LMFDB - Embedded newform 900.6.a.r.1.1

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{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ 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)

$f(q)$	$=$	$q-147.745 q^{7} +O(q^{10})$	$q-147.745 q^{7} -662.235 q^{11} -629.980 q^{13} -1082.24 q^{17} -2521.20 q^{19} -1802.24 q^{23} -6237.53 q^{29} +5617.75 q^{31} -1199.41 q^{37} -10016.2 q^{41} +10734.6 q^{43} -3803.65 q^{47} +5021.61 q^{49} +15800.9 q^{53} -1734.24 q^{59} +11199.7 q^{61} -48604.8 q^{67} -68965.9 q^{71} +80654.7 q^{73} +97842.0 q^{77} +82056.9 q^{79} -85683.4 q^{83} -25845.6 q^{89} +93076.5 q^{91} +9935.27 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 22 q^{7}+O(q^{10}) $ 2 * q + 22 * q^7	$ 2 q + 22 q^{7} - 372 q^{11} + 10 q^{13} - 1212 q^{17} - 1550 q^{19} - 2652 q^{23} + 1812 q^{29} + 10918 q^{31} + 3316 q^{37} - 19080 q^{41} + 12262 q^{43} + 15252 q^{47} + 17028 q^{49} - 20784 q^{53} - 18708 q^{59} + 35734 q^{61} - 98162 q^{67} - 114120 q^{71} + 109876 q^{73} + 147108 q^{77} + 95536 q^{79} - 3732 q^{83} - 93600 q^{89} + 201710 q^{91} - 91886 q^{97}+O(q^{100}) $ 2 * q + 22 * q^7 - 372 * q^11 + 10 * q^13 - 1212 * q^17 - 1550 * q^19 - 2652 * q^23 + 1812 * q^29 + 10918 * q^31 + 3316 * q^37 - 19080 * q^41 + 12262 * q^43 + 15252 * q^47 + 17028 * q^49 - 20784 * q^53 - 18708 * q^59 + 35734 * q^61 - 98162 * q^67 - 114120 * q^71 + 109876 * q^73 + 147108 * q^77 + 95536 * q^79 - 3732 * q^83 - 93600 * q^89 + 201710 * q^91 - 91886 * 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
$7$	−147.745	−1.13964	−0.569820	−	0.821769i	$-0.692987\pi$
$7$	−147.745	−1.13964	−0.569820	+	0.821769i	$0.692987\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−662.235	−1.65018	−0.825089	−	0.565003i	$-0.808875\pi$
$11$	−662.235	−1.65018	−0.825089	+	0.565003i	$0.808875\pi$
$12$	0	0
$13$	−629.980	−1.03388	−0.516938	−	0.856023i	$-0.672928\pi$
$13$	−629.980	−1.03388	−0.516938	+	0.856023i	$0.672928\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1082.24	−0.908237	−0.454119	−	0.890941i	$-0.650046\pi$
$17$	−1082.24	−0.908237	−0.454119	+	0.890941i	$0.650046\pi$
$18$	0	0
$19$	−2521.20	−1.60222	−0.801111	−	0.598516i	$-0.795757\pi$
$19$	−2521.20	−1.60222	−0.801111	+	0.598516i	$0.795757\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1802.24	−0.710382	−0.355191	−	0.934794i	$-0.615584\pi$
$23$	−1802.24	−0.710382	−0.355191	+	0.934794i	$0.615584\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6237.53	−1.37727	−0.688633	−	0.725110i	$-0.741789\pi$
$29$	−6237.53	−1.37727	−0.688633	+	0.725110i	$0.741789\pi$
$30$	0	0
$31$	5617.75	1.04992	0.524962	−	0.851126i	$-0.324080\pi$
$31$	5617.75	1.04992	0.524962	+	0.851126i	$0.324080\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1199.41	−0.144034	−0.0720168	−	0.997403i	$-0.522944\pi$
$37$	−1199.41	−0.144034	−0.0720168	+	0.997403i	$0.522944\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10016.2	−0.930561	−0.465281	−	0.885163i	$-0.654047\pi$
$41$	−10016.2	−0.930561	−0.465281	+	0.885163i	$0.654047\pi$
$42$	0	0
$43$	10734.6	0.885350	0.442675	−	0.896682i	$-0.354030\pi$
$43$	10734.6	0.885350	0.442675	+	0.896682i	$0.354030\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3803.65	−0.251163	−0.125581	−	0.992083i	$-0.540080\pi$
$47$	−3803.65	−0.251163	−0.125581	+	0.992083i	$0.540080\pi$
$48$	0	0
$49$	5021.61	0.298781
$50$	0	0
$51$	0	0
$52$	0	0
$53$	15800.9	0.772668	0.386334	−	0.922359i	$-0.373741\pi$
$53$	15800.9	0.772668	0.386334	+	0.922359i	$0.373741\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1734.24	−0.0648602	−0.0324301	−	0.999474i	$-0.510325\pi$
$59$	−1734.24	−0.0648602	−0.0324301	+	0.999474i	$0.510325\pi$
$60$	0	0
$61$	11199.7	0.385374	0.192687	−	0.981260i	$-0.438280\pi$
$61$	11199.7	0.385374	0.192687	+	0.981260i	$0.438280\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−48604.8	−1.32279	−0.661396	−	0.750037i	$-0.730036\pi$
$67$	−48604.8	−1.32279	−0.661396	+	0.750037i	$0.730036\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−68965.9	−1.62364	−0.811818	−	0.583911i	$-0.801522\pi$
$71$	−68965.9	−1.62364	−0.811818	+	0.583911i	$0.801522\pi$
$72$	0	0
$73$	80654.7	1.77142	0.885712	−	0.464235i	$-0.153671\pi$
$73$	80654.7	1.77142	0.885712	+	0.464235i	$0.153671\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	97842.0	1.88061
