LMFDB - Embedded newform 2400.3.e.i.1951.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,3,Mod(1951,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2400.1951");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2400.e (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$65.3952634465$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 49 x^{10} - 190 x^{9} + 792 x^{8} - 2094 x^{7} + 5517 x^{6} - 9954 x^{5} + \cdots + 5584 $ x^12 - 6x^11 + 49x^10 - 190x^9 + 792x^8 - 2094x^7 + 5517x^6 - 9954x^5 + 17189x^4 - 19884x^3 + 20996x^2 - 12416*x + 5584
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1951.8
Root			$0.500000 + 2.65258i$ of defining polynomial
Character	$\chi$	$=$	2400.1951
Dual form			2400.3.e.i.1951.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -7.06571i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -7.06571i q^{7} -3.00000 q^{9} +1.70305i q^{11} -19.3844 q^{13} -13.0069 q^{17} -13.2314i q^{19} +12.2382 q^{21} -5.26392i q^{23} -5.19615i q^{27} -33.0071 q^{29} +60.5509i q^{31} -2.94977 q^{33} +46.9515 q^{37} -33.5747i q^{39} +65.3233 q^{41} +20.2303i q^{43} +40.0636i q^{47} -0.924196 q^{49} -22.5286i q^{51} +11.2001 q^{53} +22.9174 q^{57} -39.8862i q^{59} +32.2030 q^{61} +21.1971i q^{63} -2.76692i q^{67} +9.11738 q^{69} +67.6353i q^{71} +81.8653 q^{73} +12.0333 q^{77} +143.514i q^{79} +9.00000 q^{81} -126.585i q^{83} -57.1699i q^{87} -11.2563 q^{89} +136.964i q^{91} -104.877 q^{93} +49.5443 q^{97} -5.10916i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 36 q^{9}+O(q^{10}) $ 12 * q - 36 * q^9	$ 12 q - 36 q^{9} + 80 q^{17} + 40 q^{29} + 320 q^{37} + 136 q^{41} - 212 q^{49} - 176 q^{53} + 48 q^{57} + 40 q^{61} + 448 q^{73} + 448 q^{77} + 108 q^{81} + 8 q^{89} - 144 q^{93} + 224 q^{97}+O(q^{100}) $ 12 * q - 36 * q^9 + 80 * q^17 + 40 * q^29 + 320 * q^37 + 136 * q^41 - 212 * q^49 - 176 * q^53 + 48 * q^57 + 40 * q^61 + 448 * q^73 + 448 * q^77 + 108 * q^81 + 8 * q^89 - 144 * q^93 + 224 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 7.06571i	− 1.00939i	−0.863299	−	0.504693i	$-0.831606\pi$
$7$	− 7.06571i	− 1.00939i	0.863299	−	0.504693i	$-0.168394\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	1.70305i	0.154823i	0.996999	+	0.0774115i	$0.0246655\pi$
$11$	1.70305i	0.154823i	−0.996999	+	0.0774115i	$0.975334\pi$
$12$	0	0
$13$	−19.3844	−1.49111	−0.745553	−	0.666447i	$-0.767814\pi$
$13$	−19.3844	−1.49111	−0.745553	+	0.666447i	$0.767814\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−13.0069	−0.765113	−0.382556	−	0.923932i	$-0.624956\pi$
$17$	−13.0069	−0.765113	−0.382556	+	0.923932i	$0.624956\pi$
$18$	0	0
$19$	− 13.2314i	− 0.696389i	−0.937422	−	0.348194i	$-0.886795\pi$
$19$	− 13.2314i	− 0.696389i	0.937422	−	0.348194i	$-0.113205\pi$
$20$	0	0
$21$	12.2382	0.582770
$22$	0	0
$23$	− 5.26392i	− 0.228866i	−0.993431	−	0.114433i	$-0.963495\pi$
$23$	− 5.26392i	− 0.228866i	0.993431	−	0.114433i	$-0.0365051\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−33.0071	−1.13817	−0.569087	−	0.822277i	$-0.692703\pi$
$29$	−33.0071	−1.13817	−0.569087	+	0.822277i	$0.692703\pi$
$30$	0	0
$31$	60.5509i	1.95325i	0.214941	+	0.976627i	$0.431044\pi$
$31$	60.5509i	1.95325i	−0.214941	+	0.976627i	$0.568956\pi$
$32$	0	0
$33$	−2.94977	−0.0893871
$34$	0	0
$35$	0	0
$36$	0	0
$37$	46.9515	1.26896	0.634479	−	0.772940i	$-0.281215\pi$
$37$	46.9515	1.26896	0.634479	+	0.772940i	$0.281215\pi$
$38$	0	0
$39$	− 33.5747i	− 0.860890i
$40$	0	0
$41$	65.3233	1.59325	0.796626	−	0.604472i	$-0.206616\pi$
$41$	65.3233	1.59325	0.796626	+	0.604472i	$0.206616\pi$
$42$	0	0
$43$	20.2303i	0.470472i	0.971938	+	0.235236i	$0.0755863\pi$
$43$	20.2303i	0.470472i	−0.971938	+	0.235236i	$0.924414\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	40.0636i	0.852417i	0.904625	+	0.426209i	$0.140151\pi$
$47$	40.0636i	0.852417i	−0.904625	+	0.426209i	$0.859849\pi$
$48$	0	0
$49$	−0.924196	−0.0188612
$50$	0	0
$51$	− 22.5286i	− 0.441738i
$52$	0	0
$53$	11.2001	0.211323	0.105662	−	0.994402i	$-0.466304\pi$
$53$	11.2001	0.211323	0.105662	+	0.994402i	$0.466304\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	22.9174	0.402060
$58$	0	0
$59$	− 39.8862i	− 0.676038i	−0.941139	−	0.338019i	$-0.890243\pi$
$59$	− 39.8862i	− 0.676038i	0.941139	−	0.338019i	$-0.109757\pi$
$60$	0	0
$61$	32.2030	0.527918	0.263959	−	0.964534i	$-0.414972\pi$
$61$	32.2030	0.527918	0.263959	+	0.964534i	$0.414972\pi$
$62$	0	0
$63$	21.1971i	0.336462i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.76692i	− 0.0412973i	−0.999787	−	0.0206486i	$-0.993427\pi$
$67$	− 2.76692i	− 0.0412973i	0.999787	−	0.0206486i	$-0.00657314\pi$
$68$	0	0
$69$	9.11738	0.132136
$70$	0	0
$71$	67.6353i	0.952609i	0.879280	+	0.476305i	$0.158024\pi$
$71$	67.6353i	0.952609i	−0.879280	+	0.476305i	$0.841976\pi$
$72$	0	0
