i	$-0.207321\pi$
$479$	7.98716e6	1.59057	0.795286	+	0.606234i	$0.207321\pi$
$480$	0	0
$481$	−3.63099e6	−0.715587
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.73043e6	−1.09488	−0.547438	−	0.836846i	$-0.684397\pi$
$487$	−5.73043e6	−1.09488	−0.547438	+	0.836846i	$0.684397\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.73224e6	−0.698660	−0.349330	−	0.937000i	$-0.613591\pi$
$491$	−3.73224e6	−0.698660	−0.349330	+	0.937000i	$0.613591\pi$
$492$	0	0
$493$	1.08337e6	0.200753
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.45091e6	0.808273
$498$	0	0
$499$	−2.15482e6	−0.387400	−0.193700	−	0.981061i	$-0.562049\pi$
$499$	−2.15482e6	−0.387400	−0.193700	+	0.981061i	$0.562049\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.07480e6	−0.365643	−0.182821	−	0.983146i	$-0.558523\pi$
$503$	−2.07480e6	−0.365643	−0.182821	+	0.983146i	$0.558523\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.66770e6	1.48289	0.741446	−	0.671013i	$-0.234140\pi$
$509$	8.66770e6	1.48289	0.741446	+	0.671013i	$0.234140\pi$
$510$	0	0
$511$	−1.04788e7	−1.77525
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−8.85619e6	−1.45720
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−178943.	−0.0288815	−0.0144407	−	0.999896i	$-0.504597\pi$
$521$	−178943.	−0.0288815	−0.0144407	+	0.999896i	$0.504597\pi$
$522$	0	0
$523$	965997.	0.154426	0.0772132	−	0.997015i	$-0.475398\pi$
$523$	965997.	0.154426	0.0772132	+	0.997015i	$0.475398\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.40164e6	1.31776
$528$	0	0
$529$	4.61401e6	0.716868
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.77797e6	0.423554
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.22068e7	−3.29241
$540$	0	0
$541$	−7.53777e6	−1.10726	−0.553630	−	0.832763i	$-0.686758\pi$
$541$	−7.53777e6	−1.10726	−0.553630	+	0.832763i	$0.686758\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−583191.	−0.0833379	−0.0416689	−	0.999131i	$-0.513267\pi$
$547$	−583191.	−0.0833379	−0.0416689	+	0.999131i	$0.513267\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−835898.	−0.117294
$552$	0	0
$553$	−4.67216e6	−0.649689
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.12743e6	0.973408	0.486704	−	0.873567i	$-0.338199\pi$
$557$	7.12743e6	0.973408	0.486704	+	0.873567i	$0.338199\pi$
$558$	0	0
$559$	−7.04515e6	−0.953587
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.27840e6	0.834791	0.417396	−	0.908725i	$-0.362943\pi$
$563$	6.27840e6	0.834791	0.417396	+	0.908725i	$0.362943\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.70911e6	1.25718	0.628592	−	0.777735i	$-0.283632\pi$
$569$	9.70911e6	1.25718	0.628592	+	0.777735i	$0.283632\pi$
$570$	0	0
$571$	8.43263e6	1.08236	0.541181	−	0.840906i	$-0.317977\pi$
$571$	8.43263e6	1.08236	0.541181	+	0.840906i	$0.317977\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−300096.	−0.0375251	−0.0187625	−	0.999824i	$-0.505973\pi$
$577$	−300096.	−0.0375251	−0.0187625	+	0.999824i	$0.505973\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.00671e7	−2.46629
$582$	0	0
$583$	−2.83588e6	−0.345555
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.60235e6	1.15022	0.575112	−	0.818075i	$-0.304959\pi$
$587$	9.60235e6	1.15022	0.575112	+	0.818075i	$0.304959\pi$
$588$	0	0
$589$	−6.48245e6	−0.769929
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.15863e7	1.35303	0.676517	−	0.736427i	$-0.263489\pi$
$593$	1.15863e7	1.35303	0.676517	+	0.736427i	$0.263489\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	588504.	0.0670166	0.0335083	−	0.999438i	$-0.489332\pi$
