LMFDB - Embedded newform 1428.2.d.c.169.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1428,2,Mod(169,1428)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1428, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1428.169");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1428.d (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:	$11.4026374086$
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} + 43x^{10} + 647x^{8} + 4049x^{6} + 10288x^{4} + 9088x^{2} + 2304 $ x^12 + 43x^10 + 647x^8 + 4049x^6 + 10288x^4 + 9088*x^2 + 2304
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.6
Root			$3.91259i$ of defining polynomial
Character	$\chi$	$=$	1428.169
Dual form			1428.2.d.c.169.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.91259i q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.91259i q^{5} +1.00000i q^{7} -1.00000 q^{9} +4.44087i q^{11} +4.13795 q^{13} +3.91259 q^{15} +(-2.43375 - 3.32819i) q^{17} -3.38215 q^{19} +1.00000 q^{21} +2.97132i q^{23} -10.3084 q^{25} +1.00000i q^{27} +1.24420i q^{29} -7.09502i q^{31} +4.44087 q^{33} -3.91259 q^{35} -0.739392i q^{37} -4.13795i q^{39} +11.3248i q^{41} -2.01363 q^{43} -3.91259i q^{45} -7.09502 q^{47} -1.00000 q^{49} +(-3.32819 + 2.43375i) q^{51} -11.4463 q^{53} -17.3753 q^{55} +3.38215i q^{57} +4.15158 q^{59} +12.8080i q^{61} -1.00000i q^{63} +16.1901i q^{65} +4.86372 q^{67} +2.97132 q^{69} -6.42724i q^{71} -0.225362i q^{73} +10.3084i q^{75} -4.44087 q^{77} -17.2715i q^{79} +1.00000 q^{81} -3.39416 q^{83} +(13.0219 - 9.52226i) q^{85} +1.24420 q^{87} +8.44528 q^{89} +4.13795i q^{91} -7.09502 q^{93} -13.2330i q^{95} +15.8757i q^{97} -4.44087i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} - 6 q^{13} + 2 q^{15} - 2 q^{17} + 6 q^{19} + 12 q^{21} - 26 q^{25} + 10 q^{33} - 2 q^{35} - 18 q^{43} - 20 q^{47} - 12 q^{49} - 2 q^{51} + 16 q^{53} + 22 q^{55} - 12 q^{59} + 32 q^{67} - 6 q^{69} - 10 q^{77} + 12 q^{81} - 16 q^{83} + 14 q^{85} + 24 q^{87} + 4 q^{89} - 20 q^{93}+O(q^{100}) $ 12 * q - 12 * q^9 - 6 * q^13 + 2 * q^15 - 2 * q^17 + 6 * q^19 + 12 * q^21 - 26 * q^25 + 10 * q^33 - 2 * q^35 - 18 * q^43 - 20 * q^47 - 12 * q^49 - 2 * q^51 + 16 * q^53 + 22 * q^55 - 12 * q^59 + 32 * q^67 - 6 * q^69 - 10 * q^77 + 12 * q^81 - 16 * q^83 + 14 * q^85 + 24 * q^87 + 4 * q^89 - 20 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1428\mathbb{Z}\right)^\times$.

$n$	$409$	$715$	$953$	$1261$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$			3.91259i			1.74976i	0.484336	+	0.874882i	$0.339061\pi$
$5$			3.91259i			1.74976i	−0.484336	+	0.874882i	$0.660939\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			4.44087i			1.33897i	0.742824	+	0.669487i	$0.233486\pi$
$11$			4.44087i			1.33897i	−0.742824	+	0.669487i	$0.766514\pi$
$12$		0			0
$13$	4.13795			1.14766			0.573831	−	0.818974i	$-0.305457\pi$
$13$	4.13795			1.14766			0.573831	+	0.818974i	$0.305457\pi$
$14$		0			0
$15$	3.91259			1.01023
$16$		0			0
$17$	−2.43375	−	3.32819i	−0.590271	−	0.807205i
$17$	−2.43375	−	3.32819i	−0.590271	−	0.807205i
$18$		0			0
$19$	−3.38215			−0.775918			−0.387959	−	0.921677i	$-0.626820\pi$
$19$	−3.38215			−0.775918			−0.387959	+	0.921677i	$0.626820\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$			2.97132i			0.619562i	0.950808	+	0.309781i	$0.100256\pi$
$23$			2.97132i			0.619562i	−0.950808	+	0.309781i	$0.899744\pi$
$24$		0			0
$25$	−10.3084			−2.06167
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$			1.24420i			0.231042i	0.993305	+	0.115521i	$0.0368537\pi$
$29$			1.24420i			0.231042i	−0.993305	+	0.115521i	$0.963146\pi$
$30$		0			0
$31$		−	7.09502i		−	1.27430i	−0.770739	−	0.637151i	$-0.780113\pi$
$31$		−	7.09502i		−	1.27430i	0.770739	−	0.637151i	$-0.219887\pi$
$32$		0			0
$33$	4.44087			0.773057
$34$		0			0
$35$	−3.91259			−0.661349
$36$		0			0
$37$		−	0.739392i		−	0.121555i	−0.998151	−	0.0607777i	$-0.980642\pi$
$37$		−	0.739392i		−	0.121555i	0.998151	−	0.0607777i	$-0.0193581\pi$
$38$		0			0
$39$		−	4.13795i		−	0.662603i
$40$		0			0
$41$			11.3248i			1.76863i	0.466887	+	0.884317i	$0.345375\pi$
$41$			11.3248i			1.76863i	−0.466887	+	0.884317i	$0.654625\pi$
$42$		0			0
$43$	−2.01363			−0.307076			−0.153538	−	0.988143i	$-0.549067\pi$
$43$	−2.01363			−0.307076			−0.153538	+	0.988143i	$0.549067\pi$
$44$		0			0
$45$		−	3.91259i		−	0.583255i
$46$		0			0
$47$	−7.09502			−1.03491			−0.517457	−	0.855709i	$-0.673121\pi$
$47$	−7.09502			−1.03491			−0.517457	+	0.855709i	$0.673121\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−3.32819	+	2.43375i	−0.466040	+	0.340793i
$52$		0			0
$53$	−11.4463			−1.57227			−0.786137	−	0.618053i	$-0.787922\pi$
$53$	−11.4463			−1.57227			−0.786137	+	0.618053i	$0.787922\pi$
$54$		0			0
$55$	−17.3753			−2.34289
$56$		0			0
$57$			3.38215i			0.447977i
$58$		0			0
$59$	4.15158			0.540490			0.270245	−	0.962792i	$-0.412895\pi$
$59$	4.15158			0.540490			0.270245	+	0.962792i	$0.412895\pi$
$60$		0			0
$61$			12.8080i			1.63989i	0.572441	+	0.819946i	$0.305997\pi$
$61$			12.8080i			1.63989i	−0.572441	+	0.819946i	$0.694003\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$			16.1901i			2.00814i
$66$		0			0
$67$	4.86372			0.594197			0.297099	−	0.954847i	$-0.403981\pi$
$67$	4.86372			0.594197			0.297099	+	0.954847i	$0.403981\pi$
$68$		0			0
$69$	2.97132			0.357704
$70$		0			0