$73$	81.8653	1.12144	0.560721	−	0.828005i	$-0.310524\pi$
$73$	81.8653	1.12144	0.560721	+	0.828005i	$0.310524\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.0333	0.156276
$78$	0	0
$79$	143.514i	1.81664i	0.418279	+	0.908319i	$0.362634\pi$
$79$	143.514i	1.81664i	−0.418279	+	0.908319i	$0.637366\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 126.585i	− 1.52512i	−0.646918	−	0.762560i	$-0.723942\pi$
$83$	− 126.585i	− 1.52512i	0.646918	−	0.762560i	$-0.276058\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 57.1699i	− 0.657125i
$88$	0	0
$89$	−11.2563	−0.126475	−0.0632374	−	0.997999i	$-0.520143\pi$
$89$	−11.2563	−0.126475	−0.0632374	+	0.997999i	$0.520143\pi$
$90$	0	0
$91$	136.964i	1.50510i
$92$	0	0
$93$	−104.877	−1.12771
$94$	0	0
$95$	0	0
$96$	0	0
$97$	49.5443	0.510766	0.255383	−	0.966840i	$-0.417798\pi$
$97$	49.5443	0.510766	0.255383	+	0.966840i	$0.417798\pi$
$98$	0	0
$99$	− 5.10916i	− 0.0516076i
$100$	0	0
$101$	57.1327	0.565671	0.282835	−	0.959169i	$-0.408725\pi$
$101$	57.1327	0.565671	0.282835	+	0.959169i	$0.408725\pi$
$102$	0	0
$103$	− 63.1664i	− 0.613266i	−0.951828	−	0.306633i	$-0.900798\pi$
$103$	− 63.1664i	− 0.613266i	0.951828	−	0.306633i	$-0.0992024\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 59.1537i	− 0.552838i	−0.961037	−	0.276419i	$-0.910852\pi$
$107$	− 59.1537i	− 0.552838i	0.961037	−	0.276419i	$-0.0891478\pi$
$108$	0	0
$109$	159.657	1.46474	0.732370	−	0.680906i	$-0.238414\pi$
$109$	159.657	1.46474	0.732370	+	0.680906i	$0.238414\pi$
$110$	0	0
$111$	81.3223i	0.732634i
$112$	0	0
$113$	145.680	1.28920	0.644601	−	0.764519i	$-0.277024\pi$
$113$	145.680	1.28920	0.644601	+	0.764519i	$0.277024\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	58.1531	0.497035
$118$	0	0
$119$	91.9031i	0.772295i
$120$	0	0
$121$	118.100	0.976030
$122$	0	0
$123$	113.143i	0.919865i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 236.822i	− 1.86474i	−0.361506	−	0.932370i	$-0.617737\pi$
$127$	− 236.822i	− 1.86474i	0.361506	−	0.932370i	$-0.382263\pi$
$128$	0	0
$129$	−35.0399	−0.271627
$130$	0	0
$131$	− 106.346i	− 0.811802i	−0.913917	−	0.405901i	$-0.866958\pi$
$131$	− 106.346i	− 0.811802i	0.913917	−	0.405901i	$-0.133042\pi$
$132$	0	0
$133$	−93.4891	−0.702925
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−138.800	−1.01314	−0.506570	−	0.862199i	$-0.669087\pi$
$137$	−138.800	−1.01314	−0.506570	+	0.862199i	$0.669087\pi$
$138$	0	0
$139$	− 24.3036i	− 0.174846i	−0.996171	−	0.0874229i	$-0.972137\pi$
$139$	− 24.3036i	− 0.174846i	0.996171	−	0.0874229i	$-0.0278631\pi$
$140$	0	0
$141$	−69.3922	−0.492143
$142$	0	0
$143$	− 33.0126i	− 0.230857i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.60076i	− 0.0108895i
$148$	0	0
$149$	178.349	1.19698	0.598488	−	0.801132i	$-0.295768\pi$
$149$	178.349	1.19698	0.598488	+	0.801132i	$0.295768\pi$
$150$	0	0
$151$	− 170.810i	− 1.13119i	−0.824682	−	0.565596i	$-0.808646\pi$
$151$	− 170.810i	− 1.13119i	0.824682	−	0.565596i	$-0.191354\pi$
$152$	0	0
$153$	39.0208	0.255038
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−83.1986	−0.529927	−0.264964	−	0.964258i	$-0.585360\pi$
$157$	−83.1986	−0.529927	−0.264964	+	0.964258i	$0.585360\pi$
$158$	0	0
$159$	19.3992i	0.122007i
$160$	0	0
$161$	−37.1933	−0.231014
$162$	0	0
$163$	− 81.2870i	− 0.498693i	−0.968414	−	0.249346i	$-0.919784\pi$
$163$	− 81.2870i	− 0.498693i	0.968414	−	0.249346i	$-0.0802158\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	314.667i	1.88423i	0.335288	+	0.942116i	$0.391166\pi$
$167$	314.667i	1.88423i	−0.335288	+	0.942116i	$0.608834\pi$
$168$	0	0
$169$	206.754	1.22339
$170$	0	0
$171$	39.6942i	0.232130i
$172$	0	0
$173$	143.143	0.827419	0.413709	−	0.910409i	$-0.364233\pi$
$173$	143.143	0.827419	0.413709	+	0.910409i	$0.364233\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	69.0850	0.390310
$178$	0	0
$179$	229.059i	1.27966i	0.768517	+	0.639829i	$0.220995\pi$
$179$	229.059i	1.27966i	−0.768517	+	0.639829i	$0.779005\pi$
$180$	0	0
$181$	155.190	0.857401	0.428700	−	0.903447i	$-0.358972\pi$
$181$	155.190	0.857401	0.428700	+	0.903447i	$0.358972\pi$
$182$	0	0
$183$	55.7773i	0.304794i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 22.1515i	− 0.118457i
$188$	0	0
$189$	−36.7145	−0.194257
$190$	0	0
$191$	299.037i	1.56564i	0.622249	+	0.782819i	$0.286219\pi$
$191$	299.037i	1.56564i	−0.622249	+	0.782819i	$0.713781\pi$
$192$	0	0
$193$	112.150	0.581090	0.290545	−	0.956861i	$-0.406163\pi$
$193$	112.150	0.581090	0.290545	+	0.956861i	$0.406163\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−348.881	−1.77097	−0.885485	−	0.464668i	$-0.846174\pi$
$197$	−348.881	−1.77097	−0.885485	+	0.464668i	$0.846174\pi$
$198$	0	0
$199$	217.793i	1.09444i	0.836990	+	0.547218i	$0.184313\pi$
$199$	217.793i	1.09444i	−0.836990	+	0.547218i	$0.815687\pi$
$200$	0	0