$599$	588504.	0.0670166	0.0335083	+	0.999438i	$0.489332\pi$
$600$	0	0
$601$	−8.40664e6	−0.949371	−0.474686	−	0.880155i	$-0.657438\pi$
$601$	−8.40664e6	−0.949371	−0.474686	+	0.880155i	$0.657438\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.16211e6	0.238180	0.119090	−	0.992883i	$-0.462002\pi$
$607$	2.16211e6	0.238180	0.119090	+	0.992883i	$0.462002\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.34782e6	−0.579527
$612$	0	0
$613$	4.53869e6	0.487842	0.243921	−	0.969795i	$-0.421566\pi$
$613$	4.53869e6	0.487842	0.243921	+	0.969795i	$0.421566\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.44047e7	−1.52332	−0.761658	−	0.647979i	$-0.775615\pi$
$617$	−1.44047e7	−1.52332	−0.761658	+	0.647979i	$0.775615\pi$
$618$	0	0
$619$	−4.67826e6	−0.490747	−0.245374	−	0.969429i	$-0.578911\pi$
$619$	−4.67826e6	−0.490747	−0.245374	+	0.969429i	$0.578911\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.23413e6	0.953183
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.62541e6	−0.768487
$630$	0	0
$631$	−4.90334e6	−0.490251	−0.245125	−	0.969491i	$-0.578829\pi$
$631$	−4.90334e6	−0.490251	−0.245125	+	0.969491i	$0.578829\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.34096e7	−1.30938
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.64399e6	−0.254164	−0.127082	−	0.991892i	$-0.540561\pi$
$641$	−2.64399e6	−0.254164	−0.127082	+	0.991892i	$0.540561\pi$
$642$	0	0
$643$	1.07250e7	1.02299	0.511493	−	0.859287i	$-0.329093\pi$
$643$	1.07250e7	1.02299	0.511493	+	0.859287i	$0.329093\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.70981e7	1.60578	0.802892	−	0.596124i	$-0.203293\pi$
$647$	1.70981e7	1.60578	0.802892	+	0.596124i	$0.203293\pi$
$648$	0	0
$649$	−1.94043e7	−1.80837
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.65234e7	1.51641	0.758206	−	0.652015i	$-0.226076\pi$
$653$	1.65234e7	1.51641	0.758206	+	0.652015i	$0.226076\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4.72555e6	−0.423876	−0.211938	−	0.977283i	$-0.567977\pi$
$659$	−4.72555e6	−0.423876	−0.211938	+	0.977283i	$0.567977\pi$
$660$	0	0
$661$	−1.71053e6	−0.152275	−0.0761374	−	0.997097i	$-0.524259\pi$
$661$	−1.71053e6	−0.152275	−0.0761374	+	0.997097i	$0.524259\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.15849e6	−0.361927
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.05612e6	−0.605006
$672$	0	0
$673$	−1.28565e7	−1.09417	−0.547085	−	0.837077i	$-0.684263\pi$
$673$	−1.28565e7	−1.09417	−0.547085	+	0.837077i	$0.684263\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.86277e6	0.827042	0.413521	−	0.910495i	$-0.364299\pi$
$677$	9.86277e6	0.827042	0.413521	+	0.910495i	$0.364299\pi$
$678$	0	0
$679$	1.25143e7	1.04167
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.48059e6	−0.695623	−0.347812	−	0.937564i	$-0.613075\pi$
$683$	−8.48059e6	−0.695623	−0.347812	+	0.937564i	$0.613075\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.71245e6	−0.137426
$690$	0	0
$691$	1.20827e7	0.962650	0.481325	−	0.876542i	$-0.340156\pi$
$691$	1.20827e7	0.962650	0.481325	+	0.876542i	$0.340156\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.83398e6	0.454865
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.01619e7	1.54966	0.774829	−	0.632170i	$-0.217836\pi$
$701$	2.01619e7	1.54966	0.774829	+	0.632170i	$0.217836\pi$
$702$	0	0
$703$	5.88353e6	0.449003
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.21978e6	0.242258
$708$	0	0
$709$	−4.59767e6	−0.343496	−0.171748	−	0.985141i	$-0.554941\pi$