$78$	0	0
$79$	82056.9	1.47927	0.739635	−	0.673008i	$-0.234998\pi$
$79$	82056.9	1.47927	0.739635	+	0.673008i	$0.234998\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−85683.4	−1.36522	−0.682608	−	0.730785i	$-0.739154\pi$
$83$	−85683.4	−1.36522	−0.682608	+	0.730785i	$0.739154\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−25845.6	−0.345870	−0.172935	−	0.984933i	$-0.555325\pi$
$89$	−25845.6	−0.345870	−0.172935	+	0.984933i	$0.555325\pi$
$90$	0	0
$91$	93076.5	1.17825
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9935.27	0.107214	0.0536068	−	0.998562i	$-0.482928\pi$
$97$	9935.27	0.107214	0.0536068	+	0.998562i	$0.482928\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−44176.5	−0.430911	−0.215456	−	0.976514i	$-0.569124\pi$
$101$	−44176.5	−0.430911	−0.215456	+	0.976514i	$0.569124\pi$
$102$	0	0
$103$	−131554.	−1.22183	−0.610914	−	0.791697i	$-0.709198\pi$
$103$	−131554.	−1.22183	−0.610914	+	0.791697i	$0.709198\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	170825.	1.44242	0.721210	−	0.692716i	$-0.243586\pi$
$107$	170825.	1.44242	0.721210	+	0.692716i	$0.243586\pi$
$108$	0	0
$109$	−28667.9	−0.231115	−0.115558	−	0.993301i	$-0.536866\pi$
$109$	−28667.9	−0.231115	−0.115558	+	0.993301i	$0.536866\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−85548.0	−0.630251	−0.315126	−	0.949050i	$-0.602047\pi$
$113$	−85548.0	−0.630251	−0.315126	+	0.949050i	$0.602047\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	159895.	1.03506
$120$	0	0
$121$	277505.	1.72308
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−109853.	−0.604368	−0.302184	−	0.953250i	$-0.597716\pi$
$127$	−109853.	−0.604368	−0.302184	+	0.953250i	$0.597716\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−390140.	−1.98629	−0.993145	−	0.116892i	$-0.962707\pi$
$131$	−390140.	−1.98629	−0.993145	+	0.116892i	$0.962707\pi$
$132$	0	0
$133$	372494.	1.82596
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−302596.	−1.37741	−0.688703	−	0.725044i	$-0.741820\pi$
$137$	−302596.	−1.37741	−0.688703	+	0.725044i	$0.741820\pi$
$138$	0	0
$139$	125596.	0.551367	0.275683	−	0.961249i	$-0.411096\pi$
$139$	125596.	0.551367	0.275683	+	0.961249i	$0.411096\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	417195.	1.70608
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	84368.4	0.311325	0.155662	−	0.987810i	$-0.450249\pi$
$149$	84368.4	0.311325	0.155662	+	0.987810i	$0.450249\pi$
$150$	0	0
$151$	−454944.	−1.62374	−0.811869	−	0.583840i	$-0.801550\pi$
$151$	−454944.	−1.62374	−0.811869	+	0.583840i	$0.801550\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	109519.	0.354602	0.177301	−	0.984157i	$-0.443263\pi$
$157$	109519.	0.354602	0.177301	+	0.984157i	$0.443263\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	266271.	0.809580
$162$	0	0
$163$	420668.	1.24014	0.620070	−	0.784547i	$-0.287104\pi$
$163$	420668.	1.24014	0.620070	+	0.784547i	$0.287104\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	207627.	0.576094	0.288047	−	0.957616i	$-0.406994\pi$
$167$	207627.	0.576094	0.288047	+	0.957616i	$0.406994\pi$
$168$	0	0
$169$	25582.2	0.0689003
$170$	0	0
$171$	0	0
$172$	0	0
$173$	665522.	1.69063	0.845313	−	0.534272i	$-0.179414\pi$
$173$	665522.	1.69063	0.845313	+	0.534272i	$0.179414\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−431788.	−1.00725	−0.503626	−	0.863922i	$-0.668001\pi$
$179$	−431788.	−1.00725	−0.503626	+	0.863922i	$0.668001\pi$
$180$	0	0
$181$	−588263.	−1.33467	−0.667336	−	0.744756i	$-0.732566\pi$
$181$	−588263.	−1.33467	−0.667336	+	0.744756i	$0.732566\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	716694.	1.49875
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−653915.	−1.29699	−0.648496	−	0.761218i	$-0.724602\pi$
$191$	−653915.	−1.29699	−0.648496	+	0.761218i	$0.724602\pi$
$192$	0	0
$193$	−293969.	−0.568078	−0.284039	−	0.958813i	$-0.591674\pi$
$193$	−293969.	−0.568078	−0.284039	+	0.958813i	$0.591674\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	692255.	1.27087	0.635434	−	0.772155i	$-0.280821\pi$
$197$	692255.	1.27087	0.635434	+	0.772155i	$0.280821\pi$
$198$	0	0
$199$	−89145.7	−0.159576	−0.0797880	−	0.996812i	$-0.525424\pi$
$199$	−89145.7	−0.159576	−0.0797880	+	0.996812i	$0.525424\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	921564.	1.56959
