LMFDB - Embedded newform 450.3.d.a.251.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [450,3,Mod(251,450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("450.251"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	450.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-4,0,0,-22,0,0,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.2616118962$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	450.251
Dual form			450.3.d.a.251.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -11.0000 q^{7} -2.82843i q^{8} +4.24264i q^{11} +7.00000 q^{13} -15.5563i q^{14} +4.00000 q^{16} -12.7279i q^{17} +29.0000 q^{19} -6.00000 q^{22} -38.1838i q^{23} +9.89949i q^{26} +22.0000 q^{28} -46.6690i q^{29} +29.0000 q^{31} +5.65685i q^{32} +18.0000 q^{34} -56.0000 q^{37} +41.0122i q^{38} -67.8823i q^{41} -5.00000 q^{43} -8.48528i q^{44} +54.0000 q^{46} +63.6396i q^{47} +72.0000 q^{49} -14.0000 q^{52} -67.8823i q^{53} +31.1127i q^{56} +66.0000 q^{58} +29.6985i q^{59} -55.0000 q^{61} +41.0122i q^{62} -8.00000 q^{64} +37.0000 q^{67} +25.4558i q^{68} -33.9411i q^{71} +16.0000 q^{73} -79.1960i q^{74} -58.0000 q^{76} -46.6690i q^{77} +104.000 q^{79} +96.0000 q^{82} -29.6985i q^{83} -7.07107i q^{86} +12.0000 q^{88} +135.765i q^{89} -77.0000 q^{91} +76.3675i q^{92} -90.0000 q^{94} -41.0000 q^{97} +101.823i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 22 q^{7} + 14 q^{13} + 8 q^{16} + 58 q^{19} - 12 q^{22} + 44 q^{28} + 58 q^{31} + 36 q^{34} - 112 q^{37} - 10 q^{43} + 108 q^{46} + 144 q^{49} - 28 q^{52} + 132 q^{58} - 110 q^{61} - 16 q^{64}+ \cdots - 82 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 - 22 * q^7 + 14 * q^13 + 8 * q^16 + 58 * q^19 - 12 * q^22 + 44 * q^28 + 58 * q^31 + 36 * q^34 - 112 * q^37 - 10 * q^43 + 108 * q^46 + 144 * q^49 - 28 * q^52 + 132 * q^58 - 110 * q^61 - 16 * q^64 + 74 * q^67 + 32 * q^73 - 116 * q^76 + 208 * q^79 + 192 * q^82 + 24 * q^88 - 154 * q^91 - 180 * q^94 - 82 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−11.0000	−1.57143	−0.785714	−	0.618590i	$-0.787704\pi$
$7$	−11.0000	−1.57143	−0.785714	+	0.618590i	$0.787704\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	4.24264i	0.385695i	0.981229	+	0.192847i	$0.0617722\pi$
$11$	4.24264i	0.385695i	−0.981229	+	0.192847i	$0.938228\pi$
$12$	0	0
$13$	7.00000	0.538462	0.269231	−	0.963076i	$-0.413231\pi$
$13$	7.00000	0.538462	0.269231	+	0.963076i	$0.413231\pi$
$14$	− 15.5563i	− 1.11117i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 12.7279i	− 0.748701i	−0.927287	−	0.374351i	$-0.877866\pi$
$17$	− 12.7279i	− 0.748701i	0.927287	−	0.374351i	$-0.122134\pi$
$18$	0	0
$19$	29.0000	1.52632	0.763158	−	0.646212i	$-0.223648\pi$
$19$	29.0000	1.52632	0.763158	+	0.646212i	$0.223648\pi$
$20$	0	0
$21$	0	0
$22$	−6.00000	−0.272727
$23$	− 38.1838i	− 1.66016i	−0.557642	−	0.830082i	$-0.688294\pi$
$23$	− 38.1838i	− 1.66016i	0.557642	−	0.830082i	$-0.311706\pi$
$24$	0	0
$25$	0	0
$26$	9.89949i	0.380750i
$27$	0	0
$28$	22.0000	0.785714
$29$	− 46.6690i	− 1.60928i	−0.593765	−	0.804639i	$-0.702359\pi$
$29$	− 46.6690i	− 1.60928i	0.593765	−	0.804639i	$-0.297641\pi$
$30$	0	0
$31$	29.0000	0.935484	0.467742	−	0.883865i	$-0.345068\pi$
$31$	29.0000	0.935484	0.467742	+	0.883865i	$0.345068\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	18.0000	0.529412
$35$	0	0
$36$	0	0
$37$	−56.0000	−1.51351	−0.756757	−	0.653697i	$-0.773217\pi$
$37$	−56.0000	−1.51351	−0.756757	+	0.653697i	$0.773217\pi$
$38$	41.0122i	1.07927i
$39$	0	0
$40$	0	0
$41$	− 67.8823i	− 1.65566i	−0.560976	−	0.827832i	$-0.689574\pi$
$41$	− 67.8823i	− 1.65566i	0.560976	−	0.827832i	$-0.310426\pi$
$42$	0	0
$43$	−5.00000	−0.116279	−0.0581395	−	0.998308i	$-0.518517\pi$
$43$	−5.00000	−0.116279	−0.0581395	+	0.998308i	$0.518517\pi$
$44$	− 8.48528i	− 0.192847i
$45$	0	0
$46$	54.0000	1.17391
$47$	63.6396i	1.35403i	0.735967	+	0.677017i	$0.236728\pi$
$47$	63.6396i	1.35403i	−0.735967	+	0.677017i	$0.763272\pi$
$48$	0	0
$49$	72.0000	1.46939
$50$	0	0
$51$	0	0
$52$	−14.0000	−0.269231
$53$	− 67.8823i	− 1.28080i	−0.768043	−	0.640399i	$-0.778769\pi$
$53$	− 67.8823i	− 1.28080i	0.768043	−	0.640399i	$-0.221231\pi$
$54$	0	0
$55$	0	0
$56$	31.1127i	0.555584i
$57$	0	0
$58$	66.0000	1.13793
$59$	29.6985i	0.503364i	0.967810	+	0.251682i	$0.0809837\pi$
$59$	29.6985i	0.503364i	−0.967810	+	0.251682i	$0.919016\pi$
$60$	0	0
$61$	−55.0000	−0.901639	−0.450820	−	0.892615i	$-0.648868\pi$
$61$	−55.0000	−0.901639	−0.450820	+	0.892615i	$0.648868\pi$
$62$	41.0122i	0.661487i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	37.0000	0.552239	0.276119	−	0.961123i	$-0.410951\pi$
$67$	37.0000	0.552239	0.276119	+	0.961123i	$0.410951\pi$
$68$	25.4558i	0.374351i
$69$	0	0
$70$	0	0
$71$	− 33.9411i	− 0.478044i	−0.971014	−	0.239022i	$-0.923173\pi$
$71$	− 33.9411i	− 0.478044i	0.971014	−	0.239022i	$-0.0768268\pi$
$72$	0	0
$73$	16.0000	0.219178	0.109589	−	0.993977i	$-0.465047\pi$