$201$	4.79244	0.0238430
$202$	0	0
$203$	233.218i	1.14886i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	15.7918i	0.0762887i
$208$	0	0
$209$	22.5337	0.107817
$210$	0	0
$211$	318.221i	1.50816i	0.656785	+	0.754078i	$0.271916\pi$
$211$	318.221i	1.50816i	−0.656785	+	0.754078i	$0.728084\pi$
$212$	0	0
$213$	−117.148	−0.549989
$214$	0	0
$215$	0	0
$216$	0	0
$217$	427.835	1.97159
$218$	0	0
$219$	141.795i	0.647465i
$220$	0	0
$221$	252.131	1.14086
$222$	0	0
$223$	185.718i	0.832816i	0.909178	+	0.416408i	$0.136711\pi$
$223$	185.718i	0.832816i	−0.909178	+	0.416408i	$0.863289\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	286.576i	1.26245i	0.775600	+	0.631225i	$0.217448\pi$
$227$	286.576i	1.26245i	−0.775600	+	0.631225i	$0.782552\pi$
$228$	0	0
$229$	−109.601	−0.478607	−0.239304	−	0.970945i	$-0.576919\pi$
$229$	−109.601	−0.478607	−0.239304	+	0.970945i	$0.576919\pi$
$230$	0	0
$231$	20.8422i	0.0902261i
$232$	0	0
$233$	−404.676	−1.73681	−0.868404	−	0.495857i	$-0.834854\pi$
$233$	−404.676	−1.73681	−0.868404	+	0.495857i	$0.834854\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−248.574	−1.04884
$238$	0	0
$239$	− 101.788i	− 0.425890i	−0.977064	−	0.212945i	$-0.931694\pi$
$239$	− 101.788i	− 0.425890i	0.977064	−	0.212945i	$-0.0683056\pi$
$240$	0	0
$241$	111.710	0.463526	0.231763	−	0.972772i	$-0.425551\pi$
$241$	111.710	0.463526	0.231763	+	0.972772i	$0.425551\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	256.482i	1.03839i
$248$	0	0
$249$	219.251	0.880528
$250$	0	0
$251$	168.763i	0.672363i	0.941797	+	0.336182i	$0.109136\pi$
$251$	168.763i	0.672363i	−0.941797	+	0.336182i	$0.890864\pi$
$252$	0	0
$253$	8.96473	0.0354337
$254$	0	0
$255$	0	0
$256$	0	0
$257$	320.743	1.24803	0.624013	−	0.781414i	$-0.285501\pi$
$257$	320.743	1.24803	0.624013	+	0.781414i	$0.285501\pi$
$258$	0	0
$259$	− 331.745i	− 1.28087i
$260$	0	0
$261$	99.0212	0.379392
$262$	0	0
$263$	− 58.2208i	− 0.221372i	−0.993855	−	0.110686i	$-0.964695\pi$
$263$	− 58.2208i	− 0.221372i	0.993855	−	0.110686i	$-0.0353048\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 19.4964i	− 0.0730203i
$268$	0	0
$269$	−259.743	−0.965586	−0.482793	−	0.875734i	$-0.660378\pi$
$269$	−259.743	−0.965586	−0.482793	+	0.875734i	$0.660378\pi$
$270$	0	0
$271$	− 62.7802i	− 0.231661i	−0.993269	−	0.115831i	$-0.963047\pi$
$271$	− 62.7802i	− 0.231661i	0.993269	−	0.115831i	$-0.0369530\pi$
$272$	0	0
$273$	−237.229	−0.868971
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−419.214	−1.51341	−0.756705	−	0.653757i	$-0.773192\pi$
$277$	−419.214	−1.51341	−0.756705	+	0.653757i	$0.773192\pi$
$278$	0	0
$279$	− 181.653i	− 0.651085i
$280$	0	0
$281$	429.741	1.52933	0.764664	−	0.644430i	$-0.222905\pi$
$281$	429.741	1.52933	0.764664	+	0.644430i	$0.222905\pi$
$282$	0	0
$283$	− 207.152i	− 0.731988i	−0.930617	−	0.365994i	$-0.880729\pi$
$283$	− 207.152i	− 0.731988i	0.930617	−	0.365994i	$-0.119271\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 461.556i	− 1.60821i
$288$	0	0
$289$	−119.820	−0.414602
$290$	0	0
$291$	85.8132i	0.294891i
$292$	0	0
$293$	−157.986	−0.539200	−0.269600	−	0.962972i	$-0.586891\pi$
$293$	−157.986	−0.539200	−0.269600	+	0.962972i	$0.586891\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8.84932	0.0297957
$298$	0	0
$299$	102.038i	0.341263i
$300$	0	0
$301$	142.941	0.474888
$302$	0	0
$303$	98.9568i	0.326590i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.9945i	0.0553568i	0.999617	+	0.0276784i	$0.00881143\pi$
$307$	16.9945i	0.0553568i	−0.999617	+	0.0276784i	$0.991189\pi$
$308$	0	0
$309$	109.407	0.354069
$310$	0	0
$311$	430.263i	1.38348i	0.722146	+	0.691741i	$0.243156\pi$
$311$	430.263i	1.38348i	−0.722146	+	0.691741i	$0.756844\pi$
$312$	0	0
$313$	340.367	1.08743	0.543717	−	0.839268i	$-0.317016\pi$
$313$	340.367	1.08743	0.543717	+	0.839268i	$0.317016\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−347.597	−1.09652	−0.548260	−	0.836308i	$-0.684709\pi$
$317$	−347.597	−1.09652	−0.548260	+	0.836308i	$0.684709\pi$
$318$	0	0
$319$	− 56.2128i	− 0.176216i
$320$	0	0
$321$	102.457	0.319181
$322$	0	0
$323$	172.100i	0.532816i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	276.534i	0.845668i
$328$	0	0
$329$	283.078	0.860418
$330$	0	0
$331$	521.268i	1.57483i	0.616424	+	0.787414i	$0.288581\pi$
$331$	521.268i	1.57483i	−0.616424	+	0.787414i	$0.711419\pi$
$332$	0	0
$333$	−140.854	−0.422986
$334$	0	0
$335$	0	0
$336$	0	0
$337$	124.784	0.370278	0.185139	−	0.982712i	$-0.440726\pi$
$337$	124.784	0.370278	0.185139	+	0.982712i	$0.440726\pi$
$338$	0	0
$339$	252.325i	0.744321i
$340$	0	0
$341$	−103.121	−0.302409
$342$	0	0
$343$	− 339.689i	− 0.990348i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 133.616i	− 0.385060i	−0.981291	−	0.192530i	$-0.938331\pi$
$347$	− 133.616i	− 0.385060i	0.981291	−	0.192530i	$-0.0616693\pi$