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.66962e6	2.64395
$210$	0	0
$211$	−300548.	−0.464737	−0.232369	−	0.972628i	$-0.574648\pi$
$211$	−300548.	−0.464737	−0.232369	+	0.972628i	$0.574648\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−829994.	−1.19654
$218$	0	0
$219$	0	0
$220$	0	0
$221$	681787.	0.939005
$222$	0	0
$223$	−1.22316e6	−1.64710	−0.823549	−	0.567245i	$-0.808009\pi$
$223$	−1.22316e6	−1.64710	−0.823549	+	0.567245i	$0.808009\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.35994e6	−1.75168	−0.875839	−	0.482603i	$-0.839692\pi$
$227$	−1.35994e6	−1.75168	−0.875839	+	0.482603i	$0.839692\pi$
$228$	0	0
$229$	171691.	0.216351	0.108176	−	0.994132i	$-0.465499\pi$
$229$	171691.	0.216351	0.108176	+	0.994132i	$0.465499\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	741362.	0.894624	0.447312	−	0.894378i	$-0.352381\pi$
$233$	741362.	0.894624	0.447312	+	0.894378i	$0.352381\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	309086.	0.350013	0.175006	−	0.984567i	$-0.444005\pi$
$239$	309086.	0.350013	0.175006	+	0.984567i	$0.444005\pi$
$240$	0	0
$241$	−1.44908e6	−1.60712	−0.803561	−	0.595223i	$-0.797064\pi$
$241$	−1.44908e6	−1.60712	−0.803561	+	0.595223i	$0.797064\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.58830e6	1.65650
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.14053e6	1.14268	0.571338	−	0.820715i	$-0.306425\pi$
$251$	1.14053e6	1.14268	0.571338	+	0.820715i	$0.306425\pi$
$252$	0	0
$253$	1.19350e6	1.17226
$254$	0	0
$255$	0	0
$256$	0	0
$257$	646450.	0.610523	0.305261	−	0.952269i	$-0.401256\pi$
$257$	646450.	0.610523	0.305261	+	0.952269i	$0.401256\pi$
$258$	0	0
$259$	177207.	0.164147
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.06070e6	0.945591	0.472795	−	0.881172i	$-0.343245\pi$
$263$	1.06070e6	0.945591	0.472795	+	0.881172i	$0.343245\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.19054e6	1.84574	0.922868	−	0.385116i	$-0.125838\pi$
$269$	2.19054e6	1.84574	0.922868	+	0.385116i	$0.125838\pi$
$270$	0	0
$271$	874765.	0.723550	0.361775	−	0.932265i	$-0.382171\pi$
$271$	874765.	0.723550	0.361775	+	0.932265i	$0.382171\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.32986e6	1.04137	0.520687	−	0.853748i	$-0.325676\pi$
$277$	1.32986e6	1.04137	0.520687	+	0.853748i	$0.325676\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.81200e6	−1.36896	−0.684482	−	0.729030i	$-0.739971\pi$
$281$	−1.81200e6	−1.36896	−0.684482	+	0.729030i	$0.739971\pi$
$282$	0	0
$283$	1.98848e6	1.47589	0.737946	−	0.674860i	$-0.235796\pi$
$283$	1.98848e6	1.47589	0.737946	+	0.674860i	$0.235796\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.47985e6	1.06051
$288$	0	0
$289$	−248624.	−0.175105
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.22419e6	1.51357	0.756785	−	0.653664i	$-0.226769\pi$
$293$	2.22419e6	1.51357	0.756785	+	0.653664i	$0.226769\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.13537e6	0.734447
$300$	0	0
$301$	−1.58599e6	−1.00898
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	535556.	0.324309	0.162155	−	0.986765i	$-0.448156\pi$
$307$	535556.	0.324309	0.162155	+	0.986765i	$0.448156\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.14998e6	−1.26047	−0.630236	−	0.776403i	$-0.717042\pi$
$311$	−2.14998e6	−1.26047	−0.630236	+	0.776403i	$0.717042\pi$
$312$	0	0
$313$	−319986.	−0.184616	−0.0923081	−	0.995730i	$-0.529424\pi$
$313$	−319986.	−0.184616	−0.0923081	+	0.995730i	$0.529424\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.04672e6	0.585033	0.292517	−	0.956260i	$-0.405507\pi$
$317$	1.04672e6	0.585033	0.292517	+	0.956260i	$0.405507\pi$
$318$	0	0
$319$	4.13071e6	2.27273
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.72853e6	1.45520
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	561970.	0.286235
$330$	0	0
$331$	−1.47074e6	−0.737845	−0.368923	−	0.929460i	$-0.620273\pi$
$331$	−1.47074e6	−0.737845	−0.368923	+	0.929460i	$0.620273\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.05286e6	−0.984657	−0.492329	−	0.870409i	$-0.663854\pi$
$337$	−2.05286e6	−0.984657	−0.492329	+	0.870409i	$0.663854\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.72027e6	−1.73256
$342$	0	0
$343$	1.74123e6	0.799138
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.44297e6	−0.643330	−0.321665	−	0.946854i	$-0.604242\pi$
$347$	−1.44297e6	−0.643330	−0.321665	+	0.946854i	$0.604242\pi$