$71$		−	6.42724i		−	0.762774i	−0.924416	−	0.381387i	$-0.875447\pi$
$71$		−	6.42724i		−	0.762774i	0.924416	−	0.381387i	$-0.124553\pi$
$72$		0			0
$73$		−	0.225362i		−	0.0263766i	−0.999913	−	0.0131883i	$-0.995802\pi$
$73$		−	0.225362i		−	0.0263766i	0.999913	−	0.0131883i	$-0.00419809\pi$
$74$		0			0
$75$			10.3084i			1.19031i
$76$		0			0
$77$	−4.44087			−0.506085
$78$		0			0
$79$		−	17.2715i		−	1.94320i	−0.236637	−	0.971598i	$-0.576045\pi$
$79$		−	17.2715i		−	1.94320i	0.236637	−	0.971598i	$-0.423955\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−3.39416			−0.372557			−0.186279	−	0.982497i	$-0.559643\pi$
$83$	−3.39416			−0.372557			−0.186279	+	0.982497i	$0.559643\pi$
$84$		0			0
$85$	13.0219	−	9.52226i	1.41242	−	1.03283i
$86$		0			0
$87$	1.24420			0.133392
$88$		0			0
$89$	8.44528			0.895198			0.447599	−	0.894234i	$-0.352279\pi$
$89$	8.44528			0.895198			0.447599	+	0.894234i	$0.352279\pi$
$90$		0			0
$91$			4.13795i			0.433775i
$92$		0			0
$93$	−7.09502			−0.735719
$94$		0			0
$95$		−	13.2330i		−	1.35767i
$96$		0			0
$97$			15.8757i			1.61194i	0.591959	+	0.805968i	$0.298355\pi$
$97$			15.8757i			1.61194i	−0.591959	+	0.805968i	$0.701645\pi$
$98$		0			0
$99$		−	4.44087i		−	0.446325i
$100$		0			0
$101$	18.9345			1.88405			0.942024	−	0.335545i	$-0.108920\pi$
$101$	18.9345			1.88405			0.942024	+	0.335545i	$0.108920\pi$
$102$		0			0
$103$	1.91259			0.188453			0.0942266	−	0.995551i	$-0.469962\pi$
$103$	1.91259			0.188453			0.0942266	+	0.995551i	$0.469962\pi$
$104$		0			0
$105$			3.91259i			0.381830i
$106$		0			0
$107$			13.1298i			1.26930i	0.772798	+	0.634652i	$0.218856\pi$
$107$			13.1298i			1.26930i	−0.772798	+	0.634652i	$0.781144\pi$
$108$		0			0
$109$			19.9675i			1.91254i	0.292480	+	0.956272i	$0.405520\pi$
$109$			19.9675i			1.91254i	−0.292480	+	0.956272i	$0.594480\pi$
$110$		0			0
$111$	−0.739392			−0.0701800
$112$		0			0
$113$		−	5.62392i		−	0.529054i	−0.964378	−	0.264527i	$-0.914784\pi$
$113$		−	5.62392i		−	0.529054i	0.964378	−	0.264527i	$-0.0852158\pi$
$114$		0			0
$115$	−11.6255			−1.08409
$116$		0			0
$117$	−4.13795			−0.382554
$118$		0			0
$119$	3.32819	−	2.43375i	0.305095	−	0.223101i
$120$		0			0
$121$	−8.72136			−0.792851
$122$		0			0
$123$	11.3248			1.02112
$124$		0			0
$125$		−	20.7695i		−	1.85768i
$126$		0			0
$127$	−4.06177			−0.360424			−0.180212	−	0.983628i	$-0.557678\pi$
$127$	−4.06177			−0.360424			−0.180212	+	0.983628i	$0.557678\pi$
$128$		0			0
$129$			2.01363i			0.177290i
$130$		0			0
$131$			9.51241i			0.831103i	0.909570	+	0.415552i	$0.136411\pi$
$131$			9.51241i			0.831103i	−0.909570	+	0.415552i	$0.863589\pi$
$132$		0			0
$133$		−	3.38215i		−	0.293270i
$134$		0			0
$135$	−3.91259			−0.336742
$136$		0			0
$137$	−2.28867			−0.195534			−0.0977671	−	0.995209i	$-0.531170\pi$
$137$	−2.28867			−0.195534			−0.0977671	+	0.995209i	$0.531170\pi$
$138$		0			0
$139$		−	9.48759i		−	0.804727i	−0.915480	−	0.402363i	$-0.868189\pi$
$139$		−	9.48759i		−	0.804727i	0.915480	−	0.402363i	$-0.131811\pi$
$140$		0			0
$141$			7.09502i			0.597508i
$142$		0			0
$143$			18.3761i			1.53669i
$144$		0			0
$145$	−4.86803			−0.404268
$146$		0			0
$147$			1.00000i			0.0824786i
$148$		0			0
$149$	14.2759			1.16953			0.584764	−	0.811204i	$-0.301187\pi$
$149$	14.2759			1.16953			0.584764	+	0.811204i	$0.301187\pi$
$150$		0			0
$151$	−13.7258			−1.11699			−0.558493	−	0.829509i	$-0.688620\pi$
$151$	−13.7258			−1.11699			−0.558493	+	0.829509i	$0.688620\pi$
$152$		0			0
$153$	2.43375	+	3.32819i	0.196757	+	0.269068i
$154$		0			0
$155$	27.7599			2.22973
$156$		0			0
$157$	−11.9812			−0.956201			−0.478101	−	0.878305i	$-0.658675\pi$
$157$	−11.9812			−0.956201			−0.478101	+	0.878305i	$0.658675\pi$
$158$		0			0
$159$			11.4463i			0.907752i
$160$		0			0
$161$	−2.97132			−0.234172
$162$		0			0
$163$		−	14.1660i		−	1.10957i	−0.831994	−	0.554784i	$-0.812801\pi$
$163$		−	14.1660i		−	1.10957i	0.831994	−	0.554784i	$-0.187199\pi$
$164$		0			0
$165$			17.3753i			1.35267i
$166$		0			0
$167$			11.0289i			0.853444i	0.904383	+	0.426722i	$0.140332\pi$
$167$			11.0289i			0.853444i	−0.904383	+	0.426722i	$0.859668\pi$
$168$		0			0
$169$	4.12265			0.317127
$170$		0			0
$171$	3.38215			0.258639
$172$		0			0
$173$			22.3545i			1.69958i	0.527120	+	0.849791i	$0.323272\pi$
$173$			22.3545i			1.69958i	−0.527120	+	0.849791i	$0.676728\pi$
$174$		0			0
$175$		−	10.3084i		−	0.779240i
$176$		0			0
$177$		−	4.15158i		−	0.312052i
$178$		0			0
$179$	17.0718			1.27600			0.638002	−	0.770034i	$-0.279761\pi$
$179$	17.0718			1.27600			0.638002	+	0.770034i	$0.279761\pi$
$180$		0			0
$181$			12.5013i			0.929212i	0.885518	+	0.464606i	$0.153804\pi$
$181$			12.5013i			0.929212i	−0.885518	+	0.464606i	$0.846196\pi$
$182$		0			0
$183$	12.8080			0.946792
$184$		0			0
$185$	2.89294			0.212693
$186$		0			0
$187$	14.7801	−	10.8080i	1.08083	−	0.790357i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	−1.82087			−0.131753			−0.0658766	−	0.997828i	$-0.520984\pi$
$191$	−1.82087			−0.131753			−0.0658766	+	0.997828i	$0.520984\pi$
$192$		0			0
$193$		−	9.62546i		−	0.692856i	−0.938077	−	0.346428i	$-0.887395\pi$
$193$		−	9.62546i		−	0.692856i	0.938077	−	0.346428i	$-0.112605\pi$
$194$		0			0
$195$	16.1901			1.15940
$196$		0			0
$197$		−	23.9691i		−	1.70773i	−0.520498	−	0.853863i	$-0.674254\pi$
$197$		−	23.9691i		−	1.70773i	0.520498	−	0.853863i	$-0.325746\pi$