$348$	0	0
$349$	−265.623	−0.761098	−0.380549	−	0.924761i	$-0.624265\pi$
$349$	−265.623	−0.761098	−0.380549	+	0.924761i	$0.624265\pi$
$350$	0	0
$351$	100.724i	0.286963i
$352$	0	0
$353$	−196.286	−0.556050	−0.278025	−	0.960574i	$-0.589680\pi$
$353$	−196.286	−0.556050	−0.278025	+	0.960574i	$0.589680\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−159.181	−0.445885
$358$	0	0
$359$	139.938i	0.389800i	0.980823	+	0.194900i	$0.0624383\pi$
$359$	139.938i	0.389800i	−0.980823	+	0.194900i	$0.937562\pi$
$360$	0	0
$361$	185.930	0.515043
$362$	0	0
$363$	204.555i	0.563511i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6.22040i	0.0169493i	0.999964	+	0.00847465i	$0.00269760\pi$
$367$	6.22040i	0.0169493i	−0.999964	+	0.00847465i	$0.997302\pi$
$368$	0	0
$369$	−195.970	−0.531084
$370$	0	0
$371$	− 79.1368i	− 0.213307i
$372$	0	0
$373$	687.926	1.84431	0.922153	−	0.386825i	$-0.126428\pi$
$373$	687.926	1.84431	0.922153	+	0.386825i	$0.126428\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	639.821	1.69714
$378$	0	0
$379$	− 264.922i	− 0.699004i	−0.936936	−	0.349502i	$-0.886351\pi$
$379$	− 264.922i	− 0.699004i	0.936936	−	0.349502i	$-0.113649\pi$
$380$	0	0
$381$	410.188	1.07661
$382$	0	0
$383$	335.299i	0.875454i	0.899108	+	0.437727i	$0.144216\pi$
$383$	335.299i	0.875454i	−0.899108	+	0.437727i	$0.855784\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 60.6909i	− 0.156824i
$388$	0	0
$389$	355.544	0.913996	0.456998	−	0.889468i	$-0.348925\pi$
$389$	355.544	0.913996	0.456998	+	0.889468i	$0.348925\pi$
$390$	0	0
$391$	68.4674i	0.175108i
$392$	0	0
$393$	184.197	0.468694
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−238.927	−0.601832	−0.300916	−	0.953651i	$-0.597292\pi$
$397$	−238.927	−0.601832	−0.300916	+	0.953651i	$0.597292\pi$
$398$	0	0
$399$	− 161.928i	− 0.405834i
$400$	0	0
$401$	44.9170	0.112013	0.0560063	−	0.998430i	$-0.482163\pi$
$401$	44.9170	0.112013	0.0560063	+	0.998430i	$0.482163\pi$
$402$	0	0
$403$	− 1173.74i	− 2.91251i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	79.9608i	0.196464i
$408$	0	0
$409$	−484.137	−1.18371	−0.591854	−	0.806045i	$-0.701604\pi$
$409$	−484.137	−1.18371	−0.591854	+	0.806045i	$0.701604\pi$
$410$	0	0
$411$	− 240.409i	− 0.584937i
$412$	0	0
$413$	−281.824	−0.682383
$414$	0	0
$415$	0	0
$416$	0	0
$417$	42.0950	0.100947
$418$	0	0
$419$	− 374.895i	− 0.894739i	−0.894349	−	0.447369i	$-0.852361\pi$
$419$	− 374.895i	− 0.894739i	0.894349	−	0.447369i	$-0.147639\pi$
$420$	0	0
$421$	16.8787	0.0400919	0.0200459	−	0.999799i	$-0.493619\pi$
$421$	16.8787	0.0400919	0.0200459	+	0.999799i	$0.493619\pi$
$422$	0	0
$423$	− 120.191i	− 0.284139i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 227.537i	− 0.532874i
$428$	0	0
$429$	57.1795	0.133285
$430$	0	0
$431$	29.0390i	0.0673759i	0.999432	+	0.0336879i	$0.0107252\pi$
$431$	29.0390i	0.0673759i	−0.999432	+	0.0336879i	$0.989275\pi$
$432$	0	0
$433$	128.629	0.297064	0.148532	−	0.988908i	$-0.452545\pi$
$433$	128.629	0.297064	0.148532	+	0.988908i	$0.452545\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−69.6490	−0.159380
$438$	0	0
$439$	67.3591i	0.153438i	0.997053	+	0.0767188i	$0.0244444\pi$
$439$	67.3591i	0.153438i	−0.997053	+	0.0767188i	$0.975556\pi$
$440$	0	0
$441$	2.77259	0.00628705
$442$	0	0
$443$	507.136i	1.14478i	0.819983	+	0.572388i	$0.193983\pi$
$443$	507.136i	1.14478i	−0.819983	+	0.572388i	$0.806017\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	308.910i	0.691074i
$448$	0	0
$449$	−328.869	−0.732447	−0.366224	−	0.930527i	$-0.619349\pi$
$449$	−328.869	−0.732447	−0.366224	+	0.930527i	$0.619349\pi$
$450$	0	0
$451$	111.249i	0.246672i
$452$	0	0
$453$	295.852	0.653094
$454$	0	0
$455$	0	0
$456$	0	0
$457$	640.422	1.40136	0.700681	−	0.713475i	$-0.252880\pi$
$457$	640.422	1.40136	0.700681	+	0.713475i	$0.252880\pi$
$458$	0	0
$459$	67.5859i	0.147246i
$460$	0	0
$461$	447.177	0.970014	0.485007	−	0.874510i	$-0.338817\pi$
$461$	447.177	0.970014	0.485007	+	0.874510i	$0.338817\pi$
$462$	0	0
$463$	− 892.367i	− 1.92736i	−0.267064	−	0.963679i	$-0.586053\pi$
$463$	− 892.367i	− 1.92736i	0.267064	−	0.963679i	$-0.413947\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	371.767i	0.796075i	0.917369	+	0.398038i	$0.130309\pi$
$467$	371.767i	0.796075i	−0.917369	+	0.398038i	$0.869691\pi$
$468$	0	0
$469$	−19.5502	−0.0416849
$470$	0	0
$471$	− 144.104i	− 0.305954i
$472$	0	0
$473$	−34.4532	−0.0728398
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−33.6004	−0.0704411
$478$	0	0
$479$	− 105.771i	− 0.220817i	−0.993886	−	0.110408i	$-0.964784\pi$
$479$	− 105.771i	− 0.220817i	0.993886	−	0.110408i	$-0.0352159\pi$
$480$	0	0
$481$	−910.124	−1.89215
$482$	0	0
$483$	− 64.4207i	− 0.133376i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 519.395i	− 1.06652i	−0.845952	−	0.533259i	$-0.820967\pi$