$348$	0	0
$349$	−3.53038e6	−1.55152	−0.775761	−	0.631027i	$-0.782634\pi$
$349$	−3.53038e6	−1.55152	−0.775761	+	0.631027i	$0.782634\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.46715e6	−1.05380	−0.526900	−	0.849927i	$-0.676646\pi$
$353$	−2.46715e6	−1.05380	−0.526900	+	0.849927i	$0.676646\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.26610e6	−1.33750	−0.668748	−	0.743489i	$-0.733170\pi$
$359$	−3.26610e6	−1.33750	−0.668748	+	0.743489i	$0.733170\pi$
$360$	0	0
$361$	3.88033e6	1.56711
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2.16639e6	−0.839596	−0.419798	−	0.907617i	$-0.637899\pi$
$367$	−2.16639e6	−0.839596	−0.419798	+	0.907617i	$0.637899\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.33451e6	−0.880564
$372$	0	0
$373$	4.68582e6	1.74387	0.871933	−	0.489625i	$-0.162866\pi$
$373$	4.68582e6	1.74387	0.871933	+	0.489625i	$0.162866\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.92952e6	1.42392
$378$	0	0
$379$	3.99853e6	1.42989	0.714944	−	0.699182i	$-0.246452\pi$
$379$	3.99853e6	1.42989	0.714944	+	0.699182i	$0.246452\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.83762e6	0.640115	0.320057	−	0.947398i	$-0.396298\pi$
$383$	1.83762e6	0.640115	0.320057	+	0.947398i	$0.396298\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.57983e6	−1.19947	−0.599734	−	0.800200i	$-0.704727\pi$
$389$	−3.57983e6	−1.19947	−0.599734	+	0.800200i	$0.704727\pi$
$390$	0	0
$391$	1.95044e6	0.645195
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.43403e6	−0.775086	−0.387543	−	0.921852i	$-0.626676\pi$
$397$	−2.43403e6	−0.775086	−0.387543	+	0.921852i	$0.626676\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.04925e6	0.636406	0.318203	−	0.948023i	$-0.396921\pi$
$401$	2.04925e6	0.636406	0.318203	+	0.948023i	$0.396921\pi$
$402$	0	0
$403$	−3.53907e6	−1.08549
$404$	0	0
$405$	0	0
$406$	0	0
$407$	794293.	0.237681
$408$	0	0
$409$	−4.67579e6	−1.38212	−0.691061	−	0.722796i	$-0.742856\pi$
$409$	−4.67579e6	−1.38212	−0.691061	+	0.722796i	$0.742856\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	256225.	0.0739173
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.86934e6	0.798449	0.399225	−	0.916853i	$-0.369279\pi$
$419$	2.86934e6	0.798449	0.399225	+	0.916853i	$0.369279\pi$
$420$	0	0
$421$	−495614.	−0.136282	−0.0681410	−	0.997676i	$-0.521707\pi$
$421$	−495614.	−0.136282	−0.0681410	+	0.997676i	$0.521707\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.65470e6	−0.439188
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.66876e6	−0.951319	−0.475659	−	0.879630i	$-0.657790\pi$
$431$	−3.66876e6	−0.951319	−0.475659	+	0.879630i	$0.657790\pi$
$432$	0	0
$433$	1.62199e6	0.415746	0.207873	−	0.978156i	$-0.433346\pi$
$433$	1.62199e6	0.415746	0.207873	+	0.978156i	$0.433346\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.54379e6	1.13819
$438$	0	0
$439$	3.66935e6	0.908716	0.454358	−	0.890819i	$-0.349869\pi$
$439$	3.66935e6	0.908716	0.454358	+	0.890819i	$0.349869\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.01410e6	0.245512	0.122756	−	0.992437i	$-0.460827\pi$
$443$	1.01410e6	0.245512	0.122756	+	0.992437i	$0.460827\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.43185e6	1.27154	0.635772	−	0.771877i	$-0.280682\pi$
$449$	5.43185e6	1.27154	0.635772	+	0.771877i	$0.280682\pi$
$450$	0	0
$451$	6.63310e6	1.53559
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.58390e6	1.25068	0.625341	−	0.780351i	$-0.284960\pi$
$457$	5.58390e6	1.25068	0.625341	+	0.780351i	$0.284960\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	420777.	0.0922145	0.0461073	−	0.998936i	$-0.485318\pi$
$461$	420777.	0.0922145	0.0461073	+	0.998936i	$0.485318\pi$
$462$	0	0
$463$	−512658.	−0.111141	−0.0555706	−	0.998455i	$-0.517698\pi$
$463$	−512658.	−0.111141	−0.0555706	+	0.998455i	$0.517698\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.14897e6	−1.94124	−0.970621	−	0.240612i	$-0.922652\pi$
$467$	−9.14897e6	−1.94124	−0.970621	+	0.240612i	$0.922652\pi$
$468$	0	0
$469$	7.18111e6	1.50751
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.10884e6	−1.46098
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.15726e6	0.230458	0.115229	−	0.993339i	$-0.463240\pi$
$479$	1.15726e6	0.230458	0.115229	+	0.993339i	$0.463240\pi$
$480$	0	0
$481$	755606.	0.148913
