LMFDB - Embedded newform 444.2.e.a.73.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [444,2,Mod(73,444)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("444.73"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 444 = 2^{2} \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	444.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$3.54535784974$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 4 $ x^4 + 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			73.3
Root			$-0.874032i$ of defining polynomial
Character	$\chi$	$=$	444.73
Dual form			444.2.e.a.73.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.540182i q^{5} -1.23607 q^{7} +1.00000 q^{9} +2.47214 q^{11} +4.57649i q^{13} -0.540182i q^{15} +3.36861i q^{17} -1.74806i q^{19} +1.23607 q^{21} +8.61280i q^{23} +4.70820 q^{25} -1.00000 q^{27} +7.94510i q^{29} -6.32456i q^{31} -2.47214 q^{33} -0.667701i q^{35} +(-2.23607 + 5.65685i) q^{37} -4.57649i q^{39} +2.00000 q^{41} -2.82843i q^{43} +0.540182i q^{45} +4.00000 q^{47} -5.47214 q^{49} -3.36861i q^{51} +10.9443 q^{53} +1.33540i q^{55} +1.74806i q^{57} +2.28825i q^{59} -5.65685i q^{61} -1.23607 q^{63} -2.47214 q^{65} -7.70820 q^{67} -8.61280i q^{69} -8.94427 q^{71} +3.23607 q^{73} -4.70820 q^{75} -3.05573 q^{77} -9.56564i q^{79} +1.00000 q^{81} +1.52786 q^{83} -1.81966 q^{85} -7.94510i q^{87} -8.61280i q^{89} -5.65685i q^{91} +6.32456i q^{93} +0.944272 q^{95} -13.7295i q^{97} +2.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9} - 8 q^{11} - 4 q^{21} - 8 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{41} + 16 q^{47} - 4 q^{49} + 8 q^{53} + 4 q^{63} + 8 q^{65} - 4 q^{67} + 4 q^{73} + 8 q^{75} - 48 q^{77} + 4 q^{81}+ \cdots - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9 - 8 * q^11 - 4 * q^21 - 8 * q^25 - 4 * q^27 + 8 * q^33 + 8 * q^41 + 16 * q^47 - 4 * q^49 + 8 * q^53 + 4 * q^63 + 8 * q^65 - 4 * q^67 + 4 * q^73 + 8 * q^75 - 48 * q^77 + 4 * q^81 + 24 * q^83 - 52 * q^85 - 32 * q^95 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/444\mathbb{Z}\right)^\times$.

$n$	$149$	$223$	$409$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			0.540182i			0.241577i	0.992678	+	0.120788i	$0.0385422\pi$
$5$			0.540182i			0.241577i	−0.992678	+	0.120788i	$0.961458\pi$
$6$		0			0
$7$	−1.23607			−0.467190			−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607			−0.467190			−0.233595	+	0.972334i	$0.575049\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	2.47214			0.745377			0.372689	−	0.927957i	$-0.378436\pi$
$11$	2.47214			0.745377			0.372689	+	0.927957i	$0.378436\pi$
$12$		0			0
$13$			4.57649i			1.26929i	0.772804	+	0.634645i	$0.218854\pi$
$13$			4.57649i			1.26929i	−0.772804	+	0.634645i	$0.781146\pi$
$14$		0			0
$15$		−	0.540182i		−	0.139474i
$16$		0			0
$17$			3.36861i			0.817008i	0.912757	+	0.408504i	$0.133949\pi$
$17$			3.36861i			0.817008i	−0.912757	+	0.408504i	$0.866051\pi$
$18$		0			0
$19$		−	1.74806i		−	0.401033i	−0.979690	−	0.200517i	$-0.935738\pi$
$19$		−	1.74806i		−	0.401033i	0.979690	−	0.200517i	$-0.0642621\pi$
$20$		0			0
$21$	1.23607			0.269732
$22$		0			0
$23$			8.61280i			1.79589i	0.440104	+	0.897947i	$0.354941\pi$
$23$			8.61280i			1.79589i	−0.440104	+	0.897947i	$0.645059\pi$
$24$		0			0
$25$	4.70820			0.941641
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$			7.94510i			1.47537i	0.675146	+	0.737684i	$0.264081\pi$
$29$			7.94510i			1.47537i	−0.675146	+	0.737684i	$0.735919\pi$
$30$		0			0
$31$		−	6.32456i		−	1.13592i	−0.823055	−	0.567962i	$-0.807732\pi$
$31$		−	6.32456i		−	1.13592i	0.823055	−	0.567962i	$-0.192268\pi$
$32$		0			0
$33$	−2.47214			−0.430344
$34$		0			0
$35$		−	0.667701i		−	0.112862i
$36$		0			0
$37$	−2.23607	+	5.65685i	−0.367607	+	0.929981i
$37$	−2.23607	+	5.65685i	−0.367607	+	0.929981i
$38$		0			0
$39$		−	4.57649i		−	0.732825i
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$		−	2.82843i		−	0.431331i	−0.976467	−	0.215666i	$-0.930808\pi$
$43$		−	2.82843i		−	0.431331i	0.976467	−	0.215666i	$-0.0691921\pi$
$44$		0			0
$45$			0.540182i			0.0805255i
$46$		0			0
$47$	4.00000			0.583460			0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000			0.583460			0.291730	+	0.956501i	$0.405769\pi$
$48$		0			0
$49$	−5.47214			−0.781734
$50$		0			0
$51$		−	3.36861i		−	0.471700i
$52$		0			0
$53$	10.9443			1.50331			0.751656	−	0.659556i	$-0.229256\pi$
$53$	10.9443			1.50331			0.751656	+	0.659556i	$0.229256\pi$
$54$		0			0
$55$			1.33540i			0.180066i
$56$		0			0
$57$			1.74806i			0.231537i
$58$		0			0
$59$			2.28825i			0.297904i	0.988844	+	0.148952i	$0.0475900\pi$
$59$			2.28825i			0.297904i	−0.988844	+	0.148952i	$0.952410\pi$
$60$		0			0
$61$		−	5.65685i		−	0.724286i	−0.932123	−	0.362143i	$-0.882045\pi$
$61$		−	5.65685i		−	0.724286i	0.932123	−	0.362143i	$-0.117955\pi$
$62$		0			0
$63$	−1.23607			−0.155730
$64$		0			0
$65$	−2.47214			−0.306631
$66$		0			0
$67$	−7.70820			−0.941707			−0.470853	−	0.882211i	$-0.656054\pi$
$67$	−7.70820			−0.941707			−0.470853	+	0.882211i	$0.656054\pi$
$68$		0			0
$69$		−	8.61280i		−	1.03686i
$70$		0			0
$71$	−8.94427			−1.06149			−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427			−1.06149			−0.530745	+	0.847532i	$0.678088\pi$
$72$		0			0
$73$	3.23607			0.378753			0.189377	−	0.981905i	$-0.439353\pi$