$198$		0			0
$199$		−	8.52253i		−	0.604146i	−0.953285	−	0.302073i	$-0.902321\pi$
$199$		−	8.52253i		−	0.604146i	0.953285	−	0.302073i	$-0.0976786\pi$
$200$		0			0
$201$		−	4.86372i		−	0.343060i
$202$		0			0
$203$	−1.24420			−0.0873255
$204$		0			0
$205$	−44.3092			−3.09469
$206$		0			0
$207$		−	2.97132i		−	0.206521i
$208$		0			0
$209$		−	15.0197i		−	1.03893i
$210$		0			0
$211$			13.1432i			0.904812i	0.891812	+	0.452406i	$0.149434\pi$
$211$			13.1432i			0.904812i	−0.891812	+	0.452406i	$0.850566\pi$
$212$		0			0
$213$	−6.42724			−0.440388
$214$		0			0
$215$		−	7.87851i		−	0.537310i
$216$		0			0
$217$	7.09502			0.481641
$218$		0			0
$219$	−0.225362			−0.0152285
$220$		0			0
$221$	−10.0707	−	13.7719i	−0.677431	−	0.926399i
$222$		0			0
$223$	20.0983			1.34588			0.672941	−	0.739696i	$-0.265031\pi$
$223$	20.0983			1.34588			0.672941	+	0.739696i	$0.265031\pi$
$224$		0			0
$225$	10.3084			0.687225
$226$		0			0
$227$		−	19.8594i		−	1.31811i	−0.752093	−	0.659057i	$-0.770955\pi$
$227$		−	19.8594i		−	1.31811i	0.752093	−	0.659057i	$-0.229045\pi$
$228$		0			0
$229$	−1.15598			−0.0763895			−0.0381947	−	0.999270i	$-0.512161\pi$
$229$	−1.15598			−0.0763895			−0.0381947	+	0.999270i	$0.512161\pi$
$230$		0			0
$231$			4.44087i			0.292188i
$232$		0			0
$233$			17.6135i			1.15390i	0.816781	+	0.576948i	$0.195756\pi$
$233$			17.6135i			1.15390i	−0.816781	+	0.576948i	$0.804244\pi$
$234$		0			0
$235$		−	27.7599i		−	1.81086i
$236$		0			0
$237$	−17.2715			−1.12190
$238$		0			0
$239$	−19.0481			−1.23212			−0.616060	−	0.787699i	$-0.711272\pi$
$239$	−19.0481			−1.23212			−0.616060	+	0.787699i	$0.711272\pi$
$240$		0			0
$241$		−	9.11143i		−	0.586919i	−0.955972	−	0.293459i	$-0.905193\pi$
$241$		−	9.11143i		−	0.586919i	0.955972	−	0.293459i	$-0.0948065\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$		−	3.91259i		−	0.249966i
$246$		0			0
$247$	−13.9952			−0.890492
$248$		0			0
$249$			3.39416i			0.215096i
$250$		0			0
$251$	9.18845			0.579970			0.289985	−	0.957031i	$-0.406350\pi$
$251$	9.18845			0.579970			0.289985	+	0.957031i	$0.406350\pi$
$252$		0			0
$253$	−13.1952			−0.829578
$254$		0			0
$255$	−9.52226	−	13.0219i	−0.596307	−	0.815460i
$256$		0			0
$257$	21.9051			1.36641			0.683203	−	0.730229i	$-0.260587\pi$
$257$	21.9051			1.36641			0.683203	+	0.730229i	$0.260587\pi$
$258$		0			0
$259$	0.739392			0.0459436
$260$		0			0
$261$		−	1.24420i		−	0.0770138i
$262$		0			0
$263$	−9.42142			−0.580949			−0.290475	−	0.956883i	$-0.593813\pi$
$263$	−9.42142			−0.580949			−0.290475	+	0.956883i	$0.593813\pi$
$264$		0			0
$265$		−	44.7848i		−	2.75111i
$266$		0			0
$267$		−	8.44528i		−	0.516843i
$268$		0			0
$269$		−	7.07613i		−	0.431439i	−0.976455	−	0.215720i	$-0.930790\pi$
$269$		−	7.07613i		−	0.431439i	0.976455	−	0.215720i	$-0.0692098\pi$
$270$		0			0
$271$	24.0891			1.46331			0.731654	−	0.681676i	$-0.238749\pi$
$271$	24.0891			1.46331			0.731654	+	0.681676i	$0.238749\pi$
$272$		0			0
$273$	4.13795			0.250440
$274$		0			0
$275$		−	45.7782i		−	2.76053i
$276$		0			0
$277$			3.15107i			0.189330i	0.995509	+	0.0946648i	$0.0301779\pi$
$277$			3.15107i			0.189330i	−0.995509	+	0.0946648i	$0.969822\pi$
$278$		0			0
$279$			7.09502i			0.424768i
$280$		0			0
$281$	−9.39416			−0.560408			−0.280204	−	0.959940i	$-0.590402\pi$
$281$	−9.39416			−0.560408			−0.280204	+	0.959940i	$0.590402\pi$
$282$		0			0
$283$		−	9.40859i		−	0.559283i	−0.960105	−	0.279641i	$-0.909784\pi$
$283$		−	9.40859i		−	0.559283i	0.960105	−	0.279641i	$-0.0902156\pi$
$284$		0			0
$285$	−13.2330			−0.783853
$286$		0			0
$287$	−11.3248			−0.668481
$288$		0			0
$289$	−5.15374	+	16.2000i	−0.303161	+	0.952939i
$290$		0			0
$291$	15.8757			0.930652
$292$		0			0
$293$	23.0260			1.34519			0.672595	−	0.740011i	$-0.265180\pi$
$293$	23.0260			1.34519			0.672595	+	0.740011i	$0.265180\pi$
$294$		0			0
$295$			16.2434i			0.945730i
$296$		0			0
$297$	−4.44087			−0.257686
$298$		0			0
$299$			12.2952i			0.711048i
$300$		0			0
$301$		−	2.01363i		−	0.116064i
$302$		0			0
$303$		−	18.9345i		−	1.08776i
$304$		0			0
$305$	−50.1123			−2.86942
$306$		0			0
$307$	3.90299			0.222755			0.111378	−	0.993778i	$-0.464474\pi$
$307$	3.90299			0.222755			0.111378	+	0.993778i	$0.464474\pi$
$308$		0			0
$309$		−	1.91259i		−	0.108804i
$310$		0			0
$311$			31.7471i			1.80021i	0.435668	+	0.900107i	$0.356512\pi$
$311$			31.7471i			1.80021i	−0.435668	+	0.900107i	$0.643488\pi$
$312$		0			0
$313$			12.1736i			0.688094i	0.938952	+	0.344047i	$0.111798\pi$
$313$			12.1736i			0.688094i	−0.938952	+	0.344047i	$0.888202\pi$
$314$		0			0
$315$	3.91259			0.220450
$316$		0			0
$317$		−	26.0614i		−	1.46376i	−0.681435	−	0.731878i	$-0.738644\pi$
$317$		−	26.0614i		−	1.46376i	0.681435	−	0.731878i	$-0.261356\pi$
$318$		0			0
$319$	−5.52532			−0.309359
$320$		0			0
$321$	13.1298			0.732832
$322$		0			0
$323$	8.23130	+	11.2564i	0.458002	+	0.626326i
$324$		0			0
$325$	−42.6556			−2.36610
$326$		0			0
$327$	19.9675			1.10421
$328$		0			0
$329$		−	7.09502i		−	0.391161i
$330$		0			0
$331$	−0.296669			−0.0163064			−0.00815321	−	0.999967i	$-0.502595\pi$
$331$	−0.296669			−0.0163064			−0.00815321	+	0.999967i	$0.502595\pi$
$332$		0			0
$333$			0.739392i			0.0405185i
$334$		0			0
$335$			19.0297i			1.03971i
$336$		0			0
$337$		−	1.76870i		−	0.0963472i	−0.998839	−	0.0481736i	$-0.984660\pi$