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.75578e6	0.335465	0.167732	−	0.985833i	$-0.446356\pi$
$487$	1.75578e6	0.335465	0.167732	+	0.985833i	$0.446356\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.64300e6	−0.869150	−0.434575	−	0.900636i	$-0.643101\pi$
$491$	−4.64300e6	−0.869150	−0.434575	+	0.900636i	$0.643101\pi$
$492$	0	0
$493$	6.75047e6	1.25088
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.01894e7	1.85036
$498$	0	0
$499$	−3.50099e6	−0.629419	−0.314709	−	0.949188i	$-0.601907\pi$
$499$	−3.50099e6	−0.629419	−0.314709	+	0.949188i	$0.601907\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.80449e6	−1.19916	−0.599578	−	0.800316i	$-0.704665\pi$
$503$	−6.80449e6	−1.19916	−0.599578	+	0.800316i	$0.704665\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	803645.	0.137490	0.0687448	−	0.997634i	$-0.478101\pi$
$509$	803645.	0.137490	0.0687448	+	0.997634i	$0.478101\pi$
$510$	0	0
$511$	−1.19163e7	−2.01879
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.51891e6	0.414463
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.15940e6	0.832731	0.416366	−	0.909197i	$-0.363304\pi$
$521$	5.15940e6	0.832731	0.416366	+	0.909197i	$0.363304\pi$
$522$	0	0
$523$	−4.23531e6	−0.677065	−0.338533	−	0.940955i	$-0.609931\pi$
$523$	−4.23531e6	−0.677065	−0.338533	+	0.940955i	$0.609931\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.07972e6	−0.953580
$528$	0	0
$529$	−3.18829e6	−0.495358
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.31003e6	0.962085
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.32549e6	−0.493041
$540$	0	0
$541$	−8.22222e6	−1.20780	−0.603901	−	0.797059i	$-0.706388\pi$
$541$	−8.22222e6	−1.20780	−0.603901	+	0.797059i	$0.706388\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	372072.	0.0531690	0.0265845	−	0.999647i	$-0.491537\pi$
$547$	372072.	0.0531690	0.0265845	+	0.999647i	$0.491537\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.57260e7	2.20668
$552$	0	0
$553$	−1.21235e7	−1.68584
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.23214e7	−1.68276	−0.841378	−	0.540447i	$-0.818255\pi$
$557$	−1.23214e7	−1.68276	−0.841378	+	0.540447i	$0.818255\pi$
$558$	0	0
$559$	−6.76259e6	−0.915342
$560$	0	0
$561$	0	0
$562$	0	0
$563$	577762.	0.0768207	0.0384104	−	0.999262i	$-0.487771\pi$
$563$	577762.	0.0768207	0.0384104	+	0.999262i	$0.487771\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.41678e6	−1.08985	−0.544923	−	0.838486i	$-0.683441\pi$
$569$	−8.41678e6	−1.08985	−0.544923	+	0.838486i	$0.683441\pi$
$570$	0	0
$571$	9.88483e6	1.26876	0.634379	−	0.773022i	$-0.281256\pi$
$571$	9.88483e6	1.26876	0.634379	+	0.773022i	$0.281256\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.80006e6	−0.600215	−0.300108	−	0.953905i	$-0.597023\pi$
$577$	−4.80006e6	−0.600215	−0.300108	+	0.953905i	$0.597023\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.26593e7	1.55586
$582$	0	0
$583$	−1.04639e7	−1.27504
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.20764e6	0.264443	0.132222	−	0.991220i	$-0.457789\pi$
$587$	2.20764e6	0.264443	0.132222	+	0.991220i	$0.457789\pi$
$588$	0	0
$589$	−1.41634e7	−1.68221
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.87399e6	−0.569178	−0.284589	−	0.958650i	$-0.591857\pi$
$593$	−4.87399e6	−0.569178	−0.284589	+	0.958650i	$0.591857\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.18365e7	−1.34789	−0.673945	−	0.738781i	$-0.735402\pi$
$599$	−1.18365e7	−1.34789	−0.673945	+	0.738781i	$0.735402\pi$
$600$	0	0
$601$	7.54130e6	0.851648	0.425824	−	0.904806i	$-0.359984\pi$
$601$	7.54130e6	0.851648	0.425824	+	0.904806i	$0.359984\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4.89856e6	−0.539631	−0.269815	−	0.962912i	$-0.586963\pi$
$607$	−4.89856e6	−0.539631	−0.269815	+	0.962912i	$0.586963\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.39622e6	0.259671
$612$	0	0
$613$	−9.92286e6	−1.06656	−0.533281	−	0.845938i	$-0.679041\pi$
$613$	−9.92286e6	−1.06656	−0.533281	+	0.845938i	$0.679041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.25453e6	−0.767179	−0.383589	−	0.923504i	$-0.625312\pi$
$617$	−7.25453e6	−0.767179	−0.383589	+	0.923504i	$0.625312\pi$
$618$	0	0
$619$	2.81096e6	0.294869	0.147434	−	0.989072i	$-0.452898\pi$
$619$	2.81096e6	0.294869	0.147434	+	0.989072i	$0.452898\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.81857e6	0.394167