$73$	3.23607			0.378753			0.189377	+	0.981905i	$0.439353\pi$
$74$		0			0
$75$	−4.70820			−0.543657
$76$		0			0
$77$	−3.05573			−0.348233
$78$		0			0
$79$		−	9.56564i		−	1.07622i	−0.842875	−	0.538110i	$-0.819139\pi$
$79$		−	9.56564i		−	1.07622i	0.842875	−	0.538110i	$-0.180861\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	1.52786			0.167705			0.0838524	−	0.996478i	$-0.473278\pi$
$83$	1.52786			0.167705			0.0838524	+	0.996478i	$0.473278\pi$
$84$		0			0
$85$	−1.81966			−0.197370
$86$		0			0
$87$		−	7.94510i		−	0.851804i
$88$		0			0
$89$		−	8.61280i		−	0.912955i	−0.889735	−	0.456478i	$-0.849111\pi$
$89$		−	8.61280i		−	0.912955i	0.889735	−	0.456478i	$-0.150889\pi$
$90$		0			0
$91$		−	5.65685i		−	0.592999i
$92$		0			0
$93$			6.32456i			0.655826i
$94$		0			0
$95$	0.944272			0.0968803
$96$		0			0
$97$		−	13.7295i		−	1.39402i	−0.717063	−	0.697008i	$-0.754514\pi$
$97$		−	13.7295i		−	1.39402i	0.717063	−	0.697008i	$-0.245486\pi$
$98$		0			0
$99$	2.47214			0.248459
$100$		0			0
$101$	−9.41641			−0.936968			−0.468484	−	0.883472i	$-0.655200\pi$
$101$	−9.41641			−0.936968			−0.468484	+	0.883472i	$0.655200\pi$
$102$		0			0
$103$		−	10.9010i		−	1.07411i	−0.843547	−	0.537056i	$-0.819536\pi$
$103$		−	10.9010i		−	1.07411i	0.843547	−	0.537056i	$-0.180464\pi$
$104$		0			0
$105$			0.667701i			0.0651610i
$106$		0			0
$107$	2.47214			0.238990			0.119495	−	0.992835i	$-0.461872\pi$
$107$	2.47214			0.238990			0.119495	+	0.992835i	$0.461872\pi$
$108$		0			0
$109$			13.7295i			1.31505i	0.753435	+	0.657523i	$0.228396\pi$
$109$			13.7295i			1.31505i	−0.753435	+	0.657523i	$0.771604\pi$
$110$		0			0
$111$	2.23607	−	5.65685i	0.212238	−	0.536925i
$112$		0			0
$113$		−	1.62054i		−	0.152448i	−0.997091	−	0.0762240i	$-0.975714\pi$
$113$		−	1.62054i		−	0.152448i	0.997091	−	0.0762240i	$-0.0242864\pi$
$114$		0			0
$115$	−4.65248			−0.433846
$116$		0			0
$117$			4.57649i			0.423097i
$118$		0			0
$119$		−	4.16383i		−	0.381698i
$120$		0			0
$121$	−4.88854			−0.444413
$122$		0			0
$123$	−2.00000			−0.180334
$124$		0			0
$125$			5.24419i			0.469055i
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$			2.82843i			0.249029i
$130$		0			0
$131$			0.127520i			0.0111414i	0.999984	+	0.00557072i	$0.00177322\pi$
$131$			0.127520i			0.0111414i	−0.999984	+	0.00557072i	$0.998227\pi$
$132$		0			0
$133$			2.16073i			0.187359i
$134$		0			0
$135$		−	0.540182i		−	0.0464914i
$136$		0			0
$137$	−0.472136			−0.0403373			−0.0201686	−	0.999797i	$-0.506420\pi$
$137$	−0.472136			−0.0403373			−0.0201686	+	0.999797i	$0.506420\pi$
$138$		0			0
$139$	7.70820			0.653801			0.326901	−	0.945059i	$-0.393996\pi$
$139$	7.70820			0.653801			0.326901	+	0.945059i	$0.393996\pi$
$140$		0			0
$141$	−4.00000			−0.336861
$142$		0			0
$143$			11.3137i			0.946100i
$144$		0			0
$145$	−4.29180			−0.356414
$146$		0			0
$147$	5.47214			0.451334
$148$		0			0
$149$	8.47214			0.694064			0.347032	−	0.937853i	$-0.387189\pi$
$149$	8.47214			0.694064			0.347032	+	0.937853i	$0.387189\pi$
$150$		0			0
$151$	−20.9443			−1.70442			−0.852210	−	0.523199i	$-0.824738\pi$
$151$	−20.9443			−1.70442			−0.852210	+	0.523199i	$0.824738\pi$
$152$		0			0
$153$			3.36861i			0.272336i
$154$		0			0
$155$	3.41641			0.274412
$156$		0			0
$157$	24.1803			1.92980			0.964901	−	0.262615i	$-0.0845850\pi$
$157$	24.1803			1.92980			0.964901	+	0.262615i	$0.0845850\pi$
$158$		0			0
$159$	−10.9443			−0.867937
$160$		0			0
$161$		−	10.6460i		−	0.839023i
$162$		0			0
$163$			10.6460i			0.833860i	0.908939	+	0.416930i	$0.136894\pi$
$163$			10.6460i			0.833860i	−0.908939	+	0.416930i	$0.863106\pi$
$164$		0			0
$165$		−	1.33540i		−	0.103961i
$166$		0			0
$167$		−	5.78437i		−	0.447608i	−0.974634	−	0.223804i	$-0.928152\pi$
$167$		−	5.78437i		−	0.447608i	0.974634	−	0.223804i	$-0.0718476\pi$
$168$		0			0
$169$	−7.94427			−0.611098
$170$		0			0
$171$		−	1.74806i		−	0.133678i
$172$		0			0
$173$	11.5279			0.876447			0.438224	−	0.898866i	$-0.355608\pi$
$173$	11.5279			0.876447			0.438224	+	0.898866i	$0.355608\pi$
$174$		0			0
$175$	−5.81966			−0.439925
$176$		0			0
$177$		−	2.28825i		−	0.171995i
$178$		0			0
$179$			13.1893i			0.985814i	0.870082	+	0.492907i	$0.164066\pi$
$179$			13.1893i			0.985814i	−0.870082	+	0.492907i	$0.835934\pi$
$180$		0			0
$181$	7.23607			0.537853			0.268926	−	0.963161i	$-0.413331\pi$
$181$	7.23607			0.537853			0.268926	+	0.963161i	$0.413331\pi$
$182$		0			0
$183$			5.65685i			0.418167i
$184$		0			0
$185$	−3.05573	−	1.20788i	−0.224662	−	0.0888053i
$186$		0			0
$187$			8.32766i			0.608979i
$188$		0			0
$189$	1.23607			0.0899107
$190$		0			0
$191$		−	5.78437i		−	0.418543i	−0.977858	−	0.209271i	$-0.932891\pi$
$191$		−	5.78437i		−	0.418543i	0.977858	−	0.209271i	$-0.0671092\pi$
$192$		0			0
$193$			11.3137i			0.814379i	0.913344	+	0.407189i	$0.133491\pi$
$193$			11.3137i			0.814379i	−0.913344	+	0.407189i	$0.866509\pi$
$194$		0			0
$195$	2.47214			0.177033
$196$		0			0
$197$	−17.4164			−1.24087			−0.620434	−	0.784259i	$-0.713043\pi$
$197$	−17.4164			−1.24087			−0.620434	+	0.784259i	$0.713043\pi$
$198$		0			0
$199$			2.82843i			0.200502i	0.994962	+	0.100251i	$0.0319646\pi$
$199$			2.82843i			0.200502i	−0.994962	+	0.100251i	$0.968035\pi$
$200$		0			0
$201$	7.70820			0.543695
$202$		0			0