$709$	−4.59767e6	−0.343496	−0.171748	+	0.985141i	$0.554941\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.22494e7	−2.37573
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.07916e7	−1.49991	−0.749955	−	0.661489i	$-0.769925\pi$
$719$	−2.07916e7	−1.49991	−0.749955	+	0.661489i	$0.769925\pi$
$720$	0	0
$721$	1.90584e7	1.36536
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.68618e7	−1.18323	−0.591614	−	0.806222i	$-0.701509\pi$
$727$	−1.68618e7	−1.18323	−0.591614	+	0.806222i	$0.701509\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.47954e7	−1.02408
$732$	0	0
$733$	9.77813e6	0.672196	0.336098	−	0.941827i	$-0.390893\pi$
$733$	9.77813e6	0.672196	0.336098	+	0.941827i	$0.390893\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.50506e7	−3.05514
$738$	0	0
$739$	−5.37772e6	−0.362232	−0.181116	−	0.983462i	$-0.557971\pi$
$739$	−5.37772e6	−0.362232	−0.181116	+	0.983462i	$0.557971\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.27197e6	−0.416804	−0.208402	−	0.978043i	$-0.566826\pi$
$743$	−6.27197e6	−0.416804	−0.208402	+	0.978043i	$0.566826\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.32089e6	−0.541958
$750$	0	0
$751$	−8.46198e6	−0.547485	−0.273742	−	0.961803i	$-0.588262\pi$
$751$	−8.46198e6	−0.547485	−0.273742	+	0.961803i	$0.588262\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.36314e7	0.864570	0.432285	−	0.901737i	$-0.357708\pi$
$757$	1.36314e7	0.864570	0.432285	+	0.901737i	$0.357708\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.20925e7	−1.38288	−0.691438	−	0.722436i	$-0.743022\pi$
$761$	−2.20925e7	−1.38288	−0.691438	+	0.722436i	$0.743022\pi$
$762$	0	0
$763$	−9.46490e6	−0.588579
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.17173e7	−0.719184
$768$	0	0
$769$	2.47673e7	1.51030	0.755149	−	0.655553i	$-0.227564\pi$
$769$	2.47673e7	1.51030	0.755149	+	0.655553i	$0.227564\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.27774e7	−1.37105	−0.685527	−	0.728047i	$-0.740428\pi$
$773$	−2.27774e7	−1.37105	−0.685527	+	0.728047i	$0.740428\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.50132e6	−0.265764
$780$	0	0
$781$	−1.36861e7	−0.802882
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	7.95915e6	0.458068	0.229034	−	0.973418i	$-0.426443\pi$
$787$	7.95915e6	0.458068	0.229034	+	0.973418i	$0.426443\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.02022e7	−2.28459
$792$	0	0
$793$	−4.26084e6	−0.240609
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.56623e7	0.873393	0.436697	−	0.899609i	$-0.356148\pi$
$797$	1.56623e7	0.873393	0.436697	+	0.899609i	$0.356148\pi$
$798$	0	0
$799$	−1.12309e7	−0.622368
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.22213e7	1.76341
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.63200e7	1.95108	0.975539	−	0.219826i	$-0.0705490\pi$
$809$	3.63200e7	1.95108	0.975539	+	0.219826i	$0.0705490\pi$
$810$	0	0
$811$	−3.31051e7	−1.76743	−0.883716	−	0.468023i	$-0.844967\pi$
$811$	−3.31051e7	−1.76743	−0.883716	+	0.468023i	$0.844967\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.14157e7	0.598339
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.53113e6	−0.338166	−0.169083	−	0.985602i	$-0.554081\pi$
$821$	−6.53113e6	−0.338166	−0.169083	+	0.985602i	$0.554081\pi$
$822$	0	0
$823$	−2.91165e7	−1.49844	−0.749221	−	0.662320i	$-0.769572\pi$
$823$	−2.91165e7	−1.49844	−0.749221	+	0.662320i	$0.769572\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.40322e6	−0.0713450	−0.0356725	−	0.999364i	$-0.511357\pi$
$827$	−1.40322e6	−0.0713450	−0.0356725	+	0.999364i	$0.511357\pi$
$828$	0	0
$829$	9.92867e6	0.501770	0.250885	−	0.968017i	$-0.419278\pi$