$73$	16.0000	0.219178	0.109589	+	0.993977i	$0.465047\pi$
$74$	− 79.1960i	− 1.07022i
$75$	0	0
$76$	−58.0000	−0.763158
$77$	− 46.6690i	− 0.606092i
$78$	0	0
$79$	104.000	1.31646	0.658228	−	0.752819i	$-0.271306\pi$
$79$	104.000	1.31646	0.658228	+	0.752819i	$0.271306\pi$
$80$	0	0
$81$	0	0
$82$	96.0000	1.17073
$83$	− 29.6985i	− 0.357813i	−0.983866	−	0.178907i	$-0.942744\pi$
$83$	− 29.6985i	− 0.357813i	0.983866	−	0.178907i	$-0.0572560\pi$
$84$	0	0
$85$	0	0
$86$	− 7.07107i	− 0.0822217i
$87$	0	0
$88$	12.0000	0.136364
$89$	135.765i	1.52544i	0.646727	+	0.762722i	$0.276137\pi$
$89$	135.765i	1.52544i	−0.646727	+	0.762722i	$0.723863\pi$
$90$	0	0
$91$	−77.0000	−0.846154
$92$	76.3675i	0.830082i
$93$	0	0
$94$	−90.0000	−0.957447
$95$	0	0
$96$	0	0
$97$	−41.0000	−0.422680	−0.211340	−	0.977413i	$-0.567783\pi$
$97$	−41.0000	−0.422680	−0.211340	+	0.977413i	$0.567783\pi$
$98$	101.823i	1.03901i
$99$	0	0
$100$	0	0
$101$	33.9411i	0.336051i	0.985783	+	0.168025i	$0.0537391\pi$
$101$	33.9411i	0.336051i	−0.985783	+	0.168025i	$0.946261\pi$
$102$	0	0
$103$	−62.0000	−0.601942	−0.300971	−	0.953633i	$-0.597311\pi$
$103$	−62.0000	−0.601942	−0.300971	+	0.953633i	$0.597311\pi$
$104$	− 19.7990i	− 0.190375i
$105$	0	0
$106$	96.0000	0.905660
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−169.000	−1.55046	−0.775229	−	0.631680i	$-0.782366\pi$
$109$	−169.000	−1.55046	−0.775229	+	0.631680i	$0.782366\pi$
$110$	0	0
$111$	0	0
$112$	−44.0000	−0.392857
$113$	33.9411i	0.300364i	0.988658	+	0.150182i	$0.0479860\pi$
$113$	33.9411i	0.300364i	−0.988658	+	0.150182i	$0.952014\pi$
$114$	0	0
$115$	0	0
$116$	93.3381i	0.804639i
$117$	0	0
$118$	−42.0000	−0.355932
$119$	140.007i	1.17653i
$120$	0	0
$121$	103.000	0.851240
$122$	− 77.7817i	− 0.637555i
$123$	0	0
$124$	−58.0000	−0.467742
$125$	0	0
$126$	0	0
$127$	88.0000	0.692913	0.346457	−	0.938066i	$-0.387385\pi$
$127$	88.0000	0.692913	0.346457	+	0.938066i	$0.387385\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	− 169.706i	− 1.29546i	−0.761869	−	0.647731i	$-0.775718\pi$
$131$	− 169.706i	− 1.29546i	0.761869	−	0.647731i	$-0.224282\pi$
$132$	0	0
$133$	−319.000	−2.39850
$134$	52.3259i	0.390492i
$135$	0	0
$136$	−36.0000	−0.264706
$137$	− 224.860i	− 1.64131i	−0.571421	−	0.820657i	$-0.693608\pi$
$137$	− 224.860i	− 1.64131i	0.571421	−	0.820657i	$-0.306392\pi$
$138$	0	0
$139$	−208.000	−1.49640	−0.748201	−	0.663472i	$-0.769082\pi$
$139$	−208.000	−1.49640	−0.748201	+	0.663472i	$0.769082\pi$
$140$	0	0
$141$	0	0
$142$	48.0000	0.338028
$143$	29.6985i	0.207682i
$144$	0	0
$145$	0	0
$146$	22.6274i	0.154982i
$147$	0	0
$148$	112.000	0.756757
$149$	− 190.919i	− 1.28133i	−0.767819	−	0.640667i	$-0.778658\pi$
$149$	− 190.919i	− 1.28133i	0.767819	−	0.640667i	$-0.221342\pi$
$150$	0	0
$151$	−253.000	−1.67550	−0.837748	−	0.546057i	$-0.816128\pi$
$151$	−253.000	−1.67550	−0.837748	+	0.546057i	$0.816128\pi$
$152$	− 82.0244i	− 0.539634i
$153$	0	0
$154$	66.0000	0.428571
$155$	0	0
$156$	0	0
$157$	127.000	0.808917	0.404459	−	0.914556i	$-0.367460\pi$
$157$	127.000	0.808917	0.404459	+	0.914556i	$0.367460\pi$
$158$	147.078i	0.930875i
$159$	0	0
$160$	0	0
$161$	420.021i	2.60883i
$162$	0	0
$163$	19.0000	0.116564	0.0582822	−	0.998300i	$-0.481438\pi$
$163$	19.0000	0.116564	0.0582822	+	0.998300i	$0.481438\pi$
$164$	135.765i	0.827832i
$165$	0	0
$166$	42.0000	0.253012
$167$	− 135.765i	− 0.812961i	−0.913659	−	0.406481i	$-0.866756\pi$
$167$	− 135.765i	− 0.812961i	0.913659	−	0.406481i	$-0.133244\pi$
$168$	0	0
$169$	−120.000	−0.710059
$170$	0	0
$171$	0	0
$172$	10.0000	0.0581395
$173$	− 216.375i	− 1.25072i	−0.780336	−	0.625360i	$-0.784952\pi$
$173$	− 216.375i	− 1.25072i	0.780336	−	0.625360i	$-0.215048\pi$
$174$	0	0
$175$	0	0
$176$	16.9706i	0.0964237i
$177$	0	0
$178$	−192.000	−1.07865
$179$	97.5807i	0.545144i	0.962135	+	0.272572i	$0.0878743\pi$
$179$	97.5807i	0.545144i	−0.962135	+	0.272572i	$0.912126\pi$
$180$	0	0
$181$	143.000	0.790055	0.395028	−	0.918669i	$-0.370735\pi$
$181$	143.000	0.790055	0.395028	+	0.918669i	$0.370735\pi$
$182$	− 108.894i	− 0.598321i
$183$	0	0
$184$	−108.000	−0.586957
$185$	0	0
$186$	0	0
$187$	54.0000	0.288770
$188$	− 127.279i	− 0.677017i
$189$	0	0
$190$	0	0
$191$	− 72.1249i	− 0.377617i	−0.982014	−	0.188809i	$-0.939537\pi$
$191$	− 72.1249i	− 0.377617i	0.982014	−	0.188809i	$-0.0604626\pi$
$192$	0	0
$193$	79.0000	0.409326	0.204663	−	0.978832i	$-0.434390\pi$
$193$	79.0000	0.409326	0.204663	+	0.978832i	$0.434390\pi$
$194$	− 57.9828i	− 0.298880i
$195$	0	0
$196$	−144.000	−0.734694
$197$	80.6102i	0.409189i	0.978847	+	0.204594i	$0.0655875\pi$
$197$	80.6102i	0.409189i	−0.978847	+	0.204594i	$0.934412\pi$
$198$	0	0
$199$	−85.0000	−0.427136	−0.213568	−	0.976928i	$-0.568508\pi$
$199$	−85.0000	−0.427136	−0.213568	+	0.976928i	$0.568508\pi$
$200$	0	0
$201$	0	0
$202$	−48.0000	−0.237624
$203$	513.360i	2.52886i
$204$	0	0
$205$	0	0
$206$	− 87.6812i	− 0.425637i
$207$	0	0
$208$	28.0000	0.134615