$337$		−	1.76870i		−	0.0963472i	0.998839	−	0.0481736i	$-0.0153401\pi$
$338$		0			0
$339$	−5.62392			−0.305450
$340$		0			0
$341$	31.5081			1.70626
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			11.6255i			0.625898i
$346$		0			0
$347$		−	15.7400i		−	0.844968i	−0.906370	−	0.422484i	$-0.861158\pi$
$347$		−	15.7400i		−	0.844968i	0.906370	−	0.422484i	$-0.138842\pi$
$348$		0			0
$349$	2.49358			0.133478			0.0667391	−	0.997770i	$-0.478740\pi$
$349$	2.49358			0.133478			0.0667391	+	0.997770i	$0.478740\pi$
$350$		0			0
$351$			4.13795i			0.220868i
$352$		0			0
$353$	−5.84631			−0.311168			−0.155584	−	0.987823i	$-0.549726\pi$
$353$	−5.84631			−0.311168			−0.155584	+	0.987823i	$0.549726\pi$
$354$		0			0
$355$	25.1472			1.33467
$356$		0			0
$357$	−2.43375	−	3.32819i	−0.128808	−	0.176147i
$358$		0			0
$359$	35.5431			1.87589			0.937946	−	0.346780i	$-0.112725\pi$
$359$	35.5431			1.87589			0.937946	+	0.346780i	$0.112725\pi$
$360$		0			0
$361$	−7.56106			−0.397951
$362$		0			0
$363$			8.72136i			0.457753i
$364$		0			0
$365$	0.881748			0.0461528
$366$		0			0
$367$			8.00000i			0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$			8.00000i			0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$		0			0
$369$		−	11.3248i		−	0.589545i
$370$		0			0
$371$		−	11.4463i		−	0.594263i
$372$		0			0
$373$	38.3708			1.98677			0.993383	−	0.114845i	$-0.0366373\pi$
$373$	38.3708			1.98677			0.993383	+	0.114845i	$0.0366373\pi$
$374$		0			0
$375$	−20.7695			−1.07253
$376$		0			0
$377$			5.14843i			0.265158i
$378$		0			0
$379$			35.3280i			1.81468i	0.420401	+	0.907338i	$0.361889\pi$
$379$			35.3280i			1.81468i	−0.420401	+	0.907338i	$0.638111\pi$
$380$		0			0
$381$			4.06177i			0.208091i
$382$		0			0
$383$	−4.57380			−0.233710			−0.116855	−	0.993149i	$-0.537281\pi$
$383$	−4.57380			−0.233710			−0.116855	+	0.993149i	$0.537281\pi$
$384$		0			0
$385$		−	17.3753i		−	0.885529i
$386$		0			0
$387$	2.01363			0.102359
$388$		0			0
$389$	−10.5786			−0.536355			−0.268178	−	0.963369i	$-0.586421\pi$
$389$	−10.5786			−0.536355			−0.268178	+	0.963369i	$0.586421\pi$
$390$		0			0
$391$	9.88911	−	7.23143i	0.500114	−	0.365709i
$392$		0			0
$393$	9.51241			0.479838
$394$		0			0
$395$	67.5763			3.40013
$396$		0			0
$397$		−	25.2074i		−	1.26512i	−0.774510	−	0.632562i	$-0.782003\pi$
$397$		−	25.2074i		−	1.26512i	0.774510	−	0.632562i	$-0.217997\pi$
$398$		0			0
$399$	−3.38215			−0.169319
$400$		0			0
$401$		−	1.31519i		−	0.0656777i	−0.999461	−	0.0328388i	$-0.989545\pi$
$401$		−	1.31519i		−	0.0656777i	0.999461	−	0.0328388i	$-0.0104548\pi$
$402$		0			0
$403$		−	29.3588i		−	1.46247i
$404$		0			0
$405$			3.91259i			0.194418i
$406$		0			0
$407$	3.28355			0.162759
$408$		0			0
$409$	33.3770			1.65039			0.825193	−	0.564851i	$-0.191066\pi$
$409$	33.3770			1.65039			0.825193	+	0.564851i	$0.191066\pi$
$410$		0			0
$411$			2.28867i			0.112892i
$412$		0			0
$413$			4.15158i			0.204286i
$414$		0			0
$415$		−	13.2800i		−	0.651887i
$416$		0			0
$417$	−9.48759			−0.464609
$418$		0			0
$419$		−	2.51074i		−	0.122658i	−0.998118	−	0.0613289i	$-0.980466\pi$
$419$		−	2.51074i		−	0.122658i	0.998118	−	0.0613289i	$-0.0195339\pi$
$420$		0			0
$421$	4.58299			0.223361			0.111681	−	0.993744i	$-0.464377\pi$
$421$	4.58299			0.223361			0.111681	+	0.993744i	$0.464377\pi$
$422$		0			0
$423$	7.09502			0.344972
$424$		0			0
$425$	25.0880	+	34.3082i	1.21695	+	1.66419i
$426$		0			0
$427$	−12.8080			−0.619821
$428$		0			0
$429$	18.3761			0.887208
$430$		0			0
$431$		−	3.49304i		−	0.168254i	−0.996455	−	0.0841268i	$-0.973190\pi$
$431$		−	3.49304i		−	0.168254i	0.996455	−	0.0841268i	$-0.0268101\pi$
$432$		0			0
$433$	14.8961			0.715861			0.357930	−	0.933748i	$-0.383483\pi$
$433$	14.8961			0.715861			0.357930	+	0.933748i	$0.383483\pi$
$434$		0			0
$435$			4.86803i			0.233404i
$436$		0			0
$437$		−	10.0494i		−	0.480730i
$438$		0			0
$439$		−	29.9522i		−	1.42954i	−0.699359	−	0.714771i	$-0.746531\pi$
$439$		−	29.9522i		−	1.42954i	0.699359	−	0.714771i	$-0.253469\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$	2.21494			0.105235			0.0526175	−	0.998615i	$-0.483244\pi$
$443$	2.21494			0.105235			0.0526175	+	0.998615i	$0.483244\pi$
$444$		0			0
$445$			33.0429i			1.56638i
$446$		0			0
$447$		−	14.2759i		−	0.675227i
$448$		0			0
$449$			34.7136i			1.63824i	0.573623	+	0.819119i	$0.305537\pi$
$449$			34.7136i			1.63824i	−0.573623	+	0.819119i	$0.694463\pi$
$450$		0			0
$451$	−50.2919			−2.36815
$452$		0			0
$453$			13.7258i			0.644893i
$454$		0			0
$455$	−16.1901			−0.759004
$456$		0			0
$457$	22.0110			1.02963			0.514815	−	0.857301i	$-0.327861\pi$
$457$	22.0110			1.02963			0.514815	+	0.857301i	$0.327861\pi$
$458$		0			0
$459$	3.32819	−	2.43375i	0.155347	−	0.113598i
$460$		0			0
$461$	−10.9289			−0.509009			−0.254505	−	0.967072i	$-0.581912\pi$
$461$	−10.9289			−0.509009			−0.254505	+	0.967072i	$0.581912\pi$
$462$		0			0
$463$	−15.9158			−0.739670			−0.369835	−	0.929097i	$-0.620586\pi$
$463$	−15.9158			−0.739670			−0.369835	+	0.929097i	$0.620586\pi$
$464$		0			0
$465$		−	27.7599i		−	1.28733i
$466$		0			0
$467$	11.0478			0.511230			0.255615	−	0.966779i	$-0.417722\pi$
$467$	11.0478			0.511230			0.255615	+	0.966779i	$0.417722\pi$
$468$		0			0
$469$			4.86372i			0.224586i
$470$		0			0
$471$			11.9812i			0.552063i
$472$		0			0
$473$		−	8.94227i		−	0.411166i
$474$		0			0