$203$		−	9.82068i		−	0.689277i
$204$		0			0
$205$			1.08036i			0.0754558i
$206$		0			0
$207$			8.61280i			0.598631i
$208$		0			0
$209$		−	4.32145i		−	0.298921i
$210$		0			0
$211$	17.8885			1.23150			0.615749	−	0.787942i	$-0.288854\pi$
$211$	17.8885			1.23150			0.615749	+	0.787942i	$0.288854\pi$
$212$		0			0
$213$	8.94427			0.612851
$214$		0			0
$215$	1.52786			0.104199
$216$		0			0
$217$			7.81758i			0.530692i
$218$		0			0
$219$	−3.23607			−0.218673
$220$		0			0
$221$	−15.4164			−1.03702
$222$		0			0
$223$	−3.05573			−0.204627			−0.102313	−	0.994752i	$-0.532624\pi$
$223$	−3.05573			−0.204627			−0.102313	+	0.994752i	$0.532624\pi$
$224$		0			0
$225$	4.70820			0.313880
$226$		0			0
$227$			13.1893i			0.875404i	0.899120	+	0.437702i	$0.144208\pi$
$227$			13.1893i			0.875404i	−0.899120	+	0.437702i	$0.855792\pi$
$228$		0			0
$229$	−0.472136			−0.0311996			−0.0155998	−	0.999878i	$-0.504966\pi$
$229$	−0.472136			−0.0311996			−0.0155998	+	0.999878i	$0.504966\pi$
$230$		0			0
$231$	3.05573			0.201052
$232$		0			0
$233$	−16.4721			−1.07913			−0.539563	−	0.841945i	$-0.681410\pi$
$233$	−16.4721			−1.07913			−0.539563	+	0.841945i	$0.681410\pi$
$234$		0			0
$235$			2.16073i			0.140950i
$236$		0			0
$237$			9.56564i			0.621355i
$238$		0			0
$239$			19.9265i			1.28894i	0.764630	+	0.644469i	$0.222922\pi$
$239$			19.9265i			1.28894i	−0.764630	+	0.644469i	$0.777078\pi$
$240$		0			0
$241$		−	3.24109i		−	0.208777i	−0.994537	−	0.104388i	$-0.966711\pi$
$241$		−	3.24109i		−	0.208777i	0.994537	−	0.104388i	$-0.0332885\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	2.95595i		−	0.188849i
$246$		0			0
$247$	8.00000			0.509028
$248$		0			0
$249$	−1.52786			−0.0968244
$250$		0			0
$251$		−	7.27740i		−	0.459345i	−0.973268	−	0.229673i	$-0.926234\pi$
$251$		−	7.27740i		−	0.459345i	0.973268	−	0.229673i	$-0.0737656\pi$
$252$		0			0
$253$			21.2920i			1.33862i
$254$		0			0
$255$	1.81966			0.113952
$256$		0			0
$257$		−	27.9991i		−	1.74654i	−0.487239	−	0.873269i	$-0.661996\pi$
$257$		−	27.9991i		−	1.74654i	0.487239	−	0.873269i	$-0.338004\pi$
$258$		0			0
$259$	2.76393	−	6.99226i	0.171742	−	0.434478i
$260$		0			0
$261$			7.94510i			0.491789i
$262$		0			0
$263$	27.4164			1.69057			0.845284	−	0.534317i	$-0.179431\pi$
$263$	27.4164			1.69057			0.845284	+	0.534317i	$0.179431\pi$
$264$		0			0
$265$			5.91189i			0.363165i
$266$		0			0
$267$			8.61280i			0.527095i
$268$		0			0
$269$	31.8885			1.94428			0.972139	−	0.234403i	$-0.0753137\pi$
$269$	31.8885			1.94428			0.972139	+	0.234403i	$0.0753137\pi$
$270$		0			0
$271$	−1.81966			−0.110536			−0.0552682	−	0.998472i	$-0.517601\pi$
$271$	−1.81966			−0.110536			−0.0552682	+	0.998472i	$0.517601\pi$
$272$		0			0
$273$			5.65685i			0.342368i
$274$		0			0
$275$	11.6393			0.701877
$276$		0			0
$277$		−	8.07262i		−	0.485037i	−0.970147	−	0.242518i	$-0.922027\pi$
$277$		−	8.07262i		−	0.485037i	0.970147	−	0.242518i	$-0.0779735\pi$
$278$		0			0
$279$		−	6.32456i		−	0.378641i
$280$		0			0
$281$		−	21.0069i		−	1.25316i	−0.779355	−	0.626582i	$-0.784453\pi$
$281$		−	21.0069i		−	1.25316i	0.779355	−	0.626582i	$-0.215547\pi$
$282$		0			0
$283$			24.3755i			1.44897i	0.689290	+	0.724486i	$0.257923\pi$
$283$			24.3755i			1.44897i	−0.689290	+	0.724486i	$0.742077\pi$
$284$		0			0
$285$	−0.944272			−0.0559338
$286$		0			0
$287$	−2.47214			−0.145926
$288$		0			0
$289$	5.65248			0.332499
$290$		0			0
$291$			13.7295i			0.804836i
$292$		0			0
$293$	−26.0000			−1.51894			−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000			−1.51894			−0.759468	+	0.650545i	$0.774541\pi$
$294$		0			0
$295$	−1.23607			−0.0719667
$296$		0			0
$297$	−2.47214			−0.143448
$298$		0			0
$299$	−39.4164			−2.27951
$300$		0			0
$301$			3.49613i			0.201513i
$302$		0			0
$303$	9.41641			0.540958
$304$		0			0
$305$	3.05573			0.174970
$306$		0			0
$307$	−20.9443			−1.19535			−0.597676	−	0.801737i	$-0.703909\pi$
$307$	−20.9443			−1.19535			−0.597676	+	0.801737i	$0.703909\pi$
$308$		0			0
$309$			10.9010i			0.620139i
$310$		0			0
$311$			6.19704i			0.351402i	0.984444	+	0.175701i	$0.0562191\pi$
$311$			6.19704i			0.351402i	−0.984444	+	0.175701i	$0.943781\pi$
$312$		0			0
$313$		−	7.81758i		−	0.441876i	−0.975288	−	0.220938i	$-0.929088\pi$
$313$		−	7.81758i		−	0.441876i	0.975288	−	0.220938i	$-0.0709118\pi$
$314$		0			0
$315$		−	0.667701i		−	0.0376207i
$316$		0			0
$317$	2.94427			0.165367			0.0826834	−	0.996576i	$-0.473651\pi$
$317$	2.94427			0.165367			0.0826834	+	0.996576i	$0.473651\pi$
$318$		0			0
$319$			19.6414i			1.09971i
$320$		0			0
$321$	−2.47214			−0.137981
$322$		0			0
$323$	5.88854			0.327647
$324$		0			0
$325$			21.5471i			1.19522i
$326$		0			0
$327$		−	13.7295i		−	0.759242i
$328$		0			0
$329$	−4.94427			−0.272587
$330$		0			0
$331$		−	25.7109i		−	1.41320i	−0.707614	−	0.706599i	$-0.750229\pi$
$331$		−	25.7109i		−	1.41320i	0.707614	−	0.706599i	$-0.249771\pi$
$332$		0			0
$333$	−2.23607	+	5.65685i	−0.122536	+	0.309994i
$334$		0			0
$335$		−	4.16383i		−	0.227494i
$336$		0			0
$337$	4.47214			0.243613			0.121806	−	0.992554i	$-0.461131\pi$
$337$	4.47214			0.243613			0.121806	+	0.992554i	$0.461131\pi$
$338$		0			0
$339$			1.62054i			0.0880159i
$340$		0			0
$341$		−	15.6352i		−	0.846691i
$342$		0			0
$343$	15.4164			0.832408
$344$		0			0