$829$	9.92867e6	0.501770	0.250885	+	0.968017i	$0.419278\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.81614e7	−1.40618
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.77364e6	−0.430303	−0.215152	−	0.976581i	$-0.569025\pi$
$839$	−8.77364e6	−0.430303	−0.215152	+	0.976581i	$0.569025\pi$
$840$	0	0
$841$	−1.89462e7	−0.923703
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.78080e7	3.24768
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.92698e7	1.38547
$852$	0	0
$853$	3.12765e7	1.47179	0.735895	−	0.677095i	$-0.236762\pi$
$853$	3.12765e7	1.47179	0.735895	+	0.677095i	$0.236762\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.17957e7	−1.47883	−0.739413	−	0.673252i	$-0.764897\pi$
$857$	−3.17957e7	−1.47883	−0.739413	+	0.673252i	$0.764897\pi$
$858$	0	0
$859$	−3.36126e7	−1.55424	−0.777122	−	0.629349i	$-0.783321\pi$
$859$	−3.36126e7	−1.55424	−0.777122	+	0.629349i	$0.783321\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.84379e7	0.842724	0.421362	−	0.906893i	$-0.361552\pi$
$863$	1.84379e7	0.842724	0.421362	+	0.906893i	$0.361552\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.43664e7	0.645355
$870$	0	0
$871$	−2.72038e7	−1.21502
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.20794e7	0.530330	0.265165	−	0.964203i	$-0.414574\pi$
$877$	1.20794e7	0.530330	0.265165	+	0.964203i	$0.414574\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.17740e7	0.945143	0.472572	−	0.881292i	$-0.343326\pi$
$881$	2.17740e7	0.945143	0.472572	+	0.881292i	$0.343326\pi$
$882$	0	0
$883$	−6.72651e6	−0.290327	−0.145164	−	0.989408i	$-0.546371\pi$
$883$	−6.72651e6	−0.290327	−0.145164	+	0.989408i	$0.546371\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.99152e7	−0.849916	−0.424958	−	0.905213i	$-0.639711\pi$
$887$	−1.99152e7	−0.849916	−0.424958	+	0.905213i	$0.639711\pi$
$888$	0	0
$889$	1.14968e7	0.487891
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.66541e6	0.363631
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.21361e7	0.500820
$900$	0	0
$901$	−3.59630e6	−0.147585
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.13024e7	−1.66708	−0.833541	−	0.552457i	$-0.813690\pi$
$907$	−4.13024e7	−1.66708	−0.833541	+	0.552457i	$0.813690\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.93595e7	−1.57128	−0.785640	−	0.618684i	$-0.787666\pi$
$911$	−3.93595e7	−1.57128	−0.785640	+	0.618684i	$0.787666\pi$
$912$	0	0
$913$	6.17043e7	2.44984
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.77638e7	1.48304
$918$	0	0
$919$	4.45757e7	1.74104	0.870520	−	0.492132i	$-0.163782\pi$
$919$	4.45757e7	1.74104	0.870520	+	0.492132i	$0.163782\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.26436e6	−0.319304
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.74187e7	−0.662180	−0.331090	−	0.943599i	$-0.607416\pi$
$929$	−1.74187e7	−0.662180	−0.331090	+	0.943599i	$0.607416\pi$
$930$	0	0
$931$	2.17284e7	0.821588
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.75506e6	−0.139723	−0.0698615	−	0.997557i	$-0.522256\pi$
$937$	−3.75506e6	−0.139723	−0.0698615	+	0.997557i	$0.522256\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8.21111e6	0.302293	0.151146	−	0.988511i	$-0.451704\pi$
$941$	8.21111e6	0.302293	0.151146	+	0.988511i	$0.451704\pi$
$942$	0	0
$943$	−2.23935e7	−0.820054
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.69201e6	−0.170014	−0.0850068	−	0.996380i	$-0.527091\pi$
$947$	−4.69201e6	−0.170014	−0.0850068	+	0.996380i	$0.527091\pi$
$948$	0	0
$949$	1.94569e7	0.701305