$475$	34.8645			1.59969
$476$		0			0
$477$	11.4463			0.524091
$478$		0			0
$479$			12.7967i			0.584698i	0.956312	+	0.292349i	$0.0944369\pi$
$479$			12.7967i			0.584698i	−0.956312	+	0.292349i	$0.905563\pi$
$480$		0			0
$481$		−	3.05957i		−	0.139504i
$482$		0			0
$483$			2.97132i			0.135200i
$484$		0			0
$485$	−62.1152			−2.82051
$486$		0			0
$487$			20.2716i			0.918593i	0.888283	+	0.459297i	$0.151899\pi$
$487$			20.2716i			0.918593i	−0.888283	+	0.459297i	$0.848101\pi$
$488$		0			0
$489$	−14.1660			−0.640609
$490$		0			0
$491$	−18.5049			−0.835112			−0.417556	−	0.908651i	$-0.637113\pi$
$491$	−18.5049			−0.835112			−0.417556	+	0.908651i	$0.637113\pi$
$492$		0			0
$493$	4.14093	−	3.02806i	0.186498	−	0.136377i
$494$		0			0
$495$	17.3753			0.780963
$496$		0			0
$497$	6.42724			0.288301
$498$		0			0
$499$		−	25.4594i		−	1.13972i	−0.821741	−	0.569861i	$-0.806997\pi$
$499$		−	25.4594i		−	1.13972i	0.821741	−	0.569861i	$-0.193003\pi$
$500$		0			0
$501$	11.0289			0.492736
$502$		0			0
$503$			16.6382i			0.741862i	0.928660	+	0.370931i	$0.120961\pi$
$503$			16.6382i			0.741862i	−0.928660	+	0.370931i	$0.879039\pi$
$504$		0			0
$505$			74.0828i			3.29664i
$506$		0			0
$507$		−	4.12265i		−	0.183094i
$508$		0			0
$509$	−0.537560			−0.0238269			−0.0119135	−	0.999929i	$-0.503792\pi$
$509$	−0.537560			−0.0238269			−0.0119135	+	0.999929i	$0.503792\pi$
$510$		0			0
$511$	0.225362			0.00996941
$512$		0			0
$513$		−	3.38215i		−	0.149326i
$514$		0			0
$515$			7.48319i			0.329749i
$516$		0			0
$517$		−	31.5081i		−	1.38572i
$518$		0			0
$519$	22.3545			0.981254
$520$		0			0
$521$			20.0360i			0.877794i	0.898537	+	0.438897i	$0.144631\pi$
$521$			20.0360i			0.877794i	−0.898537	+	0.438897i	$0.855369\pi$
$522$		0			0
$523$	4.88409			0.213566			0.106783	−	0.994282i	$-0.465945\pi$
$523$	4.88409			0.213566			0.106783	+	0.994282i	$0.465945\pi$
$524$		0			0
$525$	−10.3084			−0.449894
$526$		0			0
$527$	−23.6136	+	17.2675i	−1.02862	+	0.752183i
$528$		0			0
$529$	14.1713			0.616143
$530$		0			0
$531$	−4.15158			−0.180163
$532$		0			0
$533$			46.8614i			2.02979i
$534$		0			0
$535$	−51.3714			−2.22098
$536$		0			0
$537$		−	17.0718i		−	0.736702i
$538$		0			0
$539$		−	4.44087i		−	0.191282i
$540$		0			0
$541$		−	3.01890i		−	0.129792i	−0.997892	−	0.0648962i	$-0.979328\pi$
$541$		−	3.01890i		−	0.129792i	0.997892	−	0.0648962i	$-0.0206716\pi$
$542$		0			0
$543$	12.5013			0.536481
$544$		0			0
$545$	−78.1248			−3.34650
$546$		0			0
$547$		−	14.7273i		−	0.629696i	−0.949142	−	0.314848i	$-0.898046\pi$
$547$		−	14.7273i		−	0.629696i	0.949142	−	0.314848i	$-0.101954\pi$
$548$		0			0
$549$		−	12.8080i		−	0.546631i
$550$		0			0
$551$		−	4.20806i		−	0.179269i
$552$		0			0
$553$	17.2715			0.734459
$554$		0			0
$555$		−	2.89294i		−	0.122798i
$556$		0			0
$557$	3.41026			0.144497			0.0722487	−	0.997387i	$-0.476982\pi$
$557$	3.41026			0.144497			0.0722487	+	0.997387i	$0.476982\pi$
$558$		0			0
$559$	−8.33230			−0.352419
$560$		0			0
$561$	−10.8080	−	14.7801i	−0.456313	−	0.624016i
$562$		0			0
$563$	−38.0880			−1.60522			−0.802608	−	0.596506i	$-0.796555\pi$
$563$	−38.0880			−1.60522			−0.802608	+	0.596506i	$0.796555\pi$
$564$		0			0
$565$	22.0041			0.925720
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	−12.8096			−0.537007			−0.268504	−	0.963279i	$-0.586529\pi$
$569$	−12.8096			−0.537007			−0.268504	+	0.963279i	$0.586529\pi$
$570$		0			0
$571$		−	3.38984i		−	0.141860i	−0.997481	−	0.0709302i	$-0.977403\pi$
$571$		−	3.38984i		−	0.141860i	0.997481	−	0.0709302i	$-0.0225968\pi$
$572$		0			0
$573$			1.82087i			0.0760677i
$574$		0			0
$575$		−	30.6294i		−	1.27734i
$576$		0			0
$577$	39.5665			1.64718			0.823588	−	0.567188i	$-0.191969\pi$
$577$	39.5665			1.64718			0.823588	+	0.567188i	$0.191969\pi$
$578$		0			0
$579$	−9.62546			−0.400020
$580$		0			0
$581$		−	3.39416i		−	0.140813i
$582$		0			0
$583$		−	50.8317i		−	2.10523i
$584$		0			0
$585$		−	16.1901i		−	0.669379i
$586$		0			0
$587$	39.8462			1.64463			0.822314	−	0.569034i	$-0.192683\pi$
$587$	39.8462			1.64463			0.822314	+	0.569034i	$0.192683\pi$
$588$		0			0
$589$			23.9964i			0.988755i
$590$		0			0
$591$	−23.9691			−0.985956
$592$		0			0
$593$	33.9277			1.39324			0.696622	−	0.717439i	$-0.254686\pi$
$593$	33.9277			1.39324			0.696622	+	0.717439i	$0.254686\pi$
$594$		0			0
$595$	9.52226	+	13.0219i	0.390375	+	0.533844i
$596$		0			0
$597$	−8.52253			−0.348804
$598$		0			0
$599$	24.0574			0.982957			0.491479	−	0.870890i	$-0.336457\pi$
$599$	24.0574			0.982957			0.491479	+	0.870890i	$0.336457\pi$
$600$		0			0
$601$		−	1.95633i		−	0.0798004i	−0.999204	−	0.0399002i	$-0.987296\pi$
$601$		−	1.95633i		−	0.0798004i	0.999204	−	0.0399002i	$-0.0127040\pi$
$602$		0			0
$603$	−4.86372			−0.198066
$604$		0			0
$605$		−	34.1231i		−	1.38730i
$606$		0			0
$607$		−	3.36612i		−	0.136627i	−0.997664	−	0.0683133i	$-0.978238\pi$
$607$		−	3.36612i		−	0.136627i	0.997664	−	0.0683133i	$-0.0217617\pi$
$608$		0			0
$609$			1.24420i			0.0504174i
$610$		0			0
$611$	−29.3588			−1.18773
$612$		0			0
$613$	−33.5794			−1.35626			−0.678129	−	0.734943i	$-0.737209\pi$
$613$	−33.5794			−1.35626			−0.678129	+	0.734943i	$0.737209\pi$
$614$		0			0
$615$			44.3092i			1.78672i
$616$		0			0
$617$		−	5.51886i		−	0.222181i	−0.993810	−	0.111090i	$-0.964566\pi$
$617$		−	5.51886i		−	0.222181i	0.993810	−	0.111090i	$-0.0354343\pi$