$345$	4.65248			0.250481
$346$		0			0
$347$		−	26.2511i		−	1.40923i	−0.709589	−	0.704615i	$-0.751120\pi$
$347$		−	26.2511i		−	1.40923i	0.709589	−	0.704615i	$-0.248880\pi$
$348$		0			0
$349$	7.81966			0.418577			0.209288	−	0.977854i	$-0.432885\pi$
$349$	7.81966			0.418577			0.209288	+	0.977854i	$0.432885\pi$
$350$		0			0
$351$		−	4.57649i		−	0.244275i
$352$		0			0
$353$		−	26.2511i		−	1.39720i	−0.715511	−	0.698602i	$-0.753806\pi$
$353$		−	26.2511i		−	1.39720i	0.715511	−	0.698602i	$-0.246194\pi$
$354$		0			0
$355$		−	4.83153i		−	0.256431i
$356$		0			0
$357$			4.16383i			0.220373i
$358$		0			0
$359$	25.8885			1.36635			0.683173	−	0.730257i	$-0.260600\pi$
$359$	25.8885			1.36635			0.683173	+	0.730257i	$0.260600\pi$
$360$		0			0
$361$	15.9443			0.839172
$362$		0			0
$363$	4.88854			0.256582
$364$		0			0
$365$			1.74806i			0.0914979i
$366$		0			0
$367$	−12.0000			−0.626395			−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000			−0.626395			−0.313197	+	0.949688i	$0.601400\pi$
$368$		0			0
$369$	2.00000			0.104116
$370$		0			0
$371$	−13.5279			−0.702332
$372$		0			0
$373$	−14.6525			−0.758676			−0.379338	−	0.925258i	$-0.623848\pi$
$373$	−14.6525			−0.758676			−0.379338	+	0.925258i	$0.623848\pi$
$374$		0			0
$375$		−	5.24419i		−	0.270809i
$376$		0			0
$377$	−36.3607			−1.87267
$378$		0			0
$379$	13.8197			0.709868			0.354934	−	0.934891i	$-0.384503\pi$
$379$	13.8197			0.709868			0.354934	+	0.934891i	$0.384503\pi$
$380$		0			0
$381$	−12.0000			−0.614779
$382$		0			0
$383$			27.5865i			1.40960i	0.709405	+	0.704801i	$0.248964\pi$
$383$			27.5865i			1.40960i	−0.709405	+	0.704801i	$0.751036\pi$
$384$		0			0
$385$		−	1.65065i		−	0.0841248i
$386$		0			0
$387$		−	2.82843i		−	0.143777i
$388$		0			0
$389$		−	2.70091i		−	0.136941i	−0.997653	−	0.0684707i	$-0.978188\pi$
$389$		−	2.70091i		−	0.136941i	0.997653	−	0.0684707i	$-0.0218120\pi$
$390$		0			0
$391$	−29.0132			−1.46726
$392$		0			0
$393$		−	0.127520i		−	0.00643251i
$394$		0			0
$395$	5.16718			0.259989
$396$		0			0
$397$	−19.2361			−0.965431			−0.482715	−	0.875777i	$-0.660349\pi$
$397$	−19.2361			−0.965431			−0.482715	+	0.875777i	$0.660349\pi$
$398$		0			0
$399$		−	2.16073i		−	0.108172i
$400$		0			0
$401$		−	25.9960i		−	1.29818i	−0.760712	−	0.649090i	$-0.775150\pi$
$401$		−	25.9960i		−	1.29818i	0.760712	−	0.649090i	$-0.224850\pi$
$402$		0			0
$403$	28.9443			1.44182
$404$		0			0
$405$			0.540182i			0.0268418i
$406$		0			0
$407$	−5.52786	+	13.9845i	−0.274006	+	0.693187i
$408$		0			0
$409$		−	32.0354i		−	1.58405i	−0.610488	−	0.792025i	$-0.709027\pi$
$409$		−	32.0354i		−	1.58405i	0.610488	−	0.792025i	$-0.290973\pi$
$410$		0			0
$411$	0.472136			0.0232887
$412$		0			0
$413$		−	2.82843i		−	0.139178i
$414$		0			0
$415$			0.825324i			0.0405136i
$416$		0			0
$417$	−7.70820			−0.377472
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$		−	17.2256i		−	0.839524i	−0.907634	−	0.419762i	$-0.862113\pi$
$421$		−	17.2256i		−	0.839524i	0.907634	−	0.419762i	$-0.137887\pi$
$422$		0			0
$423$	4.00000			0.194487
$424$		0			0
$425$			15.8601i			0.769328i
$426$		0			0
$427$			6.99226i			0.338379i
$428$		0			0
$429$		−	11.3137i		−	0.546231i
$430$		0			0
$431$		−	12.1089i		−	0.583267i	−0.956530	−	0.291633i	$-0.905801\pi$
$431$		−	12.1089i		−	0.583267i	0.956530	−	0.291633i	$-0.0941987\pi$
$432$		0			0
$433$	3.81966			0.183561			0.0917806	−	0.995779i	$-0.470744\pi$
$433$	3.81966			0.183561			0.0917806	+	0.995779i	$0.470744\pi$
$434$		0			0
$435$	4.29180			0.205776
$436$		0			0
$437$	15.0557			0.720213
$438$		0			0
$439$		−	34.6088i		−	1.65179i	−0.563825	−	0.825895i	$-0.690671\pi$
$439$		−	34.6088i		−	1.65179i	0.563825	−	0.825895i	$-0.309329\pi$
$440$		0			0
$441$	−5.47214			−0.260578
$442$		0			0
$443$	−35.4164			−1.68268			−0.841342	−	0.540503i	$-0.818234\pi$
$443$	−35.4164			−1.68268			−0.841342	+	0.540503i	$0.818234\pi$
$444$		0			0
$445$	4.65248			0.220549
$446$		0			0
$447$	−8.47214			−0.400718
$448$		0			0
$449$		−	10.3609i		−	0.488959i	−0.969654	−	0.244480i	$-0.921383\pi$
$449$		−	10.3609i		−	0.488959i	0.969654	−	0.244480i	$-0.0786172\pi$
$450$		0			0
$451$	4.94427			0.232817
$452$		0			0
$453$	20.9443			0.984048
$454$		0			0
$455$	3.05573			0.143255
$456$		0			0
$457$		−	32.0354i		−	1.49855i	−0.662256	−	0.749277i	$-0.730401\pi$
$457$		−	32.0354i		−	1.49855i	0.662256	−	0.749277i	$-0.269599\pi$
$458$		0			0
$459$		−	3.36861i		−	0.157233i
$460$		0			0
$461$		−	25.8384i		−	1.20341i	−0.798717	−	0.601707i	$-0.794487\pi$
$461$		−	25.8384i		−	1.20341i	0.798717	−	0.601707i	$-0.205513\pi$
$462$		0			0
$463$			25.4558i			1.18303i	0.806293	+	0.591517i	$0.201471\pi$
$463$			25.4558i			1.18303i	−0.806293	+	0.591517i	$0.798529\pi$
$464$		0			0
$465$	−3.41641			−0.158432
$466$		0			0
$467$		−	34.3237i		−	1.58831i	−0.607715	−	0.794155i	$-0.707914\pi$
$467$		−	34.3237i		−	1.58831i	0.607715	−	0.794155i	$-0.292086\pi$
$468$		0			0
$469$	9.52786			0.439956
$470$		0			0
$471$	−24.1803			−1.11417
$472$		0			0
$473$		−	6.99226i		−	0.321504i
$474$		0			0
$475$		−	8.23024i		−	0.377629i
$476$		0			0
$477$	10.9443			0.501104
$478$		0			0
$479$			42.1413i			1.92548i	0.270422	+	0.962742i	$0.412837\pi$
$479$			42.1413i			1.92548i	−0.270422	+	0.962742i	$0.587163\pi$
$480$		0			0
$481$	−25.8885	−	10.2333i	−1.18042	−	0.466600i