$487$	− 519.395i	− 1.06652i	0.845952	−	0.533259i	$-0.179033\pi$
$488$	0	0
$489$	140.793	0.287921
$490$	0	0
$491$	− 805.321i	− 1.64016i	−0.572246	−	0.820082i	$-0.693928\pi$
$491$	− 805.321i	− 1.64016i	0.572246	−	0.820082i	$-0.306072\pi$
$492$	0	0
$493$	429.320	0.870832
$494$	0	0
$495$	0	0
$496$	0	0
$497$	477.891	0.961551
$498$	0	0
$499$	825.125i	1.65356i	0.562528	+	0.826778i	$0.309829\pi$
$499$	825.125i	1.65356i	−0.562528	+	0.826778i	$0.690171\pi$
$500$	0	0
$501$	−545.019	−1.08786
$502$	0	0
$503$	577.032i	1.14718i	0.819142	+	0.573590i	$0.194450\pi$
$503$	577.032i	1.14718i	−0.819142	+	0.573590i	$0.805550\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	358.108i	0.706327i
$508$	0	0
$509$	240.707	0.472902	0.236451	−	0.971643i	$-0.424016\pi$
$509$	240.707	0.472902	0.236451	+	0.971643i	$0.424016\pi$
$510$	0	0
$511$	− 578.436i	− 1.13197i
$512$	0	0
$513$	−68.7523	−0.134020
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−68.2304	−0.131974
$518$	0	0
$519$	247.932i	0.477711i
$520$	0	0
$521$	−928.796	−1.78272	−0.891359	−	0.453299i	$-0.850247\pi$
$521$	−928.796	−1.78272	−0.891359	+	0.453299i	$0.850247\pi$
$522$	0	0
$523$	1007.76i	1.92689i	0.267914	+	0.963443i	$0.413666\pi$
$523$	1007.76i	1.92689i	−0.267914	+	0.963443i	$0.586334\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 787.580i	− 1.49446i
$528$	0	0
$529$	501.291	0.947620
$530$	0	0
$531$	119.659i	0.225346i
$532$	0	0
$533$	−1266.25	−2.37571
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−396.742	−0.738811
$538$	0	0
$539$	− 1.57395i	− 0.00292014i
$540$	0	0
$541$	−733.141	−1.35516	−0.677580	−	0.735449i	$-0.736971\pi$
$541$	−733.141	−1.35516	−0.677580	+	0.735449i	$0.736971\pi$
$542$	0	0
$543$	268.796i	0.495021i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 197.440i	− 0.360950i	−0.983580	−	0.180475i	$-0.942236\pi$
$547$	− 197.440i	− 0.360950i	0.983580	−	0.180475i	$-0.0577635\pi$
$548$	0	0
$549$	−96.6091	−0.175973
$550$	0	0
$551$	436.729i	0.792612i
$552$	0	0
$553$	1014.03	1.83369
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.5650	0.0800089	0.0400045	−	0.999200i	$-0.487263\pi$
$557$	44.5650	0.0800089	0.0400045	+	0.999200i	$0.487263\pi$
$558$	0	0
$559$	− 392.151i	− 0.701523i
$560$	0	0
$561$	38.3675	0.0683912
$562$	0	0
$563$	− 166.410i	− 0.295577i	−0.989019	−	0.147788i	$-0.952785\pi$
$563$	− 166.410i	− 0.295577i	0.989019	−	0.147788i	$-0.0472154\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 63.5914i	− 0.112154i
$568$	0	0
$569$	503.775	0.885369	0.442685	−	0.896677i	$-0.354026\pi$
$569$	503.775	0.885369	0.442685	+	0.896677i	$0.354026\pi$
$570$	0	0
$571$	− 898.626i	− 1.57378i	−0.617096	−	0.786888i	$-0.711691\pi$
$571$	− 898.626i	− 1.57378i	0.617096	−	0.786888i	$-0.288309\pi$
$572$	0	0
$573$	−517.947	−0.903922
$574$	0	0
$575$	0	0
$576$	0	0
$577$	371.658	0.644121	0.322061	−	0.946719i	$-0.395624\pi$
$577$	371.658	0.644121	0.322061	+	0.946719i	$0.395624\pi$
$578$	0	0
$579$	194.250i	0.335493i
$580$	0	0
$581$	−894.412	−1.53943
$582$	0	0
$583$	19.0744i	0.0327177i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1049.99i	− 1.78873i	−0.447335	−	0.894366i	$-0.647627\pi$
$587$	− 1049.99i	− 1.78873i	0.447335	−	0.894366i	$-0.352373\pi$
$588$	0	0
$589$	801.172	1.36022
$590$	0	0
$591$	− 604.280i	− 1.02247i
$592$	0	0
$593$	553.135	0.932774	0.466387	−	0.884581i	$-0.345555\pi$
$593$	553.135	0.932774	0.466387	+	0.884581i	$0.345555\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−377.228	−0.631873
$598$	0	0
$599$	− 240.892i	− 0.402156i	−0.979575	−	0.201078i	$-0.935555\pi$
$599$	− 240.892i	− 0.402156i	0.979575	−	0.201078i	$-0.0644445\pi$
$600$	0	0
$601$	939.752	1.56365	0.781823	−	0.623500i	$-0.214290\pi$
$601$	939.752	1.56365	0.781823	+	0.623500i	$0.214290\pi$
$602$	0	0
$603$	8.30076i	0.0137658i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	159.561i	0.262868i	0.991325	+	0.131434i	$0.0419582\pi$
$607$	159.561i	0.262868i	−0.991325	+	0.131434i	$0.958042\pi$
$608$	0	0
$609$	−403.946	−0.663294
$610$	0	0
$611$	− 776.608i	− 1.27104i
$612$	0	0
$613$	−405.464	−0.661442	−0.330721	−	0.943729i	$-0.607292\pi$
$613$	−405.464	−0.661442	−0.330721	+	0.943729i	$0.607292\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−210.340	−0.340907	−0.170453	−	0.985366i	$-0.554523\pi$
$617$	−210.340	−0.340907	−0.170453	+	0.985366i	$0.554523\pi$
$618$	0	0
$619$	374.381i	0.604815i	0.953179	+	0.302408i	$0.0977904\pi$
$619$	374.381i	0.604815i	−0.953179	+	0.302408i	$0.902210\pi$
$620$	0	0
$621$	−27.3521	−0.0440453
$622$	0	0
$623$	79.5334i	0.127662i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	39.0296i	0.0622481i
$628$	0	0
$629$	−610.694	−0.970897
$630$	0	0
$631$	− 543.402i	− 0.861176i	−0.902549	−	0.430588i	$-0.858306\pi$
$631$	− 543.402i	− 0.861176i	0.902549	−	0.430588i	$-0.141694\pi$