$209$	123.037i	0.588692i
$210$	0	0
$211$	11.0000	0.0521327	0.0260664	−	0.999660i	$-0.491702\pi$
$211$	11.0000	0.0521327	0.0260664	+	0.999660i	$0.491702\pi$
$212$	135.765i	0.640399i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−319.000	−1.47005
$218$	− 239.002i	− 1.09634i
$219$	0	0
$220$	0	0
$221$	− 89.0955i	− 0.403147i
$222$	0	0
$223$	181.000	0.811659	0.405830	−	0.913949i	$-0.366983\pi$
$223$	181.000	0.811659	0.405830	+	0.913949i	$0.366983\pi$
$224$	− 62.2254i	− 0.277792i
$225$	0	0
$226$	−48.0000	−0.212389
$227$	441.235i	1.94376i	0.235467	+	0.971882i	$0.424338\pi$
$227$	441.235i	1.94376i	−0.235467	+	0.971882i	$0.575662\pi$
$228$	0	0
$229$	−7.00000	−0.0305677	−0.0152838	−	0.999883i	$-0.504865\pi$
$229$	−7.00000	−0.0305677	−0.0152838	+	0.999883i	$0.504865\pi$
$230$	0	0
$231$	0	0
$232$	−132.000	−0.568966
$233$	− 33.9411i	− 0.145670i	−0.997344	−	0.0728350i	$-0.976795\pi$
$233$	− 33.9411i	− 0.145670i	0.997344	−	0.0728350i	$-0.0232047\pi$
$234$	0	0
$235$	0	0
$236$	− 59.3970i	− 0.251682i
$237$	0	0
$238$	−198.000	−0.831933
$239$	33.9411i	0.142013i	0.997476	+	0.0710065i	$0.0226211\pi$
$239$	33.9411i	0.142013i	−0.997476	+	0.0710065i	$0.977379\pi$
$240$	0	0
$241$	47.0000	0.195021	0.0975104	−	0.995235i	$-0.468912\pi$
$241$	47.0000	0.195021	0.0975104	+	0.995235i	$0.468912\pi$
$242$	145.664i	0.601917i
$243$	0	0
$244$	110.000	0.450820
$245$	0	0
$246$	0	0
$247$	203.000	0.821862
$248$	− 82.0244i	− 0.330743i
$249$	0	0
$250$	0	0
$251$	33.9411i	0.135224i	0.997712	+	0.0676118i	$0.0215379\pi$
$251$	33.9411i	0.135224i	−0.997712	+	0.0676118i	$0.978462\pi$
$252$	0	0
$253$	162.000	0.640316
$254$	124.451i	0.489964i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 373.352i	− 1.45273i	−0.687308	−	0.726366i	$-0.741208\pi$
$257$	− 373.352i	− 1.45273i	0.687308	−	0.726366i	$-0.258792\pi$
$258$	0	0
$259$	616.000	2.37838
$260$	0	0
$261$	0	0
$262$	240.000	0.916031
$263$	33.9411i	0.129054i	0.997916	+	0.0645269i	$0.0205538\pi$
$263$	33.9411i	0.129054i	−0.997916	+	0.0645269i	$0.979446\pi$
$264$	0	0
$265$	0	0
$266$	− 451.134i	− 1.69599i
$267$	0	0
$268$	−74.0000	−0.276119
$269$	− 462.448i	− 1.71914i	−0.511021	−	0.859568i	$-0.670732\pi$
$269$	− 462.448i	− 1.71914i	0.511021	−	0.859568i	$-0.329268\pi$
$270$	0	0
$271$	200.000	0.738007	0.369004	−	0.929428i	$-0.379699\pi$
$271$	200.000	0.738007	0.369004	+	0.929428i	$0.379699\pi$
$272$	− 50.9117i	− 0.187175i
$273$	0	0
$274$	318.000	1.16058
$275$	0	0
$276$	0	0
$277$	73.0000	0.263538	0.131769	−	0.991280i	$-0.457934\pi$
$277$	73.0000	0.263538	0.131769	+	0.991280i	$0.457934\pi$
$278$	− 294.156i	− 1.05812i
$279$	0	0
$280$	0	0
$281$	224.860i	0.800213i	0.916469	+	0.400107i	$0.131027\pi$
$281$	224.860i	0.800213i	−0.916469	+	0.400107i	$0.868973\pi$
$282$	0	0
$283$	493.000	1.74205	0.871025	−	0.491239i	$-0.163456\pi$
$283$	493.000	1.74205	0.871025	+	0.491239i	$0.163456\pi$
$284$	67.8823i	0.239022i
$285$	0	0
$286$	−42.0000	−0.146853
$287$	746.705i	2.60176i
$288$	0	0
$289$	127.000	0.439446
$290$	0	0
$291$	0	0
$292$	−32.0000	−0.109589
$293$	284.257i	0.970160i	0.874470	+	0.485080i	$0.161210\pi$
$293$	284.257i	0.970160i	−0.874470	+	0.485080i	$0.838790\pi$
$294$	0	0
$295$	0	0
$296$	158.392i	0.535108i
$297$	0	0
$298$	270.000	0.906040
$299$	− 267.286i	− 0.893934i
$300$	0	0
$301$	55.0000	0.182724
$302$	− 357.796i	− 1.18476i
$303$	0	0
$304$	116.000	0.381579
$305$	0	0
$306$	0	0
$307$	235.000	0.765472	0.382736	−	0.923858i	$-0.374982\pi$
$307$	235.000	0.765472	0.382736	+	0.923858i	$0.374982\pi$
$308$	93.3381i	0.303046i
$309$	0	0
$310$	0	0
$311$	− 581.242i	− 1.86894i	−0.356036	−	0.934472i	$-0.615872\pi$
$311$	− 581.242i	− 1.86894i	0.356036	−	0.934472i	$-0.384128\pi$
$312$	0	0
$313$	169.000	0.539936	0.269968	−	0.962869i	$-0.412987\pi$
$313$	169.000	0.539936	0.269968	+	0.962869i	$0.412987\pi$
$314$	179.605i	0.571991i
$315$	0	0
$316$	−208.000	−0.658228
$317$	101.823i	0.321209i	0.987019	+	0.160605i	$0.0513444\pi$
$317$	101.823i	0.321209i	−0.987019	+	0.160605i	$0.948656\pi$
$318$	0	0
$319$	198.000	0.620690
$320$	0	0
$321$	0	0
$322$	−594.000	−1.84472
$323$	− 369.110i	− 1.14275i
$324$	0	0
$325$	0	0
$326$	26.8701i	0.0824235i
$327$	0	0
$328$	−192.000	−0.585366
$329$	− 700.036i	− 2.12777i
$330$	0	0
$331$	176.000	0.531722	0.265861	−	0.964011i	$-0.414344\pi$
$331$	176.000	0.531722	0.265861	+	0.964011i	$0.414344\pi$
$332$	59.3970i	0.178907i
$333$	0	0
$334$	192.000	0.574850
$335$	0	0
$336$	0	0
$337$	−617.000	−1.83086	−0.915430	−	0.402477i	$-0.868149\pi$
$337$	−617.000	−1.83086	−0.915430	+	0.402477i	$0.868149\pi$
$338$	− 169.706i	− 0.502088i
$339$	0	0
$340$	0	0
$341$	123.037i	0.360811i
$342$	0	0
$343$	−253.000	−0.737609
$344$	14.1421i	0.0411109i
$345$	0	0
$346$	306.000	0.884393
$347$	309.713i	0.892544i	0.894897	+	0.446272i	$0.147249\pi$
$347$	309.713i	0.892544i	−0.894897	+	0.446272i	$0.852751\pi$
$348$	0	0
$349$	−328.000	−0.939828	−0.469914	−	0.882712i	$-0.655715\pi$