$482$		0			0
$483$			10.6460i			0.484410i
$484$		0			0
$485$	7.41641			0.336762
$486$		0			0
$487$			8.74032i			0.396062i	0.980196	+	0.198031i	$0.0634546\pi$
$487$			8.74032i			0.396062i	−0.980196	+	0.198031i	$0.936545\pi$
$488$		0			0
$489$		−	10.6460i		−	0.481429i
$490$		0			0
$491$	35.4164			1.59832			0.799160	−	0.601118i	$-0.205278\pi$
$491$	35.4164			1.59832			0.799160	+	0.601118i	$0.205278\pi$
$492$		0			0
$493$	−26.7639			−1.20539
$494$		0			0
$495$			1.33540i			0.0600219i
$496$		0			0
$497$	11.0557			0.495917
$498$		0			0
$499$		−	4.16383i		−	0.186399i	−0.995647	−	0.0931993i	$-0.970291\pi$
$499$		−	4.16383i		−	0.186399i	0.995647	−	0.0931993i	$-0.0297094\pi$
$500$		0			0
$501$			5.78437i			0.258427i
$502$		0			0
$503$			4.44897i			0.198370i	0.995069	+	0.0991849i	$0.0316235\pi$
$503$			4.44897i			0.198370i	−0.995069	+	0.0991849i	$0.968376\pi$
$504$		0			0
$505$		−	5.08657i		−	0.226349i
$506$		0			0
$507$	7.94427			0.352818
$508$		0			0
$509$	9.05573			0.401388			0.200694	−	0.979654i	$-0.435680\pi$
$509$	9.05573			0.401388			0.200694	+	0.979654i	$0.435680\pi$
$510$		0			0
$511$	−4.00000			−0.176950
$512$		0			0
$513$			1.74806i			0.0771789i
$514$		0			0
$515$	5.88854			0.259480
$516$		0			0
$517$	9.88854			0.434898
$518$		0			0
$519$	−11.5279			−0.506017
$520$		0			0
$521$	−14.5836			−0.638919			−0.319459	−	0.947600i	$-0.603501\pi$
$521$	−14.5836			−0.638919			−0.319459	+	0.947600i	$0.603501\pi$
$522$		0			0
$523$			11.9814i			0.523910i	0.965080	+	0.261955i	$0.0843673\pi$
$523$			11.9814i			0.523910i	−0.965080	+	0.261955i	$0.915633\pi$
$524$		0			0
$525$	5.81966			0.253991
$526$		0			0
$527$	21.3050			0.928058
$528$		0			0
$529$	−51.1803			−2.22523
$530$		0			0
$531$			2.28825i			0.0993014i
$532$		0			0
$533$			9.15298i			0.396460i
$534$		0			0
$535$			1.33540i			0.0577345i
$536$		0			0
$537$		−	13.1893i		−	0.569160i
$538$		0			0
$539$	−13.5279			−0.582686
$540$		0			0
$541$			11.5687i			0.497379i	0.968583	+	0.248690i	$0.0799999\pi$
$541$			11.5687i			0.497379i	−0.968583	+	0.248690i	$0.920000\pi$
$542$		0			0
$543$	−7.23607			−0.310529
$544$		0			0
$545$	−7.41641			−0.317684
$546$		0			0
$547$			18.9737i			0.811255i	0.914038	+	0.405628i	$0.132947\pi$
$547$			18.9737i			0.811255i	−0.914038	+	0.405628i	$0.867053\pi$
$548$		0			0
$549$		−	5.65685i		−	0.241429i
$550$		0			0
$551$	13.8885			0.591672
$552$		0			0
$553$			11.8238i			0.502799i
$554$		0			0
$555$	3.05573	+	1.20788i	0.129708	+	0.0512718i
$556$		0			0
$557$			25.9960i			1.10149i	0.834675	+	0.550743i	$0.185656\pi$
$557$			25.9960i			1.10149i	−0.834675	+	0.550743i	$0.814344\pi$
$558$		0			0
$559$	12.9443			0.547484
$560$		0			0
$561$		−	8.32766i		−	0.351594i
$562$		0			0
$563$			26.9188i			1.13449i	0.823549	+	0.567245i	$0.191991\pi$
$563$			26.9188i			1.13449i	−0.823549	+	0.567245i	$0.808009\pi$
$564$		0			0
$565$	0.875388			0.0368279
$566$		0			0
$567$	−1.23607			−0.0519100
$568$		0			0
$569$			39.3128i			1.64808i	0.566532	+	0.824040i	$0.308285\pi$
$569$			39.3128i			1.64808i	−0.566532	+	0.824040i	$0.691715\pi$
$570$		0			0
$571$	23.7082			0.992157			0.496079	−	0.868278i	$-0.334773\pi$
$571$	23.7082			0.992157			0.496079	+	0.868278i	$0.334773\pi$
$572$		0			0
$573$			5.78437i			0.241646i
$574$		0			0
$575$			40.5508i			1.69109i
$576$		0			0
$577$			28.7943i			1.19872i	0.800478	+	0.599362i	$0.204579\pi$
$577$			28.7943i			1.19872i	−0.800478	+	0.599362i	$0.795421\pi$
$578$		0			0
$579$		−	11.3137i		−	0.470182i
$580$		0			0
$581$	−1.88854			−0.0783500
$582$		0			0
$583$	27.0557			1.12053
$584$		0			0
$585$	−2.47214			−0.102210
$586$		0			0
$587$		−	29.9048i		−	1.23430i	−0.786844	−	0.617152i	$-0.788286\pi$
$587$		−	29.9048i		−	1.23430i	0.786844	−	0.617152i	$-0.211714\pi$
$588$		0			0
$589$	−11.0557			−0.455543
$590$		0			0
$591$	17.4164			0.716415
$592$		0			0
$593$	0.111456			0.00457696			0.00228848	−	0.999997i	$-0.499272\pi$
$593$	0.111456			0.00457696			0.00228848	+	0.999997i	$0.499272\pi$
$594$		0			0
$595$	2.24922			0.0922092
$596$		0			0
$597$		−	2.82843i		−	0.115760i
$598$		0			0
$599$	−14.4721			−0.591315			−0.295658	−	0.955294i	$-0.595539\pi$
$599$	−14.4721			−0.591315			−0.295658	+	0.955294i	$0.595539\pi$
$600$		0			0
$601$	31.2361			1.27415			0.637073	−	0.770804i	$-0.280145\pi$
$601$	31.2361			1.27415			0.637073	+	0.770804i	$0.280145\pi$
$602$		0			0
$603$	−7.70820			−0.313902
$604$		0			0
$605$		−	2.64070i		−	0.107360i
$606$		0			0
$607$		−	38.3600i		−	1.55698i	−0.627654	−	0.778492i	$-0.715985\pi$
$607$		−	38.3600i		−	1.55698i	0.627654	−	0.778492i	$-0.284015\pi$
$608$		0			0
$609$			9.82068i			0.397954i
$610$		0			0
$611$			18.3060i			0.740580i
$612$		0			0
$613$	42.7214			1.72550			0.862750	−	0.505631i	$-0.168740\pi$
$613$	42.7214			1.72550			0.862750	+	0.505631i	$0.168740\pi$
$614$		0			0
$615$		−	1.08036i		−	0.0435644i
$616$		0			0
$617$	35.8885			1.44482			0.722409	−	0.691466i	$-0.243035\pi$
$617$	35.8885			1.44482			0.722409	+	0.691466i	$0.243035\pi$
$618$		0			0
$619$	−28.0000			−1.12542			−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000			−1.12542			−0.562708	+	0.826656i	$0.690240\pi$
$620$		0			0
$621$		−	8.61280i		−	0.345620i
$622$		0			0
$623$			10.6460i			0.426523i
$624$		0			0