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.29805e6	0.130817
$630$	0	0
$631$	−7.77303e6	−0.777172	−0.388586	−	0.921413i	$-0.627036\pi$
$631$	−7.77303e6	−0.777172	−0.388586	+	0.921413i	$0.627036\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.16351e6	−0.308902
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.91161e7	1.83761	0.918807	−	0.394708i	$-0.129154\pi$
$641$	1.91161e7	1.83761	0.918807	+	0.394708i	$0.129154\pi$
$642$	0	0
$643$	−3.52558e6	−0.336282	−0.168141	−	0.985763i	$-0.553776\pi$
$643$	−3.52558e6	−0.336282	−0.168141	+	0.985763i	$0.553776\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.43353e7	−1.34631	−0.673156	−	0.739501i	$-0.735062\pi$
$647$	−1.43353e7	−1.34631	−0.673156	+	0.739501i	$0.735062\pi$
$648$	0	0
$649$	1.14847e6	0.107031
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.62752e6	−0.700004	−0.350002	−	0.936749i	$-0.613819\pi$
$653$	−7.62752e6	−0.700004	−0.350002	+	0.936749i	$0.613819\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	836481.	0.0750314	0.0375157	−	0.999296i	$-0.488056\pi$
$659$	836481.	0.0750314	0.0375157	+	0.999296i	$0.488056\pi$
$660$	0	0
$661$	−1.49009e7	−1.32650	−0.663251	−	0.748397i	$-0.730824\pi$
$661$	−1.49009e7	−1.32650	−0.663251	+	0.748397i	$0.730824\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.12415e7	0.978384
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.41684e6	−0.635935
$672$	0	0
$673$	4.98768e6	0.424484	0.212242	−	0.977217i	$-0.431924\pi$
$673$	4.98768e6	0.424484	0.212242	+	0.977217i	$0.431924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.60906e7	−1.34928	−0.674639	−	0.738148i	$-0.735701\pi$
$677$	−1.60906e7	−1.34928	−0.674639	+	0.738148i	$0.735701\pi$
$678$	0	0
$679$	−1.46789e6	−0.122185
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2.10028e7	−1.72276	−0.861380	−	0.507961i	$-0.830399\pi$
$683$	−2.10028e7	−1.72276	−0.861380	+	0.507961i	$0.830399\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.95428e6	−0.798844
$690$	0	0
$691$	−8.42900e6	−0.671554	−0.335777	−	0.941941i	$-0.608999\pi$
$691$	−8.42900e6	−0.671554	−0.335777	+	0.941941i	$0.608999\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.08399e7	0.845170
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.86044e6	0.373577	0.186788	−	0.982400i	$-0.440192\pi$
$701$	4.86044e6	0.373577	0.186788	+	0.982400i	$0.440192\pi$
$702$	0	0
$703$	3.02395e6	0.230774
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.52686e6	0.491084
$708$	0	0
$709$	−14126.8	−0.00105543	−0.000527714	−	1.00000i	$-0.500168\pi$
$709$	−14126.8	−0.00105543	−0.000527714		1.00000i	$0.500168\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.01245e7	−0.745847
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.77010e6	0.488397	0.244198	−	0.969725i	$-0.421475\pi$
$719$	6.77010e6	0.488397	0.244198	+	0.969725i	$0.421475\pi$
$720$	0	0
$721$	1.94364e7	1.39244
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.46483e7	−1.02790	−0.513949	−	0.857821i	$-0.671818\pi$
$727$	−1.46483e7	−1.02790	−0.513949	+	0.857821i	$0.671818\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.16174e7	−0.804108
$732$	0	0
$733$	−3.40016e6	−0.233743	−0.116872	−	0.993147i	$-0.537287\pi$
$733$	−3.40016e6	−0.233743	−0.116872	+	0.993147i	$0.537287\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.21878e7	2.18284
$738$	0	0
$739$	−2.13728e7	−1.43962	−0.719812	−	0.694169i	$-0.755772\pi$
$739$	−2.13728e7	−1.43962	−0.719812	+	0.694169i	$0.755772\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.95381e7	−1.29841	−0.649204	−	0.760614i	$-0.724898\pi$
$743$	−1.95381e7	−1.29841	−0.649204	+	0.760614i	$0.724898\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.52385e7	−1.64384
$750$	0	0
$751$	1.50172e7	0.971605	0.485803	−	0.874069i	$-0.338527\pi$
$751$	1.50172e7	0.971605	0.485803	+	0.874069i	$0.338527\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.23095e6	0.268348	0.134174	−	0.990958i	$-0.457162\pi$
$757$	4.23095e6	0.268348	0.134174	+	0.990958i	$0.457162\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.72356e6	−0.170481	−0.0852405	−	0.996360i	$-0.527166\pi$
$761$	−2.72356e6	−0.170481	−0.0852405	+	0.996360i	$0.527166\pi$
$762$	0	0
$763$	4.23553e6	0.263389
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.09253e6	0.0670574
$768$	0	0
$769$	2.69504e7	1.64342	0.821711	−	0.569905i	$-0.193020\pi$