$349$	−328.000	−0.939828	−0.469914	+	0.882712i	$0.655715\pi$
$350$	0	0
$351$	0	0
$352$	−24.0000	−0.0681818
$353$	156.978i	0.444696i	0.974967	+	0.222348i	$0.0713721\pi$
$353$	156.978i	0.444696i	−0.974967	+	0.222348i	$0.928628\pi$
$354$	0	0
$355$	0	0
$356$	− 271.529i	− 0.762722i
$357$	0	0
$358$	−138.000	−0.385475
$359$	305.470i	0.850892i	0.904984	+	0.425446i	$0.139883\pi$
$359$	305.470i	0.850892i	−0.904984	+	0.425446i	$0.860117\pi$
$360$	0	0
$361$	480.000	1.32964
$362$	202.233i	0.558653i
$363$	0	0
$364$	154.000	0.423077
$365$	0	0
$366$	0	0
$367$	−59.0000	−0.160763	−0.0803815	−	0.996764i	$-0.525614\pi$
$367$	−59.0000	−0.160763	−0.0803815	+	0.996764i	$0.525614\pi$
$368$	− 152.735i	− 0.415041i
$369$	0	0
$370$	0	0
$371$	746.705i	2.01268i
$372$	0	0
$373$	415.000	1.11260	0.556300	−	0.830981i	$-0.312220\pi$
$373$	415.000	1.11260	0.556300	+	0.830981i	$0.312220\pi$
$374$	76.3675i	0.204191i
$375$	0	0
$376$	180.000	0.478723
$377$	− 326.683i	− 0.866534i
$378$	0	0
$379$	−235.000	−0.620053	−0.310026	−	0.950728i	$-0.600338\pi$
$379$	−235.000	−0.620053	−0.310026	+	0.950728i	$0.600338\pi$
$380$	0	0
$381$	0	0
$382$	102.000	0.267016
$383$	− 101.823i	− 0.265857i	−0.991126	−	0.132929i	$-0.957562\pi$
$383$	− 101.823i	− 0.265857i	0.991126	−	0.132929i	$-0.0424381\pi$
$384$	0	0
$385$	0	0
$386$	111.723i	0.289437i
$387$	0	0
$388$	82.0000	0.211340
$389$	− 169.706i	− 0.436261i	−0.975920	−	0.218131i	$-0.930004\pi$
$389$	− 169.706i	− 0.436261i	0.975920	−	0.218131i	$-0.0699959\pi$
$390$	0	0
$391$	−486.000	−1.24297
$392$	− 203.647i	− 0.519507i
$393$	0	0
$394$	−114.000	−0.289340
$395$	0	0
$396$	0	0
$397$	−761.000	−1.91688	−0.958438	−	0.285300i	$-0.907907\pi$
$397$	−761.000	−1.91688	−0.958438	+	0.285300i	$0.907907\pi$
$398$	− 120.208i	− 0.302031i
$399$	0	0
$400$	0	0
$401$	− 475.176i	− 1.18498i	−0.805579	−	0.592488i	$-0.798146\pi$
$401$	− 475.176i	− 1.18498i	0.805579	−	0.592488i	$-0.201854\pi$
$402$	0	0
$403$	203.000	0.503722
$404$	− 67.8823i	− 0.168025i
$405$	0	0
$406$	−726.000	−1.78818
$407$	− 237.588i	− 0.583754i
$408$	0	0
$409$	113.000	0.276284	0.138142	−	0.990412i	$-0.455887\pi$
$409$	113.000	0.276284	0.138142	+	0.990412i	$0.455887\pi$
$410$	0	0
$411$	0	0
$412$	124.000	0.300971
$413$	− 326.683i	− 0.791001i
$414$	0	0
$415$	0	0
$416$	39.5980i	0.0951875i
$417$	0	0
$418$	−174.000	−0.416268
$419$	33.9411i	0.0810051i	0.999179	+	0.0405025i	$0.0128959\pi$
$419$	33.9411i	0.0810051i	−0.999179	+	0.0405025i	$0.987104\pi$
$420$	0	0
$421$	−40.0000	−0.0950119	−0.0475059	−	0.998871i	$-0.515127\pi$
$421$	−40.0000	−0.0950119	−0.0475059	+	0.998871i	$0.515127\pi$
$422$	15.5563i	0.0368634i
$423$	0	0
$424$	−192.000	−0.452830
$425$	0	0
$426$	0	0
$427$	605.000	1.41686
$428$	0	0
$429$	0	0
$430$	0	0
$431$	470.933i	1.09265i	0.837573	+	0.546326i	$0.183974\pi$
$431$	470.933i	1.09265i	−0.837573	+	0.546326i	$0.816026\pi$
$432$	0	0
$433$	−407.000	−0.939954	−0.469977	−	0.882679i	$-0.655738\pi$
$433$	−407.000	−0.939954	−0.469977	+	0.882679i	$0.655738\pi$
$434$	− 451.134i	− 1.03948i
$435$	0	0
$436$	338.000	0.775229
$437$	− 1107.33i	− 2.53393i
$438$	0	0
$439$	461.000	1.05011	0.525057	−	0.851067i	$-0.324044\pi$
$439$	461.000	1.05011	0.525057	+	0.851067i	$0.324044\pi$
$440$	0	0
$441$	0	0
$442$	126.000	0.285068
$443$	746.705i	1.68556i	0.538255	+	0.842782i	$0.319084\pi$
$443$	746.705i	1.68556i	−0.538255	+	0.842782i	$0.680916\pi$
$444$	0	0
$445$	0	0
$446$	255.973i	0.573930i
$447$	0	0
$448$	88.0000	0.196429
$449$	101.823i	0.226778i	0.993551	+	0.113389i	$0.0361707\pi$
$449$	101.823i	0.226778i	−0.993551	+	0.113389i	$0.963829\pi$
$450$	0	0
$451$	288.000	0.638581
$452$	− 67.8823i	− 0.150182i
$453$	0	0
$454$	−624.000	−1.37445
$455$	0	0
$456$	0	0
$457$	850.000	1.85996	0.929978	−	0.367615i	$-0.119826\pi$
$457$	850.000	1.85996	0.929978	+	0.367615i	$0.119826\pi$
$458$	− 9.89949i	− 0.0216146i
$459$	0	0
$460$	0	0
$461$	576.999i	1.25163i	0.779974	+	0.625813i	$0.215232\pi$
$461$	576.999i	1.25163i	−0.779974	+	0.625813i	$0.784768\pi$
$462$	0	0
$463$	−296.000	−0.639309	−0.319654	−	0.947534i	$-0.603567\pi$
$463$	−296.000	−0.639309	−0.319654	+	0.947534i	$0.603567\pi$
$464$	− 186.676i	− 0.402319i
$465$	0	0
$466$	48.0000	0.103004
$467$	377.595i	0.808555i	0.914636	+	0.404277i	$0.132477\pi$
$467$	377.595i	0.808555i	−0.914636	+	0.404277i	$0.867523\pi$
$468$	0	0
$469$	−407.000	−0.867804
$470$	0	0
$471$	0	0
$472$	84.0000	0.177966
$473$	− 21.2132i	− 0.0448482i
$474$	0	0
$475$	0	0
$476$	− 280.014i	− 0.588265i
$477$	0	0
$478$	−48.0000	−0.100418
$479$	683.065i	1.42602i	0.701152	+	0.713012i	$0.252669\pi$
$479$	683.065i	1.42602i	−0.701152	+	0.713012i	$0.747331\pi$
$480$	0	0
$481$	−392.000	−0.814969
$482$	66.4680i	0.137900i
$483$	0	0
$484$	−206.000	−0.425620
$485$	0	0
$486$	0	0
$487$	−221.000	−0.453799	−0.226899	−	0.973918i	$-0.572859\pi$
$487$	−221.000	−0.453799	−0.226899	+	0.973918i	$0.572859\pi$
$488$	155.563i	0.318778i
$489$	0	0
$490$	0	0