$625$	20.7082			0.828328
$626$		0			0
$627$			4.32145i			0.172582i
$628$		0			0
$629$	−19.0557	−	7.53244i	−0.759802	−	0.300338i
$630$		0			0
$631$			0.412662i			0.0164278i	0.999966	+	0.00821391i	$0.00261460\pi$
$631$			0.412662i			0.0164278i	−0.999966	+	0.00821391i	$0.997385\pi$
$632$		0			0
$633$	−17.8885			−0.711006
$634$		0			0
$635$			6.48218i			0.257237i
$636$		0			0
$637$		−	25.0432i		−	0.992247i
$638$		0			0
$639$	−8.94427			−0.353830
$640$		0			0
$641$	−6.58359			−0.260036			−0.130018	−	0.991512i	$-0.541504\pi$
$641$	−6.58359			−0.260036			−0.130018	+	0.991512i	$0.541504\pi$
$642$		0			0
$643$			32.1931i			1.26957i	0.772689	+	0.634785i	$0.218911\pi$
$643$			32.1931i			1.26957i	−0.772689	+	0.634785i	$0.781089\pi$
$644$		0			0
$645$	−1.52786			−0.0601596
$646$		0			0
$647$			16.8430i			0.662168i	0.943601	+	0.331084i	$0.107414\pi$
$647$			16.8430i			0.662168i	−0.943601	+	0.331084i	$0.892586\pi$
$648$		0			0
$649$			5.65685i			0.222051i
$650$		0			0
$651$		−	7.81758i		−	0.306395i
$652$		0			0
$653$			23.8353i			0.932747i	0.884588	+	0.466374i	$0.154440\pi$
$653$			23.8353i			0.932747i	−0.884588	+	0.466374i	$0.845560\pi$
$654$		0			0
$655$	−0.0688837			−0.00269151
$656$		0			0
$657$	3.23607			0.126251
$658$		0			0
$659$	21.5279			0.838607			0.419303	−	0.907846i	$-0.362274\pi$
$659$	21.5279			0.838607			0.419303	+	0.907846i	$0.362274\pi$
$660$		0			0
$661$			13.9845i			0.543935i	0.962307	+	0.271967i	$0.0876742\pi$
$661$			13.9845i			0.543935i	−0.962307	+	0.271967i	$0.912326\pi$
$662$		0			0
$663$	15.4164			0.598724
$664$		0			0
$665$	−1.16718			−0.0452615
$666$		0			0
$667$	−68.4296			−2.64960
$668$		0			0
$669$	3.05573			0.118141
$670$		0			0
$671$		−	13.9845i		−	0.539866i
$672$		0			0
$673$	2.29180			0.0883422			0.0441711	−	0.999024i	$-0.485935\pi$
$673$	2.29180			0.0883422			0.0441711	+	0.999024i	$0.485935\pi$
$674$		0			0
$675$	−4.70820			−0.181219
$676$		0			0
$677$	2.94427			0.113158			0.0565788	−	0.998398i	$-0.481981\pi$
$677$	2.94427			0.113158			0.0565788	+	0.998398i	$0.481981\pi$
$678$		0			0
$679$			16.9706i			0.651270i
$680$		0			0
$681$		−	13.1893i		−	0.505415i
$682$		0			0
$683$		−	11.1862i		−	0.428028i	−0.976831	−	0.214014i	$-0.931346\pi$
$683$		−	11.1862i		−	0.428028i	0.976831	−	0.214014i	$-0.0686538\pi$
$684$		0			0
$685$		−	0.255039i		−	0.00974454i
$686$		0			0
$687$	0.472136			0.0180131
$688$		0			0
$689$			50.0864i			1.90814i
$690$		0			0
$691$	−20.6525			−0.785657			−0.392829	−	0.919612i	$-0.628503\pi$
$691$	−20.6525			−0.785657			−0.392829	+	0.919612i	$0.628503\pi$
$692$		0			0
$693$	−3.05573			−0.116078
$694$		0			0
$695$			4.16383i			0.157943i
$696$		0			0
$697$			6.73722i			0.255190i
$698$		0			0
$699$	16.4721			0.623033
$700$		0			0
$701$			40.2356i			1.51968i	0.650112	+	0.759838i	$0.274722\pi$
$701$			40.2356i			1.51968i	−0.650112	+	0.759838i	$0.725278\pi$
$702$		0			0
$703$	9.88854	+	3.90879i	0.372953	+	0.147423i
$704$		0			0
$705$		−	2.16073i		−	0.0813777i
$706$		0			0
$707$	11.6393			0.437742
$708$		0			0
$709$		−	42.0137i		−	1.57786i	−0.614484	−	0.788930i	$-0.710636\pi$
$709$		−	42.0137i		−	1.57786i	0.614484	−	0.788930i	$-0.289364\pi$
$710$		0			0
$711$		−	9.56564i		−	0.358740i
$712$		0			0
$713$	54.4721			2.04000
$714$		0			0
$715$	−6.11146			−0.228556
$716$		0			0
$717$		−	19.9265i		−	0.744169i
$718$		0			0
$719$	39.4164			1.46998			0.734992	−	0.678076i	$-0.237186\pi$
$719$	39.4164			1.46998			0.734992	+	0.678076i	$0.237186\pi$
$720$		0			0
$721$			13.4744i			0.501814i
$722$		0			0
$723$			3.24109i			0.120537i
$724$		0			0
$725$			37.4072i			1.38927i
$726$		0			0
$727$		−	30.2874i		−	1.12330i	−0.827376	−	0.561648i	$-0.810167\pi$
$727$		−	30.2874i		−	1.12330i	0.827376	−	0.561648i	$-0.189833\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	9.52786			0.352401
$732$		0			0
$733$	−2.94427			−0.108749			−0.0543746	−	0.998521i	$-0.517317\pi$
$733$	−2.94427			−0.108749			−0.0543746	+	0.998521i	$0.517317\pi$
$734$		0			0
$735$			2.95595i			0.109032i
$736$		0			0
$737$	−19.0557			−0.701927
$738$		0			0
$739$	−0.944272			−0.0347356			−0.0173678	−	0.999849i	$-0.505529\pi$
$739$	−0.944272			−0.0347356			−0.0173678	+	0.999849i	$0.505529\pi$
$740$		0			0
$741$	−8.00000			−0.293887
$742$		0			0
$743$	−40.9443			−1.50210			−0.751050	−	0.660246i	$-0.770452\pi$
$743$	−40.9443			−1.50210			−0.751050	+	0.660246i	$0.770452\pi$
$744$		0			0
$745$			4.57649i			0.167670i
$746$		0			0
$747$	1.52786			0.0559016
$748$		0			0
$749$	−3.05573			−0.111654
$750$		0			0
$751$	−3.70820			−0.135314			−0.0676571	−	0.997709i	$-0.521552\pi$
$751$	−3.70820			−0.135314			−0.0676571	+	0.997709i	$0.521552\pi$
$752$		0			0
$753$			7.27740i			0.265203i
$754$		0			0
$755$		−	11.3137i		−	0.411748i
$756$		0			0
$757$			39.8530i			1.44848i	0.689547	+	0.724241i	$0.257810\pi$
$757$			39.8530i			1.44848i	−0.689547	+	0.724241i	$0.742190\pi$
$758$		0			0
$759$		−	21.2920i		−	0.772851i
$760$		0			0
$761$	20.4721			0.742114			0.371057	−	0.928610i	$-0.378995\pi$
$761$	20.4721			0.742114			0.371057	+	0.928610i	$0.378995\pi$
$762$		0			0
$763$		−	16.9706i		−	0.614376i
$764$		0			0
$765$	−1.81966			−0.0657900
$766$		0			0
$767$	−10.4721			−0.378127
$768$		0			0
$769$		−	49.2610i		−	1.77640i	−0.459458	−	0.888199i	$-0.651956\pi$