$632$	0	0
$633$	−551.175	−0.870734
$634$	0	0
$635$	0	0
$636$	0	0
$637$	17.9150	0.0281240
$638$	0	0
$639$	− 202.906i	− 0.317536i
$640$	0	0
$641$	−27.0948	−0.0422696	−0.0211348	−	0.999777i	$-0.506728\pi$
$641$	−27.0948	−0.0422696	−0.0211348	+	0.999777i	$0.506728\pi$
$642$	0	0
$643$	− 933.931i	− 1.45246i	−0.687453	−	0.726229i	$-0.741271\pi$
$643$	− 933.931i	− 1.45246i	0.687453	−	0.726229i	$-0.258729\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	824.289i	1.27402i	0.770857	+	0.637009i	$0.219829\pi$
$647$	824.289i	1.27402i	−0.770857	+	0.637009i	$0.780171\pi$
$648$	0	0
$649$	67.9283	0.104666
$650$	0	0
$651$	741.031i	1.13830i
$652$	0	0
$653$	−794.099	−1.21608	−0.608039	−	0.793907i	$-0.708044\pi$
$653$	−794.099	−1.21608	−0.608039	+	0.793907i	$0.708044\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−245.596	−0.373814
$658$	0	0
$659$	251.919i	0.382275i	0.981563	+	0.191137i	$0.0612176\pi$
$659$	251.919i	0.382275i	−0.981563	+	0.191137i	$0.938782\pi$
$660$	0	0
$661$	190.361	0.287989	0.143995	−	0.989578i	$-0.454005\pi$
$661$	190.361	0.287989	0.143995	+	0.989578i	$0.454005\pi$
$662$	0	0
$663$	436.703i	0.658678i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	173.747i	0.260490i
$668$	0	0
$669$	−321.673	−0.480826
$670$	0	0
$671$	54.8434i	0.0817339i
$672$	0	0
$673$	1116.66	1.65922	0.829612	−	0.558341i	$-0.188562\pi$
$673$	1116.66	1.65922	0.829612	+	0.558341i	$0.188562\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−170.279	−0.251520	−0.125760	−	0.992061i	$-0.540137\pi$
$677$	−170.279	−0.251520	−0.125760	+	0.992061i	$0.540137\pi$
$678$	0	0
$679$	− 350.065i	− 0.515560i
$680$	0	0
$681$	−496.365	−0.728876
$682$	0	0
$683$	602.204i	0.881704i	0.897580	+	0.440852i	$0.145324\pi$
$683$	602.204i	0.881704i	−0.897580	+	0.440852i	$0.854676\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 189.835i	− 0.276324i
$688$	0	0
$689$	−217.107	−0.315105
$690$	0	0
$691$	− 268.267i	− 0.388230i	−0.980979	−	0.194115i	$-0.937816\pi$
$691$	− 268.267i	− 0.388230i	0.980979	−	0.194115i	$-0.0621836\pi$
$692$	0	0
$693$	−36.0998	−0.0520921
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−849.655	−1.21902
$698$	0	0
$699$	− 700.920i	− 1.00275i
$700$	0	0
$701$	−1009.23	−1.43970	−0.719848	−	0.694132i	$-0.755788\pi$
$701$	−1009.23	−1.43970	−0.719848	+	0.694132i	$0.755788\pi$
$702$	0	0
$703$	− 621.233i	− 0.883688i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 403.683i	− 0.570980i
$708$	0	0
$709$	−737.419	−1.04008	−0.520042	−	0.854141i	$-0.674084\pi$
$709$	−737.419	−1.04008	−0.520042	+	0.854141i	$0.674084\pi$
$710$	0	0
$711$	− 430.543i	− 0.605546i
$712$	0	0
$713$	318.735	0.447034
$714$	0	0
$715$	0	0
$716$	0	0
$717$	176.302	0.245888
$718$	0	0
$719$	747.093i	1.03907i	0.854448	+	0.519536i	$0.173895\pi$
$719$	747.093i	1.03907i	−0.854448	+	0.519536i	$0.826105\pi$
$720$	0	0
$721$	−446.315	−0.619022
$722$	0	0
$723$	193.487i	0.267617i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 209.131i	− 0.287663i	−0.989602	−	0.143832i	$-0.954058\pi$
$727$	− 209.131i	− 0.287663i	0.989602	−	0.143832i	$-0.0459424\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 263.134i	− 0.359964i
$732$	0	0
$733$	557.973	0.761218	0.380609	−	0.924736i	$-0.375714\pi$
$733$	557.973	0.761218	0.380609	+	0.924736i	$0.375714\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.71221	0.00639377
$738$	0	0
$739$	79.3912i	0.107431i	0.998556	+	0.0537153i	$0.0171063\pi$
$739$	79.3912i	0.107431i	−0.998556	+	0.0537153i	$0.982894\pi$
$740$	0	0
$741$	−444.240	−0.599514
$742$	0	0
$743$	606.208i	0.815892i	0.913006	+	0.407946i	$0.133755\pi$
$743$	606.208i	0.815892i	−0.913006	+	0.407946i	$0.866245\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	379.755i	0.508373i
$748$	0	0
$749$	−417.963	−0.558028
$750$	0	0
$751$	− 548.207i	− 0.729969i	−0.931014	−	0.364984i	$-0.881074\pi$
$751$	− 548.207i	− 0.729969i	0.931014	−	0.364984i	$-0.118926\pi$
$752$	0	0
$753$	−292.306	−0.388189
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1299.81	1.71705	0.858526	−	0.512770i	$-0.171381\pi$
$757$	1299.81	1.71705	0.858526	+	0.512770i	$0.171381\pi$
$758$	0	0
$759$	15.5274i	0.0204577i
$760$	0	0
$761$	−390.151	−0.512682	−0.256341	−	0.966586i	$-0.582517\pi$
$761$	−390.151	−0.512682	−0.256341	+	0.966586i	$0.582517\pi$
$762$	0	0
$763$	− 1128.09i	− 1.47849i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	773.169i	1.00804i
$768$	0	0
$769$	450.628	0.585992	0.292996	−	0.956114i	$-0.405348\pi$
$769$	450.628	0.585992	0.292996	+	0.956114i	$0.405348\pi$
$770$	0	0
$771$	555.543i	0.720549i
$772$	0	0
$773$	−226.657	−0.293217	−0.146608	−	0.989195i	$-0.546836\pi$
$773$	−226.657	−0.293217	−0.146608	+	0.989195i	$0.546836\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	574.600	0.739510
$778$	0	0
$779$	− 864.318i	− 1.10952i