$491$	− 916.410i	− 1.86642i	−0.359336	−	0.933208i	$-0.616997\pi$
$491$	− 916.410i	− 1.86642i	0.359336	−	0.933208i	$-0.383003\pi$
$492$	0	0
$493$	−594.000	−1.20487
$494$	287.085i	0.581144i
$495$	0	0
$496$	116.000	0.233871
$497$	373.352i	0.751212i
$498$	0	0
$499$	−661.000	−1.32465	−0.662325	−	0.749217i	$-0.730430\pi$
$499$	−661.000	−1.32465	−0.662325	+	0.749217i	$0.730430\pi$
$500$	0	0
$501$	0	0
$502$	−48.0000	−0.0956175
$503$	− 67.8823i	− 0.134955i	−0.997721	−	0.0674774i	$-0.978505\pi$
$503$	− 67.8823i	− 0.134955i	0.997721	−	0.0674774i	$-0.0214951\pi$
$504$	0	0
$505$	0	0
$506$	229.103i	0.452772i
$507$	0	0
$508$	−176.000	−0.346457
$509$	835.800i	1.64204i	0.570897	+	0.821022i	$0.306596\pi$
$509$	835.800i	1.64204i	−0.570897	+	0.821022i	$0.693404\pi$
$510$	0	0
$511$	−176.000	−0.344423
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	528.000	1.02724
$515$	0	0
$516$	0	0
$517$	−270.000	−0.522244
$518$	871.156i	1.68177i
$519$	0	0
$520$	0	0
$521$	224.860i	0.431593i	0.976438	+	0.215797i	$0.0692348\pi$
$521$	224.860i	0.431593i	−0.976438	+	0.215797i	$0.930765\pi$
$522$	0	0
$523$	−149.000	−0.284895	−0.142447	−	0.989802i	$-0.545497\pi$
$523$	−149.000	−0.284895	−0.142447	+	0.989802i	$0.545497\pi$
$524$	339.411i	0.647731i
$525$	0	0
$526$	−48.0000	−0.0912548
$527$	− 369.110i	− 0.700398i
$528$	0	0
$529$	−929.000	−1.75614
$530$	0	0
$531$	0	0
$532$	638.000	1.19925
$533$	− 475.176i	− 0.891512i
$534$	0	0
$535$	0	0
$536$	− 104.652i	− 0.195246i
$537$	0	0
$538$	654.000	1.21561
$539$	305.470i	0.566735i
$540$	0	0
$541$	17.0000	0.0314233	0.0157116	−	0.999877i	$-0.494999\pi$
$541$	17.0000	0.0314233	0.0157116	+	0.999877i	$0.494999\pi$
$542$	282.843i	0.521850i
$543$	0	0
$544$	72.0000	0.132353
$545$	0	0
$546$	0	0
$547$	454.000	0.829982	0.414991	−	0.909826i	$-0.363785\pi$
$547$	454.000	0.829982	0.414991	+	0.909826i	$0.363785\pi$
$548$	449.720i	0.820657i
$549$	0	0
$550$	0	0
$551$	− 1353.40i	− 2.45627i
$552$	0	0
$553$	−1144.00	−2.06872
$554$	103.238i	0.186349i
$555$	0	0
$556$	416.000	0.748201
$557$	− 80.6102i	− 0.144722i	−0.997379	−	0.0723610i	$-0.976947\pi$
$557$	− 80.6102i	− 0.144722i	0.997379	−	0.0723610i	$-0.0230534\pi$
$558$	0	0
$559$	−35.0000	−0.0626118
$560$	0	0
$561$	0	0
$562$	−318.000	−0.565836
$563$	674.580i	1.19819i	0.800679	+	0.599094i	$0.204472\pi$
$563$	674.580i	1.19819i	−0.800679	+	0.599094i	$0.795528\pi$
$564$	0	0
$565$	0	0
$566$	697.207i	1.23181i
$567$	0	0
$568$	−96.0000	−0.169014
$569$	12.7279i	0.0223689i	0.999937	+	0.0111845i	$0.00356020\pi$
$569$	12.7279i	0.0223689i	−0.999937	+	0.0111845i	$0.996440\pi$
$570$	0	0
$571$	−787.000	−1.37828	−0.689142	−	0.724626i	$-0.742012\pi$
$571$	−787.000	−1.37828	−0.689142	+	0.724626i	$0.742012\pi$
$572$	− 59.3970i	− 0.103841i
$573$	0	0
$574$	−1056.00	−1.83972
$575$	0	0
$576$	0	0
$577$	697.000	1.20797	0.603986	−	0.796995i	$-0.293578\pi$
$577$	697.000	1.20797	0.603986	+	0.796995i	$0.293578\pi$
$578$	179.605i	0.310736i
$579$	0	0
$580$	0	0
$581$	326.683i	0.562278i
$582$	0	0
$583$	288.000	0.493997
$584$	− 45.2548i	− 0.0774912i
$585$	0	0
$586$	−402.000	−0.686007
$587$	33.9411i	0.0578213i	0.999582	+	0.0289107i	$0.00920383\pi$
$587$	33.9411i	0.0578213i	−0.999582	+	0.0289107i	$0.990796\pi$
$588$	0	0
$589$	841.000	1.42784
$590$	0	0
$591$	0	0
$592$	−224.000	−0.378378
$593$	− 441.235i	− 0.744072i	−0.928218	−	0.372036i	$-0.878660\pi$
$593$	− 441.235i	− 0.744072i	0.928218	−	0.372036i	$-0.121340\pi$
$594$	0	0
$595$	0	0
$596$	381.838i	0.640667i
$597$	0	0
$598$	378.000	0.632107
$599$	− 33.9411i	− 0.0566630i	−0.999599	−	0.0283315i	$-0.990981\pi$
$599$	− 33.9411i	− 0.0566630i	0.999599	−	0.0283315i	$-0.00901940\pi$
$600$	0	0
$601$	−607.000	−1.00998	−0.504992	−	0.863124i	$-0.668504\pi$
$601$	−607.000	−1.00998	−0.504992	+	0.863124i	$0.668504\pi$
$602$	77.7817i	0.129206i
$603$	0	0
$604$	506.000	0.837748
$605$	0	0
$606$	0	0
$607$	568.000	0.935750	0.467875	−	0.883795i	$-0.345020\pi$
$607$	568.000	0.935750	0.467875	+	0.883795i	$0.345020\pi$
$608$	164.049i	0.269817i
$609$	0	0
$610$	0	0
$611$	445.477i	0.729095i
$612$	0	0
$613$	232.000	0.378467	0.189233	−	0.981932i	$-0.439400\pi$
$613$	232.000	0.378467	0.189233	+	0.981932i	$0.439400\pi$
$614$	332.340i	0.541271i
$615$	0	0
$616$	−132.000	−0.214286
$617$	814.587i	1.32024i	0.751161	+	0.660119i	$0.229494\pi$
$617$	814.587i	1.32024i	−0.751161	+	0.660119i	$0.770506\pi$
$618$	0	0
$619$	−109.000	−0.176090	−0.0880452	−	0.996116i	$-0.528062\pi$
$619$	−109.000	−0.176090	−0.0880452	+	0.996116i	$0.528062\pi$
$620$	0	0
$621$	0	0
$622$	822.000	1.32154
$623$	− 1493.41i	− 2.39713i
$624$	0	0
$625$	0	0
$626$	239.002i	0.381792i
$627$	0	0
$628$	−254.000	−0.404459
$629$	712.764i	1.13317i
$630$	0	0
$631$	−955.000	−1.51347	−0.756735	−	0.653721i	$-0.773207\pi$
$631$	−955.000	−1.51347	−0.756735	+	0.653721i	$0.773207\pi$
$632$	− 294.156i	− 0.465437i
$633$	0	0
$634$	−144.000	−0.227129
$635$	0	0
$636$	0	0
$637$	504.000	0.791209