$769$	2.69504e7	1.64342	0.821711	+	0.569905i	$0.193020\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.40755e7	1.44920	0.724598	−	0.689171i	$-0.242025\pi$
$773$	2.40755e7	1.44920	0.724598	+	0.689171i	$0.242025\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.52529e7	1.49097
$780$	0	0
$781$	4.56716e7	2.67929
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.52790e6	0.260591	0.130296	−	0.991475i	$-0.458407\pi$
$787$	4.52790e6	0.260591	0.130296	+	0.991475i	$0.458407\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.26393e7	0.718260
$792$	0	0
$793$	−7.05559e6	−0.398429
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8.67415e6	−0.483705	−0.241853	−	0.970313i	$-0.577755\pi$
$797$	−8.67415e6	−0.483705	−0.241853	+	0.970313i	$0.577755\pi$
$798$	0	0
$799$	4.11644e6	0.228115
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.34124e7	−2.92316
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.02725e7	−0.551830	−0.275915	−	0.961182i	$-0.588981\pi$
$809$	−1.02725e7	−0.551830	−0.275915	+	0.961182i	$0.588981\pi$
$810$	0	0
$811$	−2.51741e7	−1.34401	−0.672005	−	0.740546i	$-0.734567\pi$
$811$	−2.51741e7	−1.34401	−0.672005	+	0.740546i	$0.734567\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.70640e7	−1.41853
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.66034e6	0.293079	0.146539	−	0.989205i	$-0.453186\pi$
$821$	5.66034e6	0.293079	0.146539	+	0.989205i	$0.453186\pi$
$822$	0	0
$823$	445634.	0.0229339	0.0114670	−	0.999934i	$-0.496350\pi$
$823$	445634.	0.0229339	0.0114670	+	0.999934i	$0.496350\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.37086e7	−0.696992	−0.348496	−	0.937310i	$-0.613308\pi$
$827$	−1.37086e7	−0.696992	−0.348496	+	0.937310i	$0.613308\pi$
$828$	0	0
$829$	−1.50876e7	−0.762489	−0.381244	−	0.924474i	$-0.624504\pi$
$829$	−1.50876e7	−0.762489	−0.381244	+	0.924474i	$0.624504\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.43456e6	−0.271364
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.74393e7	0.855310	0.427655	−	0.903942i	$-0.359340\pi$
$839$	1.74393e7	0.855310	0.427655	+	0.903942i	$0.359340\pi$
$840$	0	0
$841$	1.83956e7	0.896859
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.09999e7	−1.96370
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.16162e6	0.102319
$852$	0	0
$853$	−2.83662e7	−1.33484	−0.667418	−	0.744683i	$-0.732601\pi$
$853$	−2.83662e7	−1.33484	−0.667418	+	0.744683i	$0.732601\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.85290e6	0.179199	0.0895996	−	0.995978i	$-0.471441\pi$
$857$	3.85290e6	0.179199	0.0895996	+	0.995978i	$0.471441\pi$
$858$	0	0
$859$	2.13700e7	0.988145	0.494073	−	0.869421i	$-0.335508\pi$
$859$	2.13700e7	0.988145	0.494073	+	0.869421i	$0.335508\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.18358e7	−1.45509	−0.727544	−	0.686061i	$-0.759338\pi$
$863$	−3.18358e7	−1.45509	−0.727544	+	0.686061i	$0.759338\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5.43410e7	−2.44106
$870$	0	0
$871$	3.06200e7	1.36760
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.26983e7	0.996539	0.498269	−	0.867022i	$-0.333969\pi$
$877$	2.26983e7	0.996539	0.498269	+	0.867022i	$0.333969\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.72028e7	−1.18079	−0.590396	−	0.807114i	$-0.701029\pi$
$881$	−2.72028e7	−1.18079	−0.590396	+	0.807114i	$0.701029\pi$
$882$	0	0
$883$	7.57028e6	0.326746	0.163373	−	0.986564i	$-0.447763\pi$
$883$	7.57028e6	0.326746	0.163373	+	0.986564i	$0.447763\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.48320e6	0.276682	0.138341	−	0.990385i	$-0.455823\pi$
$887$	6.48320e6	0.276682	0.138341	+	0.990385i	$0.455823\pi$
$888$	0	0
$889$	1.62302e7	0.688762
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.58974e6	0.402419
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.50408e7	−1.44602
$900$	0	0
$901$	−1.71003e7	−0.701766
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.32491e7	−1.74566	−0.872829	−	0.488027i	$-0.837717\pi$
$907$	−4.32491e7	−1.74566	−0.872829	+	0.488027i	$0.837717\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.46826e7	−1.78379	−0.891893	−	0.452247i	$-0.850623\pi$
$911$	−4.46826e7	−1.78379	−0.891893	+	0.452247i	$0.850623\pi$
$912$	0	0
$913$	5.67426e7	2.25285
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.76413e7	2.26366
$918$	0	0