$780$	0	0
$781$	−115.186	−0.147486
$782$	0	0
$783$	171.510i	0.219042i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	204.623i	0.260004i	0.991514	+	0.130002i	$0.0414984\pi$
$787$	204.623i	0.260004i	−0.991514	+	0.130002i	$0.958502\pi$
$788$	0	0
$789$	100.841	0.127809
$790$	0	0
$791$	− 1029.33i	− 1.30130i
$792$	0	0
$793$	−624.235	−0.787182
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−431.987	−0.542016	−0.271008	−	0.962577i	$-0.587357\pi$
$797$	−431.987	−0.542016	−0.271008	+	0.962577i	$0.587357\pi$
$798$	0	0
$799$	− 521.104i	− 0.652195i
$800$	0	0
$801$	33.7688	0.0421583
$802$	0	0
$803$	139.421i	0.173625i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 449.888i	− 0.557482i
$808$	0	0
$809$	−1348.23	−1.66654	−0.833270	−	0.552867i	$-0.813534\pi$
$809$	−1348.23	−1.66654	−0.833270	+	0.552867i	$0.813534\pi$
$810$	0	0
$811$	635.864i	0.784050i	0.919955	+	0.392025i	$0.128225\pi$
$811$	635.864i	0.784050i	−0.919955	+	0.392025i	$0.871775\pi$
$812$	0	0
$813$	108.738	0.133750
$814$	0	0
$815$	0	0
$816$	0	0
$817$	267.675	0.327631
$818$	0	0
$819$	− 410.893i	− 0.501700i
$820$	0	0
$821$	3.88250	0.00472899	0.00236449	−	0.999997i	$-0.499247\pi$
$821$	3.88250	0.00472899	0.00236449	+	0.999997i	$0.499247\pi$
$822$	0	0
$823$	− 799.122i	− 0.970987i	−0.874240	−	0.485494i	$-0.838640\pi$
$823$	− 799.122i	− 0.970987i	0.874240	−	0.485494i	$-0.161360\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1337.16i	1.61688i	0.588582	+	0.808438i	$0.299687\pi$
$827$	1337.16i	1.61688i	−0.588582	+	0.808438i	$0.700313\pi$
$828$	0	0
$829$	711.385	0.858124	0.429062	−	0.903275i	$-0.358844\pi$
$829$	711.385	0.858124	0.429062	+	0.903275i	$0.358844\pi$
$830$	0	0
$831$	− 726.101i	− 0.873767i
$832$	0	0
$833$	12.0209	0.0144309
$834$	0	0
$835$	0	0
$836$	0	0
$837$	314.632	0.375904
$838$	0	0
$839$	− 903.214i	− 1.07654i	−0.842774	−	0.538268i	$-0.819079\pi$
$839$	− 903.214i	− 1.07654i	0.842774	−	0.538268i	$-0.180921\pi$
$840$	0	0
$841$	248.466	0.295441
$842$	0	0
$843$	744.333i	0.882957i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 834.457i	− 0.985191i
$848$	0	0
$849$	358.799	0.422613
$850$	0	0
$851$	− 247.149i	− 0.290422i
$852$	0	0
$853$	473.826	0.555482	0.277741	−	0.960656i	$-0.410414\pi$
$853$	473.826	0.555482	0.277741	+	0.960656i	$0.410414\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−357.541	−0.417201	−0.208601	−	0.978001i	$-0.566891\pi$
$857$	−357.541	−0.417201	−0.208601	+	0.978001i	$0.566891\pi$
$858$	0	0
$859$	312.604i	0.363917i	0.983306	+	0.181958i	$0.0582436\pi$
$859$	312.604i	0.363917i	−0.983306	+	0.181958i	$0.941756\pi$
$860$	0	0
$861$	799.438	0.928499
$862$	0	0
$863$	− 1106.26i	− 1.28187i	−0.767594	−	0.640937i	$-0.778546\pi$
$863$	− 1106.26i	− 1.28187i	0.767594	−	0.640937i	$-0.221454\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 207.534i	− 0.239371i
$868$	0	0
$869$	−244.412	−0.281257
$870$	0	0
$871$	53.6350i	0.0615786i
$872$	0	0
$873$	−148.633	−0.170255
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−150.342	−0.171428	−0.0857140	−	0.996320i	$-0.527317\pi$
$877$	−150.342	−0.171428	−0.0857140	+	0.996320i	$0.527317\pi$
$878$	0	0
$879$	− 273.639i	− 0.311307i
$880$	0	0
$881$	717.680	0.814619	0.407310	−	0.913290i	$-0.366467\pi$
$881$	717.680	0.814619	0.407310	+	0.913290i	$0.366467\pi$
$882$	0	0
$883$	900.807i	1.02017i	0.860125	+	0.510083i	$0.170385\pi$
$883$	900.807i	1.02017i	−0.860125	+	0.510083i	$0.829615\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	794.320i	0.895513i	0.894155	+	0.447757i	$0.147777\pi$
$887$	794.320i	0.895513i	−0.894155	+	0.447757i	$0.852223\pi$
$888$	0	0
$889$	−1673.31	−1.88224
$890$	0	0
$891$	15.3275i	0.0172025i
$892$	0	0
$893$	530.097	0.593614
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−176.735	−0.197029
$898$	0	0
$899$	− 1998.61i	− 2.22314i
$900$	0	0
$901$	−145.679	−0.161686
$902$	0	0
$903$	247.582i	0.274177i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 340.876i	− 0.375829i	−0.982185	−	0.187914i	$-0.939827\pi$
$907$	− 340.876i	− 0.375829i	0.982185	−	0.187914i	$-0.0601727\pi$
$908$	0	0
$909$	−171.398	−0.188557
$910$	0	0
$911$	1364.47i	1.49777i	0.662697	+	0.748887i	$0.269412\pi$
$911$	1364.47i	1.49777i	−0.662697	+	0.748887i	$0.730588\pi$
$912$	0	0
$913$	215.581	0.236123
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−751.410	−0.819422
$918$	0	0
$919$	− 903.899i	− 0.983568i	−0.870717	−	0.491784i	$-0.836345\pi$
$919$	− 903.899i	− 0.983568i	0.870717	−	0.491784i	$-0.163655\pi$
$920$	0	0
$921$	−29.4354	−0.0319602
$922$	0	0
$923$	− 1311.07i	− 1.42044i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	189.499i	0.204422i
$928$	0	0
$929$	−267.291	−0.287719	−0.143860	−	0.989598i	$-0.545951\pi$
$929$	−267.291	−0.287719	−0.143860	+	0.989598i	$0.545951\pi$
$930$	0	0
$931$	12.2284i	0.0131347i