$638$	280.014i	0.438894i
$639$	0	0
$640$	0	0
$641$	− 67.8823i	− 0.105901i	−0.998597	−	0.0529503i	$-0.983138\pi$
$641$	− 67.8823i	− 0.105901i	0.998597	−	0.0529503i	$-0.0168625\pi$
$642$	0	0
$643$	−368.000	−0.572317	−0.286159	−	0.958182i	$-0.592378\pi$
$643$	−368.000	−0.572317	−0.286159	+	0.958182i	$0.592378\pi$
$644$	− 840.043i	− 1.30441i
$645$	0	0
$646$	522.000	0.808050
$647$	− 543.058i	− 0.839348i	−0.907675	−	0.419674i	$-0.862144\pi$
$647$	− 543.058i	− 0.839348i	0.907675	−	0.419674i	$-0.137856\pi$
$648$	0	0
$649$	−126.000	−0.194145
$650$	0	0
$651$	0	0
$652$	−38.0000	−0.0582822
$653$	598.212i	0.916099i	0.888927	+	0.458049i	$0.151452\pi$
$653$	598.212i	0.916099i	−0.888927	+	0.458049i	$0.848548\pi$
$654$	0	0
$655$	0	0
$656$	− 271.529i	− 0.413916i
$657$	0	0
$658$	990.000	1.50456
$659$	203.647i	0.309024i	0.987991	+	0.154512i	$0.0493805\pi$
$659$	203.647i	0.309024i	−0.987991	+	0.154512i	$0.950619\pi$
$660$	0	0
$661$	872.000	1.31921	0.659607	−	0.751611i	$-0.270723\pi$
$661$	872.000	1.31921	0.659607	+	0.751611i	$0.270723\pi$
$662$	248.902i	0.375984i
$663$	0	0
$664$	−84.0000	−0.126506
$665$	0	0
$666$	0	0
$667$	−1782.00	−2.67166
$668$	271.529i	0.406481i
$669$	0	0
$670$	0	0
$671$	− 233.345i	− 0.347757i
$672$	0	0
$673$	1120.00	1.66419	0.832095	−	0.554633i	$-0.187141\pi$
$673$	1120.00	1.66419	0.832095	+	0.554633i	$0.187141\pi$
$674$	− 872.570i	− 1.29461i
$675$	0	0
$676$	240.000	0.355030
$677$	− 700.036i	− 1.03403i	−0.855977	−	0.517013i	$-0.827044\pi$
$677$	− 700.036i	− 1.03403i	0.855977	−	0.517013i	$-0.172956\pi$
$678$	0	0
$679$	451.000	0.664212
$680$	0	0
$681$	0	0
$682$	−174.000	−0.255132
$683$	712.764i	1.04358i	0.853075	+	0.521789i	$0.174735\pi$
$683$	712.764i	1.04358i	−0.853075	+	0.521789i	$0.825265\pi$
$684$	0	0
$685$	0	0
$686$	− 357.796i	− 0.521569i
$687$	0	0
$688$	−20.0000	−0.0290698
$689$	− 475.176i	− 0.689660i
$690$	0	0
$691$	410.000	0.593343	0.296671	−	0.954980i	$-0.404123\pi$
$691$	410.000	0.593343	0.296671	+	0.954980i	$0.404123\pi$
$692$	432.749i	0.625360i
$693$	0	0
$694$	−438.000	−0.631124
$695$	0	0
$696$	0	0
$697$	−864.000	−1.23960
$698$	− 463.862i	− 0.664559i
$699$	0	0
$700$	0	0
$701$	− 190.919i	− 0.272352i	−0.990685	−	0.136176i	$-0.956519\pi$
$701$	− 190.919i	− 0.272352i	0.990685	−	0.136176i	$-0.0434813\pi$
$702$	0	0
$703$	−1624.00	−2.31010
$704$	− 33.9411i	− 0.0482118i
$705$	0	0
$706$	−222.000	−0.314448
$707$	− 373.352i	− 0.528080i
$708$	0	0
$709$	1103.00	1.55571	0.777856	−	0.628442i	$-0.216307\pi$
$709$	1103.00	1.55571	0.777856	+	0.628442i	$0.216307\pi$
$710$	0	0
$711$	0	0
$712$	384.000	0.539326
$713$	− 1107.33i	− 1.55306i
$714$	0	0
$715$	0	0
$716$	− 195.161i	− 0.272572i
$717$	0	0
$718$	−432.000	−0.601671
$719$	377.595i	0.525167i	0.964909	+	0.262583i	$0.0845745\pi$
$719$	377.595i	0.525167i	−0.964909	+	0.262583i	$0.915425\pi$
$720$	0	0
$721$	682.000	0.945908
$722$	678.823i	0.940197i
$723$	0	0
$724$	−286.000	−0.395028
$725$	0	0
$726$	0	0
$727$	−587.000	−0.807428	−0.403714	−	0.914885i	$-0.632281\pi$
$727$	−587.000	−0.807428	−0.403714	+	0.914885i	$0.632281\pi$
$728$	217.789i	0.299161i
$729$	0	0
$730$	0	0
$731$	63.6396i	0.0870583i
$732$	0	0
$733$	−296.000	−0.403820	−0.201910	−	0.979404i	$-0.564715\pi$
$733$	−296.000	−0.403820	−0.201910	+	0.979404i	$0.564715\pi$
$734$	− 83.4386i	− 0.113677i
$735$	0	0
$736$	216.000	0.293478
$737$	156.978i	0.212996i
$738$	0	0
$739$	−304.000	−0.411367	−0.205683	−	0.978619i	$-0.565942\pi$
$739$	−304.000	−0.411367	−0.205683	+	0.978619i	$0.565942\pi$
$740$	0	0
$741$	0	0
$742$	−1056.00	−1.42318
$743$	− 848.528i	− 1.14203i	−0.820940	−	0.571015i	$-0.806550\pi$
$743$	− 848.528i	− 1.14203i	0.820940	−	0.571015i	$-0.193450\pi$
$744$	0	0
$745$	0	0
$746$	586.899i	0.786727i
$747$	0	0
$748$	−108.000	−0.144385
$749$	0	0
$750$	0	0
$751$	728.000	0.969374	0.484687	−	0.874688i	$-0.338933\pi$
$751$	728.000	0.969374	0.484687	+	0.874688i	$0.338933\pi$
$752$	254.558i	0.338509i
$753$	0	0
$754$	462.000	0.612732
$755$	0	0
$756$	0	0
$757$	−815.000	−1.07662	−0.538309	−	0.842747i	$-0.680937\pi$
$757$	−815.000	−1.07662	−0.538309	+	0.842747i	$0.680937\pi$
$758$	− 332.340i	− 0.438444i
$759$	0	0
$760$	0	0
$761$	− 203.647i	− 0.267604i	−0.991008	−	0.133802i	$-0.957281\pi$
$761$	− 203.647i	− 0.267604i	0.991008	−	0.133802i	$-0.0427186\pi$
$762$	0	0
$763$	1859.00	2.43644
$764$	144.250i	0.188809i
$765$	0	0
$766$	144.000	0.187990
$767$	207.889i	0.271042i
$768$	0	0
$769$	785.000	1.02081	0.510403	−	0.859935i	$-0.329496\pi$
$769$	785.000	1.02081	0.510403	+	0.859935i	$0.329496\pi$
$770$	0	0
$771$	0	0
$772$	−158.000	−0.204663
$773$	373.352i	0.482991i	0.970402	+	0.241496i	$0.0776380\pi$
$773$	373.352i	0.482991i	−0.970402	+	0.241496i	$0.922362\pi$
$774$	0	0
$775$	0	0
$776$	115.966i	0.149440i
$777$	0	0
$778$	240.000	0.308483
$779$	− 1968.59i	− 2.52707i
$780$	0	0
$781$	144.000	0.184379
$782$	− 687.308i	− 0.878910i
$783$	0	0
$784$	288.000	0.367347
$785$	0	0