$769$		−	49.2610i		−	1.77640i	0.459458	−	0.888199i	$-0.348044\pi$
$770$		0			0
$771$			27.9991i			1.00836i
$772$		0			0
$773$	10.3607			0.372648			0.186324	−	0.982488i	$-0.440343\pi$
$773$	10.3607			0.372648			0.186324	+	0.982488i	$0.440343\pi$
$774$		0			0
$775$		−	29.7773i		−	1.06963i
$776$		0			0
$777$	−2.76393	+	6.99226i	−0.0991555	+	0.250846i
$778$		0			0
$779$		−	3.49613i		−	0.125262i
$780$		0			0
$781$	−22.1115			−0.791210
$782$		0			0
$783$		−	7.94510i		−	0.283935i
$784$		0			0
$785$			13.0618i			0.466195i
$786$		0			0
$787$	22.8328			0.813902			0.406951	−	0.913450i	$-0.366592\pi$
$787$	22.8328			0.813902			0.406951	+	0.913450i	$0.366592\pi$
$788$		0			0
$789$	−27.4164			−0.976050
$790$		0			0
$791$			2.00310i			0.0712222i
$792$		0			0
$793$	25.8885			0.919329
$794$		0			0
$795$		−	5.91189i		−	0.209673i
$796$		0			0
$797$			22.5973i			0.800438i	0.916420	+	0.400219i	$0.131066\pi$
$797$			22.5973i			0.800438i	−0.916420	+	0.400219i	$0.868934\pi$
$798$		0			0
$799$			13.4744i			0.476691i
$800$		0			0
$801$		−	8.61280i		−	0.304318i
$802$		0			0
$803$	8.00000			0.282314
$804$		0			0
$805$	5.75078			0.202688
$806$		0			0
$807$	−31.8885			−1.12253
$808$		0			0
$809$			17.2557i			0.606678i	0.952883	+	0.303339i	$0.0981015\pi$
$809$			17.2557i			0.606678i	−0.952883	+	0.303339i	$0.901898\pi$
$810$		0			0
$811$	11.3475			0.398465			0.199233	−	0.979952i	$-0.436155\pi$
$811$	11.3475			0.398465			0.199233	+	0.979952i	$0.436155\pi$
$812$		0			0
$813$	1.81966			0.0638183
$814$		0			0
$815$	−5.75078			−0.201441
$816$		0			0
$817$	−4.94427			−0.172978
$818$		0			0
$819$		−	5.65685i		−	0.197666i
$820$		0			0
$821$	−28.4721			−0.993684			−0.496842	−	0.867841i	$-0.665507\pi$
$821$	−28.4721			−0.993684			−0.496842	+	0.867841i	$0.665507\pi$
$822$		0			0
$823$	17.8885			0.623555			0.311778	−	0.950155i	$-0.399076\pi$
$823$	17.8885			0.623555			0.311778	+	0.950155i	$0.399076\pi$
$824$		0			0
$825$	−11.6393			−0.405229
$826$		0			0
$827$		−	41.4736i		−	1.44218i	−0.692843	−	0.721089i	$-0.743642\pi$
$827$		−	41.4736i		−	1.44218i	0.692843	−	0.721089i	$-0.256358\pi$
$828$		0			0
$829$		−	19.3863i		−	0.673315i	−0.941627	−	0.336657i	$-0.890704\pi$
$829$		−	19.3863i		−	0.673315i	0.941627	−	0.336657i	$-0.109296\pi$
$830$		0			0
$831$			8.07262i			0.280036i
$832$		0			0
$833$		−	18.4335i		−	0.638682i
$834$		0			0
$835$	3.12461			0.108132
$836$		0			0
$837$			6.32456i			0.218609i
$838$		0			0
$839$	29.3050			1.01172			0.505860	−	0.862616i	$-0.331175\pi$
$839$	29.3050			1.01172			0.505860	+	0.862616i	$0.331175\pi$
$840$		0			0
$841$	−34.1246			−1.17671
$842$		0			0
$843$			21.0069i			0.723515i
$844$		0			0
$845$		−	4.29135i		−	0.147627i
$846$		0			0
$847$	6.04257			0.207625
$848$		0			0
$849$		−	24.3755i		−	0.836564i
$850$		0			0
$851$	−48.7214	−	19.2588i	−1.67015	−	0.660183i
$852$		0			0
$853$			23.1375i			0.792213i	0.918205	+	0.396106i	$0.129639\pi$
$853$			23.1375i			0.792213i	−0.918205	+	0.396106i	$0.870361\pi$
$854$		0			0
$855$	0.944272			0.0322934
$856$		0			0
$857$			47.3855i			1.61866i	0.587357	+	0.809328i	$0.300169\pi$
$857$			47.3855i			1.61866i	−0.587357	+	0.809328i	$0.699831\pi$
$858$		0			0
$859$		−	16.3029i		−	0.556246i	−0.960545	−	0.278123i	$-0.910288\pi$
$859$		−	16.3029i		−	0.556246i	0.960545	−	0.278123i	$-0.0897124\pi$
$860$		0			0
$861$	2.47214			0.0842502
$862$		0			0
$863$	−36.3607			−1.23773			−0.618866	−	0.785497i	$-0.712408\pi$
$863$	−36.3607			−1.23773			−0.618866	+	0.785497i	$0.712408\pi$
$864$		0			0
$865$			6.22714i			0.211729i
$866$		0			0
$867$	−5.65248			−0.191968
$868$		0			0
$869$		−	23.6476i		−	0.802189i
$870$		0			0
$871$		−	35.2765i		−	1.19530i
$872$		0			0
$873$		−	13.7295i		−	0.464672i
$874$		0			0
$875$		−	6.48218i		−	0.219138i
$876$		0			0
$877$	19.2361			0.649556			0.324778	−	0.945790i	$-0.394710\pi$
$877$	19.2361			0.649556			0.324778	+	0.945790i	$0.394710\pi$
$878$		0			0
$879$	26.0000			0.876958
$880$		0			0
$881$	52.4721			1.76783			0.883916	−	0.467646i	$-0.154898\pi$
$881$	52.4721			1.76783			0.883916	+	0.467646i	$0.154898\pi$
$882$		0			0
$883$			29.2070i			0.982894i	0.870907	+	0.491447i	$0.163532\pi$
$883$			29.2070i			0.982894i	−0.870907	+	0.491447i	$0.836468\pi$
$884$		0			0
$885$	1.23607			0.0415500
$886$		0			0
$887$	−46.4721			−1.56038			−0.780191	−	0.625542i	$-0.784878\pi$
$887$	−46.4721			−1.56038			−0.780191	+	0.625542i	$0.784878\pi$
$888$		0			0
$889$	−14.8328			−0.497477
$890$		0			0
$891$	2.47214			0.0828197
$892$		0			0
$893$		−	6.99226i		−	0.233987i
$894$		0			0
$895$	−7.12461			−0.238150
$896$		0			0
$897$	39.4164			1.31608
$898$		0			0
$899$	50.2492			1.67591
$900$		0			0
$901$			36.8670i			1.22822i
$902$		0			0
$903$		−	3.49613i		−	0.116344i
$904$		0			0
$905$			3.90879i			0.129933i
$906$		0			0
$907$		−	39.4404i		−	1.30960i	−0.755804	−	0.654798i	$-0.772754\pi$
$907$		−	39.4404i		−	1.30960i	0.755804	−	0.654798i	$-0.227246\pi$
$908$		0			0
$909$	−9.41641			−0.312323
$910$		0			0
$911$		−	11.6963i		−	0.387515i	−0.981049	−	0.193757i	$-0.937933\pi$
$911$		−	11.6963i		−	0.387515i	0.981049	−	0.193757i	$-0.0620674\pi$
$912$		0			0
$913$	3.77709			0.125003
$914$		0			0
$915$	−3.05573			−0.101019
$916$		0			0
$917$		−	0.157623i		−	0.00520516i
$918$		0			0