$618$		0			0
$619$			23.5818i			0.947833i	0.880570	+	0.473916i	$0.157160\pi$
$619$			23.5818i			0.947833i	−0.880570	+	0.473916i	$0.842840\pi$
$620$		0			0
$621$	−2.97132			−0.119235
$622$		0			0
$623$			8.44528i			0.338353i
$624$		0			0
$625$	29.7206			1.18883
$626$		0			0
$627$	−15.0197			−0.599829
$628$		0			0
$629$	−2.46084	+	1.79949i	−0.0981202	+	0.0717506i
$630$		0			0
$631$	−25.5574			−1.01743			−0.508713	−	0.860936i	$-0.669879\pi$
$631$	−25.5574			−1.01743			−0.508713	+	0.860936i	$0.669879\pi$
$632$		0			0
$633$	13.1432			0.522394
$634$		0			0
$635$		−	15.8921i		−	0.630657i
$636$		0			0
$637$	−4.13795			−0.163952
$638$		0			0
$639$			6.42724i			0.254258i
$640$		0			0
$641$			13.5606i			0.535610i	0.963473	+	0.267805i	$0.0862983\pi$
$641$			13.5606i			0.535610i	−0.963473	+	0.267805i	$0.913702\pi$
$642$		0			0
$643$			29.7138i			1.17180i	0.810384	+	0.585899i	$0.199258\pi$
$643$			29.7138i			1.17180i	−0.810384	+	0.585899i	$0.800742\pi$
$644$		0			0
$645$	−7.87851			−0.310216
$646$		0			0
$647$	−19.2150			−0.755420			−0.377710	−	0.925924i	$-0.623288\pi$
$647$	−19.2150			−0.755420			−0.377710	+	0.925924i	$0.623288\pi$
$648$		0			0
$649$			18.4367i			0.723702i
$650$		0			0
$651$		−	7.09502i		−	0.278076i
$652$		0			0
$653$		−	5.79258i		−	0.226681i	−0.993556	−	0.113341i	$-0.963845\pi$
$653$		−	5.79258i		−	0.226681i	0.993556	−	0.113341i	$-0.0361552\pi$
$654$		0			0
$655$	−37.2182			−1.45423
$656$		0			0
$657$			0.225362i			0.00879220i
$658$		0			0
$659$	4.02491			0.156788			0.0783941	−	0.996922i	$-0.475021\pi$
$659$	4.02491			0.156788			0.0783941	+	0.996922i	$0.475021\pi$
$660$		0			0
$661$	−10.2090			−0.397085			−0.198543	−	0.980092i	$-0.563621\pi$
$661$	−10.2090			−0.397085			−0.198543	+	0.980092i	$0.563621\pi$
$662$		0			0
$663$	−13.7719	+	10.0707i	−0.534857	+	0.391115i
$664$		0			0
$665$	13.2330			0.513153
$666$		0			0
$667$	−3.69690			−0.143145
$668$		0			0
$669$		−	20.0983i		−	0.777045i
$670$		0			0
$671$	−56.8786			−2.19577
$672$		0			0
$673$			35.6268i			1.37331i	0.726982	+	0.686657i	$0.240922\pi$
$673$			35.6268i			1.37331i	−0.726982	+	0.686657i	$0.759078\pi$
$674$		0			0
$675$		−	10.3084i		−	0.396769i
$676$		0			0
$677$			30.3681i			1.16714i	0.812063	+	0.583570i	$0.198344\pi$
$677$			30.3681i			1.16714i	−0.812063	+	0.583570i	$0.801656\pi$
$678$		0			0
$679$	−15.8757			−0.609254
$680$		0			0
$681$	−19.8594			−0.761014
$682$		0			0
$683$		−	12.8762i		−	0.492695i	−0.969182	−	0.246347i	$-0.920770\pi$
$683$		−	12.8762i		−	0.492695i	0.969182	−	0.246347i	$-0.0792304\pi$
$684$		0			0
$685$		−	8.95463i		−	0.342139i
$686$		0			0
$687$			1.15598i			0.0441035i
$688$		0			0
$689$	−47.3643			−1.80444
$690$		0			0
$691$		−	14.5347i		−	0.552925i	−0.961025	−	0.276462i	$-0.910838\pi$
$691$		−	14.5347i		−	0.552925i	0.961025	−	0.276462i	$-0.0891621\pi$
$692$		0			0
$693$	4.44087			0.168695
$694$		0			0
$695$	37.1211			1.40808
$696$		0			0
$697$	37.6911	−	27.5617i	1.42765	−	1.04397i
$698$		0			0
$699$	17.6135			0.666202
$700$		0			0
$701$	3.63947			0.137461			0.0687304	−	0.997635i	$-0.478105\pi$
$701$	3.63947			0.137461			0.0687304	+	0.997635i	$0.478105\pi$
$702$		0			0
$703$			2.50074i			0.0943170i
$704$		0			0
$705$	−27.7599			−1.04550
$706$		0			0
$707$			18.9345i			0.712103i
$708$		0			0
$709$		−	8.00000i		−	0.300446i	−0.988652	−	0.150223i	$-0.952001\pi$
$709$		−	8.00000i		−	0.300446i	0.988652	−	0.150223i	$-0.0479992\pi$
$710$		0			0
$711$			17.2715i			0.647732i
$712$		0			0
$713$	21.0815			0.789510
$714$		0			0
$715$	−71.8983			−2.68884
$716$		0			0
$717$			19.0481i			0.711365i
$718$		0			0
$719$			18.0505i			0.673169i	0.941653	+	0.336585i	$0.109272\pi$
$719$			18.0505i			0.673169i	−0.941653	+	0.336585i	$0.890728\pi$
$720$		0			0
$721$			1.91259i			0.0712286i
$722$		0			0
$723$	−9.11143			−0.338858
$724$		0			0
$725$		−	12.8256i		−	0.476332i
$726$		0			0
$727$	47.2322			1.75175			0.875873	−	0.482542i	$-0.160286\pi$
$727$	47.2322			1.75175			0.875873	+	0.482542i	$0.160286\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	4.90067	+	6.70175i	0.181258	+	0.247873i
$732$		0			0
$733$	0.0333529			0.00123192			0.000615958	−	1.00000i	$-0.499804\pi$
$733$	0.0333529			0.00123192			0.000615958		1.00000i	$0.499804\pi$
$734$		0			0
$735$	−3.91259			−0.144318
$736$		0			0
$737$			21.5992i			0.795615i
$738$		0			0
$739$	18.0819			0.665153			0.332577	−	0.943076i	$-0.392082\pi$
$739$	18.0819			0.665153			0.332577	+	0.943076i	$0.392082\pi$
$740$		0			0
$741$			13.9952i			0.514126i
$742$		0			0
$743$		−	40.8270i		−	1.49780i	−0.662684	−	0.748899i	$-0.730583\pi$
$743$		−	40.8270i		−	1.49780i	0.662684	−	0.748899i	$-0.269417\pi$
$744$		0			0
$745$			55.8558i			2.04640i
$746$		0			0
$747$	3.39416			0.124186
$748$		0			0
$749$	−13.1298			−0.479751
$750$		0			0
$751$			37.8773i			1.38216i	0.722776	+	0.691082i	$0.242866\pi$
$751$			37.8773i			1.38216i	−0.722776	+	0.691082i	$0.757134\pi$
$752$		0			0
$753$		−	9.18845i		−	0.334846i
$754$		0			0
$755$		−	53.7033i		−	1.95446i
$756$		0			0
$757$	34.3043			1.24681			0.623406	−	0.781898i	$-0.285748\pi$
$757$	34.3043			1.24681			0.623406	+	0.781898i	$0.285748\pi$
$758$		0			0
$759$			13.1952i			0.478957i
$760$		0			0
$761$	−20.6687			−0.749238			−0.374619	−	0.927179i	$-0.622227\pi$
$761$	−20.6687			−0.749238			−0.374619	+	0.927179i	$0.622227\pi$