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.07861e7	0.741381	0.370691	−	0.928756i	$-0.379121\pi$
$953$	2.07861e7	0.741381	0.370691	+	0.928756i	$0.379121\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.72474e7	−0.956706
$960$	0	0
$961$	6.54875e7	2.28744
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−8.31772e6	−0.286047	−0.143024	−	0.989719i	$-0.545683\pi$
$967$	−8.31772e6	−0.286047	−0.143024	+	0.989719i	$0.545683\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7.53203e6	−0.256368	−0.128184	−	0.991750i	$-0.540915\pi$
$971$	−7.53203e6	−0.256368	−0.128184	+	0.991750i	$0.540915\pi$
$972$	0	0
$973$	−4.91417e6	−0.166406
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.54137e7	1.18696	0.593479	−	0.804849i	$-0.297754\pi$
$977$	3.54137e7	1.18696	0.593479	+	0.804849i	$0.297754\pi$
$978$	0	0
$979$	−2.83940e7	−0.946825
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.34967e7	−0.775574	−0.387787	−	0.921749i	$-0.626760\pi$
$983$	−2.34967e7	−0.775574	−0.387787	+	0.921749i	$0.626760\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.67917e7	1.84627
$990$	0	0
$991$	3.98799e6	0.128994	0.0644971	−	0.997918i	$-0.479456\pi$
$991$	3.98799e6	0.128994	0.0644971	+	0.997918i	$0.479456\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.57767e7	−1.13989	−0.569944	−	0.821684i	$-0.693035\pi$
$997$	−3.57767e7	−1.13989	−0.569944	+	0.821684i	$0.693035\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.a.x.1.3		3
3.2	odd	2		300.6.a.j.1.3		3
5.2	odd	4		180.6.d.d.109.6		6
5.3	odd	4		180.6.d.d.109.5		6
5.4	even	2		900.6.a.w.1.1		3
15.2	even	4		60.6.d.a.49.1	✓	6
15.8	even	4		60.6.d.a.49.4	yes	6
15.14	odd	2		300.6.a.i.1.1		3
20.3	even	4		720.6.f.m.289.5		6
20.7	even	4		720.6.f.m.289.6		6
60.23	odd	4		240.6.f.d.49.1		6
60.47	odd	4		240.6.f.d.49.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.d.a.49.1	✓	6	15.2	even	4
60.6.d.a.49.4	yes	6	15.8	even	4
180.6.d.d.109.5		6	5.3	odd	4
180.6.d.d.109.6		6	5.2	odd	4
240.6.f.d.49.1		6	60.23	odd	4
240.6.f.d.49.4		6	60.47	odd	4
300.6.a.i.1.1		3	15.14	odd	2
300.6.a.j.1.3		3	3.2	odd	2
720.6.f.m.289.5		6	20.3	even	4
720.6.f.m.289.6		6	20.7	even	4
900.6.a.w.1.1		3	5.4	even	2
900.6.a.x.1.3		3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+222.092 q^{7} +O(q^{10})\)	\(q+222.092 q^{7} -682.910 q^{11} -412.376 q^{13} -866.026 q^{17} +668.199 q^{19} +3324.21 q^{23} -1250.97 q^{29} -9701.37 q^{31} +8805.05 q^{37} -6736.49 q^{41} +17084.3 q^{43} +12968.3 q^{47} +32517.9 q^{49} +4152.64 q^{53} +28414.2 q^{59} +10332.4 q^{61} +65968.5 q^{67} +20040.8 q^{71} -47182.3 q^{73} -151669. q^{77} -21037.1 q^{79} -90354.9 q^{83} +41577.9 q^{89} -91585.4 q^{91} +56347.2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 88 q^{7}+O(q^{10}) \) 3 * q + 88 * q^7	\( 3 q + 88 q^{7} - 148 q^{11} - 868 q^{17} + 3000 q^{19} + 1052 q^{23} - 7962 q^{29} + 132 q^{31} + 11944 q^{37} - 8170 q^{41} + 26188 q^{43} - 37892 q^{47} + 13827 q^{49} - 11076 q^{53} + 46228 q^{59} + 3126 q^{61} + 126332 q^{67} + 80400 q^{71} + 42904 q^{73} - 234608 q^{77} - 64476 q^{79} - 126268 q^{83} + 38030 q^{89} - 49200 q^{91} + 331016 q^{97}+O(q^{100}) \) 3 * q + 88 * q^7 - 148 * q^11 - 868 * q^17 + 3000 * q^19 + 1052 * q^23 - 7962 * q^29 + 132 * q^31 + 11944 * q^37 - 8170 * q^41 + 26188 * q^43 - 37892 * q^47 + 13827 * q^49 - 11076 * q^53 + 46228 * q^59 + 3126 * q^61 + 126332 * q^67 + 80400 * q^71 + 42904 * q^73 - 234608 * q^77 - 64476 * q^79 - 126268 * q^83 + 38030 * q^89 - 49200 * q^91 + 331016 * q^97

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