$786$	0	0
$787$	925.000	1.17535	0.587675	−	0.809097i	$-0.300043\pi$
$787$	925.000	1.17535	0.587675	+	0.809097i	$0.300043\pi$
$788$	− 161.220i	− 0.204594i
$789$	0	0
$790$	0	0
$791$	− 373.352i	− 0.472000i
$792$	0	0
$793$	−385.000	−0.485498
$794$	− 1076.22i	− 1.35544i
$795$	0	0
$796$	170.000	0.213568
$797$	339.411i	0.425861i	0.977067	+	0.212931i	$0.0683008\pi$
$797$	339.411i	0.425861i	−0.977067	+	0.212931i	$0.931699\pi$
$798$	0	0
$799$	810.000	1.01377
$800$	0	0
$801$	0	0
$802$	672.000	0.837905
$803$	67.8823i	0.0845358i
$804$	0	0
$805$	0	0
$806$	287.085i	0.356185i
$807$	0	0
$808$	96.0000	0.118812
$809$	− 780.646i	− 0.964952i	−0.875909	−	0.482476i	$-0.839738\pi$
$809$	− 780.646i	− 0.964952i	0.875909	−	0.482476i	$-0.160262\pi$
$810$	0	0
$811$	635.000	0.782984	0.391492	−	0.920182i	$-0.371959\pi$
$811$	635.000	0.782984	0.391492	+	0.920182i	$0.371959\pi$
$812$	− 1026.72i	− 1.26443i
$813$	0	0
$814$	336.000	0.412776
$815$	0	0
$816$	0	0
$817$	−145.000	−0.177479
$818$	159.806i	0.195362i
$819$	0	0
$820$	0	0
$821$	− 861.256i	− 1.04903i	−0.851400	−	0.524516i	$-0.824246\pi$
$821$	− 861.256i	− 1.04903i	0.851400	−	0.524516i	$-0.175754\pi$
$822$	0	0
$823$	763.000	0.927096	0.463548	−	0.886072i	$-0.346576\pi$
$823$	763.000	0.927096	0.463548	+	0.886072i	$0.346576\pi$
$824$	175.362i	0.212819i
$825$	0	0
$826$	462.000	0.559322
$827$	− 407.294i	− 0.492495i	−0.969207	−	0.246248i	$-0.920802\pi$
$827$	− 407.294i	− 0.492495i	0.969207	−	0.246248i	$-0.0791976\pi$
$828$	0	0
$829$	182.000	0.219542	0.109771	−	0.993957i	$-0.464988\pi$
$829$	182.000	0.219542	0.109771	+	0.993957i	$0.464988\pi$
$830$	0	0
$831$	0	0
$832$	−56.0000	−0.0673077
$833$	− 916.410i	− 1.10013i
$834$	0	0
$835$	0	0
$836$	− 246.073i	− 0.294346i
$837$	0	0
$838$	−48.0000	−0.0572792
$839$	− 750.947i	− 0.895051i	−0.894271	−	0.447525i	$-0.852305\pi$
$839$	− 750.947i	− 0.895051i	0.894271	−	0.447525i	$-0.147695\pi$
$840$	0	0
$841$	−1337.00	−1.58977
$842$	− 56.5685i	− 0.0671835i
$843$	0	0
$844$	−22.0000	−0.0260664
$845$	0	0
$846$	0	0
$847$	−1133.00	−1.33766
$848$	− 271.529i	− 0.320199i
$849$	0	0
$850$	0	0
$851$	2138.29i	2.51268i
$852$	0	0
$853$	409.000	0.479484	0.239742	−	0.970837i	$-0.422937\pi$
$853$	409.000	0.479484	0.239742	+	0.970837i	$0.422937\pi$
$854$	855.599i	1.00187i
$855$	0	0
$856$	0	0
$857$	− 67.8823i	− 0.0792092i	−0.999215	−	0.0396046i	$-0.987390\pi$
$857$	− 67.8823i	− 0.0792092i	0.999215	−	0.0396046i	$-0.0126098\pi$
$858$	0	0
$859$	458.000	0.533178	0.266589	−	0.963810i	$-0.414103\pi$
$859$	458.000	0.533178	0.266589	+	0.963810i	$0.414103\pi$
$860$	0	0
$861$	0	0
$862$	−666.000	−0.772622
$863$	− 1255.82i	− 1.45518i	−0.686011	−	0.727591i	$-0.740640\pi$
$863$	− 1255.82i	− 1.45518i	0.686011	−	0.727591i	$-0.259360\pi$
$864$	0	0
$865$	0	0
$866$	− 575.585i	− 0.664648i
$867$	0	0
$868$	638.000	0.735023
$869$	441.235i	0.507750i
$870$	0	0
$871$	259.000	0.297359
$872$	478.004i	0.548170i
$873$	0	0
$874$	1566.00	1.79176
$875$	0	0
$876$	0	0
$877$	895.000	1.02052	0.510262	−	0.860019i	$-0.329548\pi$
$877$	895.000	1.02052	0.510262	+	0.860019i	$0.329548\pi$
$878$	651.952i	0.742543i
$879$	0	0
$880$	0	0
$881$	− 916.410i	− 1.04019i	−0.854107	−	0.520097i	$-0.825896\pi$
$881$	− 916.410i	− 1.04019i	0.854107	−	0.520097i	$-0.174104\pi$
$882$	0	0
$883$	781.000	0.884485	0.442242	−	0.896896i	$-0.354183\pi$
$883$	781.000	0.884485	0.442242	+	0.896896i	$0.354183\pi$
$884$	178.191i	0.201573i
$885$	0	0
$886$	−1056.00	−1.19187
$887$	538.815i	0.607458i	0.952758	+	0.303729i	$0.0982318\pi$
$887$	538.815i	0.607458i	−0.952758	+	0.303729i	$0.901768\pi$
$888$	0	0
$889$	−968.000	−1.08886
$890$	0	0
$891$	0	0
$892$	−362.000	−0.405830
$893$	1845.55i	2.06668i
$894$	0	0
$895$	0	0
$896$	124.451i	0.138896i
$897$	0	0
$898$	−144.000	−0.160356
$899$	− 1353.40i	− 1.50545i
$900$	0	0
$901$	−864.000	−0.958935
$902$	407.294i	0.451545i
$903$	0	0
$904$	96.0000	0.106195
$905$	0	0
$906$	0	0
$907$	688.000	0.758545	0.379272	−	0.925285i	$-0.376174\pi$
$907$	688.000	0.758545	0.379272	+	0.925285i	$0.376174\pi$
$908$	− 882.469i	− 0.971882i
$909$	0	0
$910$	0	0
$911$	1022.48i	1.12237i	0.827691	+	0.561184i	$0.189654\pi$
$911$	1022.48i	1.12237i	−0.827691	+	0.561184i	$0.810346\pi$
$912$	0	0
$913$	126.000	0.138007
$914$	1202.08i	1.31519i
$915$	0	0
$916$	14.0000	0.0152838
$917$	1866.76i	2.03573i
$918$	0	0
$919$	35.0000	0.0380849	0.0190424	−	0.999819i	$-0.493938\pi$
$919$	35.0000	0.0380849	0.0190424	+	0.999819i	$0.493938\pi$
$920$	0	0
$921$	0	0
$922$	−816.000	−0.885033
$923$	− 237.588i	− 0.257408i
$924$	0	0
$925$	0	0
$926$	− 418.607i	− 0.452060i
$927$	0	0
$928$	264.000	0.284483
$929$	− 284.257i	− 0.305982i	−0.988228	−	0.152991i	$-0.951110\pi$
$929$	− 284.257i	− 0.305982i	0.988228	−	0.152991i	$-0.0488905\pi$
$930$	0	0
$931$	2088.00	2.24275
$932$	67.8823i	0.0728350i
$933$	0	0
$934$	−534.000	−0.571734
$935$	0	0
$936$	0	0
$937$	−377.000	−0.402348	−0.201174	−	0.979556i	$-0.564476\pi$