$762$		0			0
$763$	−19.9675			−0.722873
$764$		0			0
$765$	−13.0219	+	9.52226i	−0.470806	+	0.344278i
$766$		0			0
$767$	17.1791			0.620300
$768$		0			0
$769$	−17.6709			−0.637228			−0.318614	−	0.947885i	$-0.603217\pi$
$769$	−17.6709			−0.637228			−0.318614	+	0.947885i	$0.603217\pi$
$770$		0			0
$771$		−	21.9051i		−	0.788894i
$772$		0			0
$773$	−14.0888			−0.506738			−0.253369	−	0.967370i	$-0.581539\pi$
$773$	−14.0888			−0.506738			−0.253369	+	0.967370i	$0.581539\pi$
$774$		0			0
$775$			73.1381i			2.62720i
$776$		0			0
$777$		−	0.739392i		−	0.0265256i
$778$		0			0
$779$		−	38.3021i		−	1.37232i
$780$		0			0
$781$	28.5426			1.02133
$782$		0			0
$783$	−1.24420			−0.0444640
$784$		0			0
$785$		−	46.8774i		−	1.67313i
$786$		0			0
$787$			40.7335i			1.45199i	0.687699	+	0.725996i	$0.258621\pi$
$787$			40.7335i			1.45199i	−0.687699	+	0.725996i	$0.741379\pi$
$788$		0			0
$789$			9.42142i			0.335411i
$790$		0			0
$791$	5.62392			0.199964
$792$		0			0
$793$			52.9988i			1.88204i
$794$		0			0
$795$	−44.7848			−1.58835
$796$		0			0
$797$	−32.7425			−1.15980			−0.579899	−	0.814688i	$-0.696908\pi$
$797$	−32.7425			−1.15980			−0.579899	+	0.814688i	$0.696908\pi$
$798$		0			0
$799$	17.2675	+	23.6136i	0.610880	+	0.835389i
$800$		0			0
$801$	−8.44528			−0.298399
$802$		0			0
$803$	1.00080			0.0353176
$804$		0			0
$805$		−	11.6255i		−	0.409747i
$806$		0			0
$807$	−7.07613			−0.249092
$808$		0			0
$809$		−	43.7899i		−	1.53957i	−0.638302	−	0.769786i	$-0.720363\pi$
$809$		−	43.7899i		−	1.53957i	0.638302	−	0.769786i	$-0.279637\pi$
$810$		0			0
$811$		−	30.7971i		−	1.08143i	−0.841205	−	0.540717i	$-0.818153\pi$
$811$		−	30.7971i		−	1.08143i	0.841205	−	0.540717i	$-0.181847\pi$
$812$		0			0
$813$		−	24.0891i		−	0.844841i
$814$		0			0
$815$	55.4258			1.94148
$816$		0			0
$817$	6.81040			0.238266
$818$		0			0
$819$		−	4.13795i		−	0.144592i
$820$		0			0
$821$		−	10.9187i		−	0.381064i	−0.981681	−	0.190532i	$-0.938979\pi$
$821$		−	10.9187i		−	0.381064i	0.981681	−	0.190532i	$-0.0610213\pi$
$822$		0			0
$823$			4.95027i			0.172556i	0.996271	+	0.0862779i	$0.0274973\pi$
$823$			4.95027i			0.172556i	−0.996271	+	0.0862779i	$0.972503\pi$
$824$		0			0
$825$	−45.7782			−1.59379
$826$		0			0
$827$		−	20.1922i		−	0.702153i	−0.936347	−	0.351076i	$-0.885816\pi$
$827$		−	20.1922i		−	0.702153i	0.936347	−	0.351076i	$-0.114184\pi$
$828$		0			0
$829$	8.36610			0.290567			0.145283	−	0.989390i	$-0.453591\pi$
$829$	8.36610			0.290567			0.145283	+	0.989390i	$0.453591\pi$
$830$		0			0
$831$	3.15107			0.109310
$832$		0			0
$833$	2.43375	+	3.32819i	0.0843244	+	0.115315i
$834$		0			0
$835$	−43.1517			−1.49333
$836$		0			0
$837$	7.09502			0.245240
$838$		0			0
$839$			42.8842i			1.48053i	0.672316	+	0.740264i	$0.265299\pi$
$839$			42.8842i			1.48053i	−0.672316	+	0.740264i	$0.734701\pi$
$840$		0			0
$841$	27.4520			0.946620
$842$		0			0
$843$			9.39416i			0.323552i
$844$		0			0
$845$			16.1303i			0.554898i
$846$		0			0
$847$		−	8.72136i		−	0.299670i
$848$		0			0
$849$	−9.40859			−0.322902
$850$		0			0
$851$	2.19697			0.0753111
$852$		0			0
$853$			0.170379i			0.00583368i	0.999996	+	0.00291684i	$0.000928461\pi$
$853$			0.170379i			0.00583368i	−0.999996	+	0.00291684i	$0.999072\pi$
$854$		0			0
$855$			13.2330i			0.452558i
$856$		0			0
$857$		−	18.3396i		−	0.626468i	−0.949676	−	0.313234i	$-0.898588\pi$
$857$		−	18.3396i		−	0.626468i	0.949676	−	0.313234i	$-0.101412\pi$
$858$		0			0
$859$	43.8565			1.49636			0.748182	−	0.663493i	$-0.230927\pi$
$859$	43.8565			1.49636			0.748182	+	0.663493i	$0.230927\pi$
$860$		0			0
$861$			11.3248i			0.385948i
$862$		0			0
$863$	−24.9639			−0.849780			−0.424890	−	0.905245i	$-0.639687\pi$
$863$	−24.9639			−0.849780			−0.424890	+	0.905245i	$0.639687\pi$
$864$		0			0
$865$	−87.4641			−2.97387
$866$		0			0
$867$	16.2000	+	5.15374i	0.550180	+	0.175030i
$868$		0			0
$869$	76.7006			2.60189
$870$		0			0
$871$	20.1258			0.681938
$872$		0			0
$873$		−	15.8757i		−	0.537312i
$874$		0			0
$875$	20.7695			0.702137
$876$		0			0
$877$		−	3.09011i		−	0.104346i	−0.998638	−	0.0521728i	$-0.983385\pi$
$877$		−	3.09011i		−	0.104346i	0.998638	−	0.0521728i	$-0.0166146\pi$
$878$		0			0
$879$		−	23.0260i		−	0.776646i
$880$		0			0
$881$			52.2506i			1.76037i	0.474632	+	0.880184i	$0.342581\pi$
$881$			52.2506i			1.76037i	−0.474632	+	0.880184i	$0.657419\pi$
$882$		0			0
$883$	−8.41782			−0.283282			−0.141641	−	0.989918i	$-0.545238\pi$
$883$	−8.41782			−0.283282			−0.141641	+	0.989918i	$0.545238\pi$
$884$		0			0
$885$	16.2434			0.546017
$886$		0			0
$887$		−	51.8084i		−	1.73956i	−0.493442	−	0.869779i	$-0.664262\pi$
$887$		−	51.8084i		−	1.73956i	0.493442	−	0.869779i	$-0.335738\pi$
$888$		0			0
$889$		−	4.06177i		−	0.136227i
$890$		0			0
$891$			4.44087i			0.148775i
$892$		0			0
$893$	23.9964			0.803009
$894$		0			0
$895$			66.7949i			2.23271i
$896$		0			0
$897$	12.2952			0.410524
$898$		0			0
$899$	8.82760			0.294417
$900$		0			0
$901$	27.8575	+	38.0956i	0.928067	+	1.26915i
$902$		0			0
$903$	−2.01363			−0.0670094
$904$		0			0
$905$	−48.9124			−1.62590
$906$		0			0
$907$			13.4094i			0.445251i	0.974904	+	0.222625i	$0.0714627\pi$
$907$			13.4094i			0.445251i	−0.974904	+	0.222625i	$0.928537\pi$
$908$		0			0
$909$	−18.9345			−0.628016
$910$		0			0
$911$			1.14780i			0.0380285i	0.999819	+	0.0190142i	$0.00605278\pi$