$932$	0	0
$933$	−745.237	−0.798753
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1612.36	1.72076	0.860382	−	0.509649i	$-0.170225\pi$
$937$	1612.36	1.72076	0.860382	+	0.509649i	$0.170225\pi$
$938$	0	0
$939$	589.533i	0.627831i
$940$	0	0
$941$	878.212	0.933276	0.466638	−	0.884448i	$-0.345465\pi$
$941$	878.212	0.933276	0.466638	+	0.884448i	$0.345465\pi$
$942$	0	0
$943$	− 343.857i	− 0.364641i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	726.974i	0.767660i	0.923404	+	0.383830i	$0.125395\pi$
$947$	726.974i	0.767660i	−0.923404	+	0.383830i	$0.874605\pi$
$948$	0	0
$949$	−1586.91	−1.67219
$950$	0	0
$951$	− 602.055i	− 0.633076i
$952$	0	0
$953$	603.417	0.633177	0.316588	−	0.948563i	$-0.397463\pi$
$953$	603.417	0.633177	0.316588	+	0.948563i	$0.397463\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	97.3633	0.101738
$958$	0	0
$959$	980.721i	1.02265i
$960$	0	0
$961$	−2705.41	−2.81520
$962$	0	0
$963$	177.461i	0.184279i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	408.083i	0.422009i	0.977485	+	0.211004i	$0.0676735\pi$
$967$	408.083i	0.422009i	−0.977485	+	0.211004i	$0.932327\pi$
$968$	0	0
$969$	−298.085	−0.307621
$970$	0	0
$971$	− 963.723i	− 0.992506i	−0.868178	−	0.496253i	$-0.834709\pi$
$971$	− 963.723i	− 0.992506i	0.868178	−	0.496253i	$-0.165291\pi$
$972$	0	0
$973$	−171.722	−0.176487
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−498.460	−0.510195	−0.255097	−	0.966915i	$-0.582108\pi$
$977$	−498.460	−0.510195	−0.255097	+	0.966915i	$0.582108\pi$
$978$	0	0
$979$	− 19.1700i	− 0.0195812i
$980$	0	0
$981$	−478.970	−0.488247
$982$	0	0
$983$	1328.10i	1.35107i	0.737328	+	0.675535i	$0.236087\pi$
$983$	1328.10i	1.35107i	−0.737328	+	0.675535i	$0.763913\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	490.305i	0.496763i
$988$	0	0
$989$	106.491	0.107675
$990$	0	0
$991$	1160.02i	1.17055i	0.810834	+	0.585277i	$0.199014\pi$
$991$	1160.02i	1.17055i	−0.810834	+	0.585277i	$0.800986\pi$
$992$	0	0
$993$	−902.863	−0.909228
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−806.436	−0.808862	−0.404431	−	0.914568i	$-0.632530\pi$
$997$	−806.436	−0.808862	−0.404431	+	0.914568i	$0.632530\pi$
$998$	0	0
$999$	− 243.967i	− 0.244211i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.3.e.i.1951.8		12
4.3	odd	2	inner	2400.3.e.i.1951.5		12
5.2	odd	4		480.3.j.a.319.8	yes	12
5.3	odd	4		480.3.j.b.319.1	yes	12
5.4	even	2		2400.3.e.h.1951.5		12
15.2	even	4		1440.3.j.d.1279.11		12
15.8	even	4		1440.3.j.c.1279.12		12
20.3	even	4		480.3.j.a.319.7	✓	12
20.7	even	4		480.3.j.b.319.2	yes	12
20.19	odd	2		2400.3.e.h.1951.8		12
40.3	even	4		960.3.j.f.319.6		12
40.13	odd	4		960.3.j.g.319.12		12
40.27	even	4		960.3.j.g.319.11		12
40.37	odd	4		960.3.j.f.319.5		12
60.23	odd	4		1440.3.j.d.1279.12		12
60.47	odd	4		1440.3.j.c.1279.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.j.a.319.7	✓	12	20.3	even	4
480.3.j.a.319.8	yes	12	5.2	odd	4
480.3.j.b.319.1	yes	12	5.3	odd	4
480.3.j.b.319.2	yes	12	20.7	even	4
960.3.j.f.319.5		12	40.37	odd	4
960.3.j.f.319.6		12	40.3	even	4
960.3.j.g.319.11		12	40.27	even	4
960.3.j.g.319.12		12	40.13	odd	4
1440.3.j.c.1279.11		12	60.47	odd	4
1440.3.j.c.1279.12		12	15.8	even	4
1440.3.j.d.1279.11		12	15.2	even	4
1440.3.j.d.1279.12		12	60.23	odd	4
2400.3.e.h.1951.5		12	5.4	even	2
2400.3.e.h.1951.8		12	20.19	odd	2
2400.3.e.i.1951.5		12	4.3	odd	2	inner
2400.3.e.i.1951.8		12	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2400.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -7.06571i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -7.06571i q^{7} -3.00000 q^{9} +1.70305i q^{11} -19.3844 q^{13} -13.0069 q^{17} -13.2314i q^{19} +12.2382 q^{21} -5.26392i q^{23} -5.19615i q^{27} -33.0071 q^{29} +60.5509i q^{31} -2.94977 q^{33} +46.9515 q^{37} -33.5747i q^{39} +65.3233 q^{41} +20.2303i q^{43} +40.0636i q^{47} -0.924196 q^{49} -22.5286i q^{51} +11.2001 q^{53} +22.9174 q^{57} -39.8862i q^{59} +32.2030 q^{61} +21.1971i q^{63} -2.76692i q^{67} +9.11738 q^{69} +67.6353i q^{71} +81.8653 q^{73} +12.0333 q^{77} +143.514i q^{79} +9.00000 q^{81} -126.585i q^{83} -57.1699i q^{87} -11.2563 q^{89} +136.964i q^{91} -104.877 q^{93} +49.5443 q^{97} -5.10916i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 36 q^{9}+O(q^{10}) \) 12 * q - 36 * q^9	\( 12 q - 36 q^{9} + 80 q^{17} + 40 q^{29} + 320 q^{37} + 136 q^{41} - 212 q^{49} - 176 q^{53} + 48 q^{57} + 40 q^{61} + 448 q^{73} + 448 q^{77} + 108 q^{81} + 8 q^{89} - 144 q^{93} + 224 q^{97}+O(q^{100}) \) 12 * q - 36 * q^9 + 80 * q^17 + 40 * q^29 + 320 * q^37 + 136 * q^41 - 212 * q^49 - 176 * q^53 + 48 * q^57 + 40 * q^61 + 448 * q^73 + 448 * q^77 + 108 * q^81 + 8 * q^89 - 144 * q^93 + 224 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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