$919$		−	23.0401i		−	0.760022i	−0.924982	−	0.380011i	$-0.875920\pi$
$919$		−	23.0401i		−	0.760022i	0.924982	−	0.380011i	$-0.124080\pi$
$920$		0			0
$921$	20.9443			0.690137
$922$		0			0
$923$		−	40.9334i		−	1.34734i
$924$		0			0
$925$	−10.5279	+	26.6336i	−0.346154	+	0.875708i
$926$		0			0
$927$		−	10.9010i		−	0.358037i
$928$		0			0
$929$	−15.8885			−0.521286			−0.260643	−	0.965435i	$-0.583935\pi$
$929$	−15.8885			−0.521286			−0.260643	+	0.965435i	$0.583935\pi$
$930$		0			0
$931$			9.56564i			0.313501i
$932$		0			0
$933$		−	6.19704i		−	0.202882i
$934$		0			0
$935$	−4.49845			−0.147115
$936$		0			0
$937$	−34.3607			−1.12251			−0.561257	−	0.827641i	$-0.689682\pi$
$937$	−34.3607			−1.12251			−0.561257	+	0.827641i	$0.689682\pi$
$938$		0			0
$939$			7.81758i			0.255117i
$940$		0			0
$941$	18.3607			0.598541			0.299271	−	0.954168i	$-0.403257\pi$
$941$	18.3607			0.598541			0.299271	+	0.954168i	$0.403257\pi$
$942$		0			0
$943$			17.2256i			0.560943i
$944$		0			0
$945$			0.667701i			0.0217203i
$946$		0			0
$947$		−	23.1676i		−	0.752846i	−0.926448	−	0.376423i	$-0.877154\pi$
$947$		−	23.1676i		−	0.752846i	0.926448	−	0.376423i	$-0.122846\pi$
$948$		0			0
$949$			14.8098i			0.480748i
$950$		0			0
$951$	−2.94427			−0.0954746
$952$		0			0
$953$	7.52786			0.243851			0.121926	−	0.992539i	$-0.461093\pi$
$953$	7.52786			0.243851			0.121926	+	0.992539i	$0.461093\pi$
$954$		0			0
$955$	3.12461			0.101110
$956$		0			0
$957$		−	19.6414i		−	0.634915i
$958$		0			0
$959$	0.583592			0.0188452
$960$		0			0
$961$	−9.00000			−0.290323
$962$		0			0
$963$	2.47214			0.0796635
$964$		0			0
$965$	−6.11146			−0.196735
$966$		0			0
$967$		−	11.9814i		−	0.385296i	−0.981268	−	0.192648i	$-0.938292\pi$
$967$		−	11.9814i		−	0.385296i	0.981268	−	0.192648i	$-0.0617076\pi$
$968$		0			0
$969$	−5.88854			−0.189167
$970$		0			0
$971$	−58.2492			−1.86931			−0.934653	−	0.355560i	$-0.884290\pi$
$971$	−58.2492			−1.86931			−0.934653	+	0.355560i	$0.884290\pi$
$972$		0			0
$973$	−9.52786			−0.305449
$974$		0			0
$975$		−	21.5471i		−	0.690058i
$976$		0			0
$977$			31.0826i			0.994420i	0.867630	+	0.497210i	$0.165642\pi$
$977$			31.0826i			0.994420i	−0.867630	+	0.497210i	$0.834358\pi$
$978$		0			0
$979$		−	21.2920i		−	0.680496i
$980$		0			0
$981$			13.7295i			0.438348i
$982$		0			0
$983$	−39.7771			−1.26869			−0.634346	−	0.773049i	$-0.718731\pi$
$983$	−39.7771			−1.26869			−0.634346	+	0.773049i	$0.718731\pi$
$984$		0			0
$985$		−	9.40802i		−	0.299764i
$986$		0			0
$987$	4.94427			0.157378
$988$		0			0
$989$	24.3607			0.774625
$990$		0			0
$991$			5.49923i			0.174689i	0.996178	+	0.0873444i	$0.0278381\pi$
$991$			5.49923i			0.174689i	−0.996178	+	0.0873444i	$0.972162\pi$
$992$		0			0
$993$			25.7109i			0.815910i
$994$		0			0
$995$	−1.52786			−0.0484365
$996$		0			0
$997$		−	42.2688i		−	1.33867i	−0.742963	−	0.669333i	$-0.766580\pi$
$997$		−	42.2688i		−	1.33867i	0.742963	−	0.669333i	$-0.233420\pi$
$998$		0			0
$999$	2.23607	−	5.65685i	0.0707461	−	0.178975i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	444.2.e.a.73.3	yes	4
3.2	odd	2		1332.2.e.e.73.2		4
4.3	odd	2		1776.2.h.f.961.3		4
12.11	even	2		5328.2.h.g.2737.2		4
37.36	even	2	inner	444.2.e.a.73.2	✓	4
111.110	odd	2		1332.2.e.e.73.3		4
148.147	odd	2		1776.2.h.f.961.2		4
444.443	even	2		5328.2.h.g.2737.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
444.2.e.a.73.2	✓	4	37.36	even	2	inner
444.2.e.a.73.3	yes	4	1.1	even	1	trivial
1332.2.e.e.73.2		4	3.2	odd	2
1332.2.e.e.73.3		4	111.110	odd	2
1776.2.h.f.961.2		4	148.147	odd	2
1776.2.h.f.961.3		4	4.3	odd	2
5328.2.h.g.2737.2		4	12.11	even	2
5328.2.h.g.2737.3		4	444.443	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:	\( N \)	\(=\)	\( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	444.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.540182i q^{5} -1.23607 q^{7} +1.00000 q^{9} +2.47214 q^{11} +4.57649i q^{13} -0.540182i q^{15} +3.36861i q^{17} -1.74806i q^{19} +1.23607 q^{21} +8.61280i q^{23} +4.70820 q^{25} -1.00000 q^{27} +7.94510i q^{29} -6.32456i q^{31} -2.47214 q^{33} -0.667701i q^{35} +(-2.23607 + 5.65685i) q^{37} -4.57649i q^{39} +2.00000 q^{41} -2.82843i q^{43} +0.540182i q^{45} +4.00000 q^{47} -5.47214 q^{49} -3.36861i q^{51} +10.9443 q^{53} +1.33540i q^{55} +1.74806i q^{57} +2.28825i q^{59} -5.65685i q^{61} -1.23607 q^{63} -2.47214 q^{65} -7.70820 q^{67} -8.61280i q^{69} -8.94427 q^{71} +3.23607 q^{73} -4.70820 q^{75} -3.05573 q^{77} -9.56564i q^{79} +1.00000 q^{81} +1.52786 q^{83} -1.81966 q^{85} -7.94510i q^{87} -8.61280i q^{89} -5.65685i q^{91} +6.32456i q^{93} +0.944272 q^{95} -13.7295i q^{97} +2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9} - 8 q^{11} - 4 q^{21} - 8 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{41} + 16 q^{47} - 4 q^{49} + 8 q^{53} + 4 q^{63} + 8 q^{65} - 4 q^{67} + 4 q^{73} + 8 q^{75} - 48 q^{77} + 4 q^{81}+ \cdots - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9 - 8 * q^11 - 4 * q^21 - 8 * q^25 - 4 * q^27 + 8 * q^33 + 8 * q^41 + 16 * q^47 - 4 * q^49 + 8 * q^53 + 4 * q^63 + 8 * q^65 - 4 * q^67 + 4 * q^73 + 8 * q^75 - 48 * q^77 + 4 * q^81 + 24 * q^83 - 52 * q^85 - 32 * q^95 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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