$911$			1.14780i			0.0380285i	−0.999819	+	0.0190142i	$0.993947\pi$
$912$		0			0
$913$		−	15.0730i		−	0.498844i
$914$		0			0
$915$			50.1123i			1.65666i
$916$		0			0
$917$	−9.51241			−0.314128
$918$		0			0
$919$	13.0713			0.431182			0.215591	−	0.976484i	$-0.430832\pi$
$919$	13.0713			0.431182			0.215591	+	0.976484i	$0.430832\pi$
$920$		0			0
$921$		−	3.90299i		−	0.128608i
$922$		0			0
$923$		−	26.5956i		−	0.875406i
$924$		0			0
$925$			7.62193i			0.250608i
$926$		0			0
$927$	−1.91259			−0.0628177
$928$		0			0
$929$			40.0442i			1.31381i	0.753974	+	0.656904i	$0.228134\pi$
$929$			40.0442i			1.31381i	−0.753974	+	0.656904i	$0.771866\pi$
$930$		0			0
$931$	3.38215			0.110845
$932$		0			0
$933$	31.7471			1.03935
$934$		0			0
$935$	42.2872	+	57.8284i	1.38294	+	1.89119i
$936$		0			0
$937$	32.0049			1.04555			0.522777	−	0.852469i	$-0.324896\pi$
$937$	32.0049			1.04555			0.522777	+	0.852469i	$0.324896\pi$
$938$		0			0
$939$	12.1736			0.397271
$940$		0			0
$941$			30.3822i			0.990432i	0.868770	+	0.495216i	$0.164911\pi$
$941$			30.3822i			0.990432i	−0.868770	+	0.495216i	$0.835089\pi$
$942$		0			0
$943$	−33.6495			−1.09578
$944$		0			0
$945$		−	3.91259i		−	0.127277i
$946$		0			0
$947$		−	13.1718i		−	0.428027i	−0.976831	−	0.214014i	$-0.931346\pi$
$947$		−	13.1718i		−	0.428027i	0.976831	−	0.214014i	$-0.0686537\pi$
$948$		0			0
$949$		−	0.932536i		−	0.0302714i
$950$		0			0
$951$	−26.0614			−0.845100
$952$		0			0
$953$	−5.37966			−0.174264			−0.0871322	−	0.996197i	$-0.527770\pi$
$953$	−5.37966			−0.174264			−0.0871322	+	0.996197i	$0.527770\pi$
$954$		0			0
$955$		−	7.12430i		−	0.230537i
$956$		0			0
$957$			5.52532i			0.178608i
$958$		0			0
$959$		−	2.28867i		−	0.0739050i
$960$		0			0
$961$	−19.3393			−0.623847
$962$		0			0
$963$		−	13.1298i		−	0.423101i
$964$		0			0
$965$	37.6605			1.21233
$966$		0			0
$967$	−13.9294			−0.447940			−0.223970	−	0.974596i	$-0.571902\pi$
$967$	−13.9294			−0.447940			−0.223970	+	0.974596i	$0.571902\pi$
$968$		0			0
$969$	11.2564	−	8.23130i	0.361609	−	0.264427i
$970$		0			0
$971$	−47.9524			−1.53887			−0.769433	−	0.638728i	$-0.779461\pi$
$971$	−47.9524			−1.53887			−0.769433	+	0.638728i	$0.779461\pi$
$972$		0			0
$973$	9.48759			0.304158
$974$		0			0
$975$			42.6556i			1.36607i
$976$		0			0
$977$	−36.9027			−1.18062			−0.590311	−	0.807176i	$-0.700995\pi$
$977$	−36.9027			−1.18062			−0.590311	+	0.807176i	$0.700995\pi$
$978$		0			0
$979$			37.5044i			1.19865i
$980$		0			0
$981$		−	19.9675i		−	0.637514i
$982$		0			0
$983$			29.1267i			0.928997i	0.885574	+	0.464498i	$0.153765\pi$
$983$			29.1267i			0.928997i	−0.885574	+	0.464498i	$0.846235\pi$
$984$		0			0
$985$	93.7812			2.98812
$986$		0			0
$987$	−7.09502			−0.225837
$988$		0			0
$989$		−	5.98313i		−	0.190252i
$990$		0			0
$991$			15.8284i			0.502806i	0.967882	+	0.251403i	$0.0808920\pi$
$991$			15.8284i			0.502806i	−0.967882	+	0.251403i	$0.919108\pi$
$992$		0			0
$993$			0.296669i			0.00941451i
$994$		0			0
$995$	33.3452			1.05711
$996$		0			0
$997$		−	5.13388i		−	0.162592i	−0.996690	−	0.0812958i	$-0.974094\pi$
$997$		−	5.13388i		−	0.162592i	0.996690	−	0.0812958i	$-0.0259059\pi$
$998$		0			0
$999$	0.739392			0.0233933

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1428.2.d.c.169.6	✓	12
3.2	odd	2		4284.2.d.f.3025.2		12
17.16	even	2	inner	1428.2.d.c.169.7	yes	12
51.50	odd	2		4284.2.d.f.3025.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1428.2.d.c.169.6	✓	12	1.1	even	1	trivial
1428.2.d.c.169.7	yes	12	17.16	even	2	inner
4284.2.d.f.3025.2		12	3.2	odd	2
4284.2.d.f.3025.11		12	51.50	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1428.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.91259i q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.91259i q^{5} +1.00000i q^{7} -1.00000 q^{9} +4.44087i q^{11} +4.13795 q^{13} +3.91259 q^{15} +(-2.43375 - 3.32819i) q^{17} -3.38215 q^{19} +1.00000 q^{21} +2.97132i q^{23} -10.3084 q^{25} +1.00000i q^{27} +1.24420i q^{29} -7.09502i q^{31} +4.44087 q^{33} -3.91259 q^{35} -0.739392i q^{37} -4.13795i q^{39} +11.3248i q^{41} -2.01363 q^{43} -3.91259i q^{45} -7.09502 q^{47} -1.00000 q^{49} +(-3.32819 + 2.43375i) q^{51} -11.4463 q^{53} -17.3753 q^{55} +3.38215i q^{57} +4.15158 q^{59} +12.8080i q^{61} -1.00000i q^{63} +16.1901i q^{65} +4.86372 q^{67} +2.97132 q^{69} -6.42724i q^{71} -0.225362i q^{73} +10.3084i q^{75} -4.44087 q^{77} -17.2715i q^{79} +1.00000 q^{81} -3.39416 q^{83} +(13.0219 - 9.52226i) q^{85} +1.24420 q^{87} +8.44528 q^{89} +4.13795i q^{91} -7.09502 q^{93} -13.2330i q^{95} +15.8757i q^{97} -4.44087i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} - 6 q^{13} + 2 q^{15} - 2 q^{17} + 6 q^{19} + 12 q^{21} - 26 q^{25} + 10 q^{33} - 2 q^{35} - 18 q^{43} - 20 q^{47} - 12 q^{49} - 2 q^{51} + 16 q^{53} + 22 q^{55} - 12 q^{59} + 32 q^{67} - 6 q^{69} - 10 q^{77} + 12 q^{81} - 16 q^{83} + 14 q^{85} + 24 q^{87} + 4 q^{89} - 20 q^{93}+O(q^{100}) \) 12 * q - 12 * q^9 - 6 * q^13 + 2 * q^15 - 2 * q^17 + 6 * q^19 + 12 * q^21 - 26 * q^25 + 10 * q^33 - 2 * q^35 - 18 * q^43 - 20 * q^47 - 12 * q^49 - 2 * q^51 + 16 * q^53 + 22 * q^55 - 12 * q^59 + 32 * q^67 - 6 * q^69 - 10 * q^77 + 12 * q^81 - 16 * q^83 + 14 * q^85 + 24 * q^87 + 4 * q^89 - 20 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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