$919$	−1.73106e7	−0.676118	−0.338059	−	0.941125i	$-0.609770\pi$
$919$	−1.73106e7	−0.676118	−0.338059	+	0.941125i	$0.609770\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.34471e7	1.67864
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.98156e6	0.189377	0.0946883	−	0.995507i	$-0.469815\pi$
$929$	4.98156e6	0.189377	0.0946883	+	0.995507i	$0.469815\pi$
$930$	0	0
$931$	−1.26605e7	−0.478713
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.25149e7	1.58195	0.790975	−	0.611849i	$-0.209574\pi$
$937$	4.25149e7	1.58195	0.790975	+	0.611849i	$0.209574\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.22797e7	0.820230	0.410115	−	0.912034i	$-0.365489\pi$
$941$	2.22797e7	0.820230	0.410115	+	0.912034i	$0.365489\pi$
$942$	0	0
$943$	1.80516e7	0.661054
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.79405e7	−1.37476	−0.687381	−	0.726297i	$-0.741240\pi$
$947$	−3.79405e7	−1.37476	−0.687381	+	0.726297i	$0.741240\pi$
$948$	0	0
$949$	−5.08109e7	−1.83143
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.55436e7	1.62441	0.812204	−	0.583374i	$-0.198268\pi$
$953$	4.55436e7	1.62441	0.812204	+	0.583374i	$0.198268\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.47071e7	1.56975
$960$	0	0
$961$	2.92991e6	0.102340
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.58055e7	0.543552	0.271776	−	0.962361i	$-0.412389\pi$
$967$	1.58055e7	0.543552	0.271776	+	0.962361i	$0.412389\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.14171e6	−0.0728976	−0.0364488	−	0.999336i	$-0.511605\pi$
$971$	−2.14171e6	−0.0728976	−0.0364488	+	0.999336i	$0.511605\pi$
$972$	0	0
$973$	−1.85563e7	−0.628360
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.92203e7	0.644204	0.322102	−	0.946705i	$-0.395611\pi$
$977$	1.92203e7	0.644204	0.322102	+	0.946705i	$0.395611\pi$
$978$	0	0
$979$	1.71159e7	0.570746
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.85343e7	0.941853	0.470926	−	0.882173i	$-0.343920\pi$
$983$	2.85343e7	0.941853	0.470926	+	0.882173i	$0.343920\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.93463e7	−0.628937
$990$	0	0
$991$	−3.83531e7	−1.24056	−0.620278	−	0.784382i	$-0.712980\pi$
$991$	−3.83531e7	−1.24056	−0.620278	+	0.784382i	$0.712980\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.31478e7	1.05613	0.528064	−	0.849205i	$-0.322918\pi$
$997$	3.31478e7	1.05613	0.528064	+	0.849205i	$0.322918\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.a.r.1.1		2
3.2	odd	2		300.6.a.h.1.1	yes	2
5.2	odd	4		900.6.d.j.649.2		4
5.3	odd	4		900.6.d.j.649.3		4
5.4	even	2		900.6.a.n.1.2		2
15.2	even	4		300.6.d.f.49.1		4
15.8	even	4		300.6.d.f.49.4		4
15.14	odd	2		300.6.a.g.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.6.a.g.1.2	✓	2	15.14	odd	2
300.6.a.h.1.1	yes	2	3.2	odd	2
300.6.d.f.49.1		4	15.2	even	4
300.6.d.f.49.4		4	15.8	even	4
900.6.a.n.1.2		2	5.4	even	2
900.6.a.r.1.1		2	1.1	even	1	trivial
900.6.d.j.649.2		4	5.2	odd	4
900.6.d.j.649.3		4	5.3	odd	4

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-147.745 q^{7} +O(q^{10})\)	\(q-147.745 q^{7} -662.235 q^{11} -629.980 q^{13} -1082.24 q^{17} -2521.20 q^{19} -1802.24 q^{23} -6237.53 q^{29} +5617.75 q^{31} -1199.41 q^{37} -10016.2 q^{41} +10734.6 q^{43} -3803.65 q^{47} +5021.61 q^{49} +15800.9 q^{53} -1734.24 q^{59} +11199.7 q^{61} -48604.8 q^{67} -68965.9 q^{71} +80654.7 q^{73} +97842.0 q^{77} +82056.9 q^{79} -85683.4 q^{83} -25845.6 q^{89} +93076.5 q^{91} +9935.27 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 22 q^{7}+O(q^{10}) \) 2 * q + 22 * q^7	\( 2 q + 22 q^{7} - 372 q^{11} + 10 q^{13} - 1212 q^{17} - 1550 q^{19} - 2652 q^{23} + 1812 q^{29} + 10918 q^{31} + 3316 q^{37} - 19080 q^{41} + 12262 q^{43} + 15252 q^{47} + 17028 q^{49} - 20784 q^{53} - 18708 q^{59} + 35734 q^{61} - 98162 q^{67} - 114120 q^{71} + 109876 q^{73} + 147108 q^{77} + 95536 q^{79} - 3732 q^{83} - 93600 q^{89} + 201710 q^{91} - 91886 q^{97}+O(q^{100}) \) 2 * q + 22 * q^7 - 372 * q^11 + 10 * q^13 - 1212 * q^17 - 1550 * q^19 - 2652 * q^23 + 1812 * q^29 + 10918 * q^31 + 3316 * q^37 - 19080 * q^41 + 12262 * q^43 + 15252 * q^47 + 17028 * q^49 - 20784 * q^53 - 18708 * q^59 + 35734 * q^61 - 98162 * q^67 - 114120 * q^71 + 109876 * q^73 + 147108 * q^77 + 95536 * q^79 - 3732 * q^83 - 93600 * q^89 + 201710 * q^91 - 91886 * q^97

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