$937$	−377.000	−0.402348	−0.201174	+	0.979556i	$0.564476\pi$
$938$	− 575.585i	− 0.613630i
$939$	0	0
$940$	0	0
$941$	− 793.374i	− 0.843118i	−0.906801	−	0.421559i	$-0.861483\pi$
$941$	− 793.374i	− 0.843118i	0.906801	−	0.421559i	$-0.138517\pi$
$942$	0	0
$943$	−2592.00	−2.74867
$944$	118.794i	0.125841i
$945$	0	0
$946$	30.0000	0.0317125
$947$	1013.99i	1.07074i	0.844618	+	0.535370i	$0.179828\pi$
$947$	1013.99i	1.07074i	−0.844618	+	0.535370i	$0.820172\pi$
$948$	0	0
$949$	112.000	0.118019
$950$	0	0
$951$	0	0
$952$	396.000	0.415966
$953$	1120.06i	1.17530i	0.809117	+	0.587648i	$0.199946\pi$
$953$	1120.06i	1.17530i	−0.809117	+	0.587648i	$0.800054\pi$
$954$	0	0
$955$	0	0
$956$	− 67.8823i	− 0.0710065i
$957$	0	0
$958$	−966.000	−1.00835
$959$	2473.46i	2.57921i
$960$	0	0
$961$	−120.000	−0.124870
$962$	− 554.372i	− 0.576270i
$963$	0	0
$964$	−94.0000	−0.0975104
$965$	0	0
$966$	0	0
$967$	−1160.00	−1.19959	−0.599793	−	0.800155i	$-0.704750\pi$
$967$	−1160.00	−1.19959	−0.599793	+	0.800155i	$0.704750\pi$
$968$	− 291.328i	− 0.300959i
$969$	0	0
$970$	0	0
$971$	− 649.124i	− 0.668511i	−0.942483	−	0.334255i	$-0.891515\pi$
$971$	− 649.124i	− 0.668511i	0.942483	−	0.334255i	$-0.108485\pi$
$972$	0	0
$973$	2288.00	2.35149
$974$	− 312.541i	− 0.320884i
$975$	0	0
$976$	−220.000	−0.225410
$977$	632.153i	0.647035i	0.946222	+	0.323518i	$0.104866\pi$
$977$	632.153i	0.647035i	−0.946222	+	0.323518i	$0.895134\pi$
$978$	0	0
$979$	−576.000	−0.588355
$980$	0	0
$981$	0	0
$982$	1296.00	1.31976
$983$	814.587i	0.828674i	0.910123	+	0.414337i	$0.135987\pi$
$983$	814.587i	0.828674i	−0.910123	+	0.414337i	$0.864013\pi$
$984$	0	0
$985$	0	0
$986$	− 840.043i	− 0.851970i
$987$	0	0
$988$	−406.000	−0.410931
$989$	190.919i	0.193042i
$990$	0	0
$991$	−157.000	−0.158426	−0.0792129	−	0.996858i	$-0.525241\pi$
$991$	−157.000	−0.158426	−0.0792129	+	0.996858i	$0.525241\pi$
$992$	164.049i	0.165372i
$993$	0	0
$994$	−528.000	−0.531187
$995$	0	0
$996$	0	0
$997$	616.000	0.617854	0.308927	−	0.951086i	$-0.400030\pi$
$997$	616.000	0.617854	0.308927	+	0.951086i	$0.400030\pi$
$998$	− 934.795i	− 0.936669i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.3.d.a.251.2	yes	2
3.2	odd	2	inner	450.3.d.a.251.1	✓	2
4.3	odd	2		3600.3.l.k.1601.1		2
5.2	odd	4		450.3.b.a.449.1		4
5.3	odd	4		450.3.b.a.449.4		4
5.4	even	2		450.3.d.g.251.1	yes	2
12.11	even	2		3600.3.l.k.1601.2		2
15.2	even	4		450.3.b.a.449.3		4
15.8	even	4		450.3.b.a.449.2		4
15.14	odd	2		450.3.d.g.251.2	yes	2
20.3	even	4		3600.3.c.f.449.1		4
20.7	even	4		3600.3.c.f.449.3		4
20.19	odd	2		3600.3.l.a.1601.1		2
60.23	odd	4		3600.3.c.f.449.2		4
60.47	odd	4		3600.3.c.f.449.4		4
60.59	even	2		3600.3.l.a.1601.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.3.b.a.449.1		4	5.2	odd	4
450.3.b.a.449.2		4	15.8	even	4
450.3.b.a.449.3		4	15.2	even	4
450.3.b.a.449.4		4	5.3	odd	4
450.3.d.a.251.1	✓	2	3.2	odd	2	inner
450.3.d.a.251.2	yes	2	1.1	even	1	trivial
450.3.d.g.251.1	yes	2	5.4	even	2
450.3.d.g.251.2	yes	2	15.14	odd	2
3600.3.c.f.449.1		4	20.3	even	4
3600.3.c.f.449.2		4	60.23	odd	4
3600.3.c.f.449.3		4	20.7	even	4
3600.3.c.f.449.4		4	60.47	odd	4
3600.3.l.a.1601.1		2	20.19	odd	2
3600.3.l.a.1601.2		2	60.59	even	2
3600.3.l.k.1601.1		2	4.3	odd	2
3600.3.l.k.1601.2		2	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	450.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -11.0000 q^{7} -2.82843i q^{8} +4.24264i q^{11} +7.00000 q^{13} -15.5563i q^{14} +4.00000 q^{16} -12.7279i q^{17} +29.0000 q^{19} -6.00000 q^{22} -38.1838i q^{23} +9.89949i q^{26} +22.0000 q^{28} -46.6690i q^{29} +29.0000 q^{31} +5.65685i q^{32} +18.0000 q^{34} -56.0000 q^{37} +41.0122i q^{38} -67.8823i q^{41} -5.00000 q^{43} -8.48528i q^{44} +54.0000 q^{46} +63.6396i q^{47} +72.0000 q^{49} -14.0000 q^{52} -67.8823i q^{53} +31.1127i q^{56} +66.0000 q^{58} +29.6985i q^{59} -55.0000 q^{61} +41.0122i q^{62} -8.00000 q^{64} +37.0000 q^{67} +25.4558i q^{68} -33.9411i q^{71} +16.0000 q^{73} -79.1960i q^{74} -58.0000 q^{76} -46.6690i q^{77} +104.000 q^{79} +96.0000 q^{82} -29.6985i q^{83} -7.07107i q^{86} +12.0000 q^{88} +135.765i q^{89} -77.0000 q^{91} +76.3675i q^{92} -90.0000 q^{94} -41.0000 q^{97} +101.823i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 22 q^{7} + 14 q^{13} + 8 q^{16} + 58 q^{19} - 12 q^{22} + 44 q^{28} + 58 q^{31} + 36 q^{34} - 112 q^{37} - 10 q^{43} + 108 q^{46} + 144 q^{49} - 28 q^{52} + 132 q^{58} - 110 q^{61} - 16 q^{64}+ \cdots - 82 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 - 22 * q^7 + 14 * q^13 + 8 * q^16 + 58 * q^19 - 12 * q^22 + 44 * q^28 + 58 * q^31 + 36 * q^34 - 112 * q^37 - 10 * q^43 + 108 * q^46 + 144 * q^49 - 28 * q^52 + 132 * q^58 - 110 * q^61 - 16 * q^64 + 74 * q^67 + 32 * q^73 - 116 * q^76 + 208 * q^79 + 192 * q^82 + 24 * q^88 - 154 * q^91 - 180 * q^94 - 82 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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