LMFDB - Embedded newform 2252.3.d.b.1125.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2252,3,Mod(1125,2252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2252.1125"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2252 = 2^{2} \cdot 563 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2252.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [76] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$61.3625555339$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1125.16
Character	$\chi$	$=$	2252.1125
Dual form			2252.3.d.b.1125.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.46678 q^{3} +1.78369i q^{5} -7.09205 q^{7} +3.01854 q^{9} +0.0787181 q^{11} -12.9657 q^{13} -6.18365i q^{15} +3.40849 q^{17} +6.75628 q^{19} +24.5866 q^{21} +35.5594 q^{23} +21.8185 q^{25} +20.7364 q^{27} -45.1580i q^{29} +46.3610i q^{31} -0.272898 q^{33} -12.6500i q^{35} +62.2734i q^{37} +44.9493 q^{39} +51.6186i q^{41} -31.5544i q^{43} +5.38413i q^{45} -40.1206 q^{47} +1.29721 q^{49} -11.8165 q^{51} -23.3327i q^{53} +0.140409i q^{55} -23.4225 q^{57} -1.25854 q^{59} +46.4689 q^{61} -21.4076 q^{63} -23.1268i q^{65} +47.0822 q^{67} -123.276 q^{69} -36.3976 q^{71} -63.9488i q^{73} -75.6397 q^{75} -0.558273 q^{77} -23.0688i q^{79} -99.0553 q^{81} +4.20676i q^{83} +6.07968i q^{85} +156.553i q^{87} -17.6508i q^{89} +91.9537 q^{91} -160.723i q^{93} +12.0511i q^{95} +70.6669i q^{97} +0.237614 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q + 4 q^{3} - 8 q^{7} + 128 q^{9} + 2 q^{11} - 6 q^{13} + 22 q^{17} + 12 q^{19} - 6 q^{21} + 24 q^{23} - 912 q^{25} + 22 q^{27} - 52 q^{33} - 70 q^{39} - 28 q^{47} - 340 q^{49} + 314 q^{51} - 98 q^{57}+ \cdots + 348 q^{99}+O(q^{100}) $ 76 * q + 4 * q^3 - 8 * q^7 + 128 * q^9 + 2 * q^11 - 6 * q^13 + 22 * q^17 + 12 * q^19 - 6 * q^21 + 24 * q^23 - 912 * q^25 + 22 * q^27 - 52 * q^33 - 70 * q^39 - 28 * q^47 - 340 * q^49 + 314 * q^51 - 98 * q^57 + 18 * q^59 - 162 * q^61 - 246 * q^63 - 178 * q^67 + 2 * q^69 + 72 * q^71 + 14 * q^75 - 498 * q^77 - 468 * q^81 + 132 * q^91 + 348 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2252\mathbb{Z}\right)^\times$.

$n$	$565$	$1127$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−3.46678			−1.15559			−0.577796	−	0.816181i	$-0.696087\pi$
$3$	−3.46678			−1.15559			−0.577796	+	0.816181i	$0.696087\pi$
$4$		0			0
$5$			1.78369i			0.356738i	0.983964	+	0.178369i	$0.0570820\pi$
$5$			1.78369i			0.356738i	−0.983964	+	0.178369i	$0.942918\pi$
$6$		0			0
$7$	−7.09205			−1.01315			−0.506575	−	0.862196i	$-0.669089\pi$
$7$	−7.09205			−1.01315			−0.506575	+	0.862196i	$0.669089\pi$
$8$		0			0
$9$	3.01854			0.335393
$10$		0			0
$11$	0.0787181			0.00715619			0.00357810	−	0.999994i	$-0.498861\pi$
$11$	0.0787181			0.00715619			0.00357810	+	0.999994i	$0.498861\pi$
$12$		0			0
$13$	−12.9657			−0.997365			−0.498682	−	0.866785i	$-0.666183\pi$
$13$	−12.9657			−0.997365			−0.498682	+	0.866785i	$0.666183\pi$
$14$		0			0
$15$		−	6.18365i		−	0.412243i
$16$		0			0
$17$	3.40849			0.200499			0.100250	−	0.994962i	$-0.468036\pi$
$17$	3.40849			0.200499			0.100250	+	0.994962i	$0.468036\pi$
$18$		0			0
$19$	6.75628			0.355594			0.177797	−	0.984067i	$-0.443103\pi$
$19$	6.75628			0.355594			0.177797	+	0.984067i	$0.443103\pi$
$20$		0			0
$21$	24.5866			1.17079
$22$		0			0
$23$	35.5594			1.54606			0.773030	−	0.634369i	$-0.218740\pi$
$23$	35.5594			1.54606			0.773030	+	0.634369i	$0.218740\pi$
$24$		0			0
$25$	21.8185			0.872738
$26$		0			0
$27$	20.7364			0.768014
$28$		0			0
$29$		−	45.1580i		−	1.55717i	−0.627538	−	0.778586i	$-0.715937\pi$
$29$		−	45.1580i		−	1.55717i	0.627538	−	0.778586i	$-0.284063\pi$
$30$		0			0
$31$			46.3610i			1.49552i	0.663972	+	0.747758i	$0.268869\pi$
$31$			46.3610i			1.49552i	−0.663972	+	0.747758i	$0.731131\pi$
$32$		0			0
$33$	−0.272898			−0.00826964
$34$		0			0
$35$		−	12.6500i		−	0.361429i
$36$		0			0
$37$			62.2734i			1.68306i	0.540207	+	0.841532i	$0.318346\pi$
$37$			62.2734i			1.68306i	−0.540207	+	0.841532i	$0.681654\pi$
$38$		0			0
$39$	44.9493			1.15255
$40$		0			0
$41$			51.6186i			1.25899i	0.777004	+	0.629495i	$0.216738\pi$
$41$			51.6186i			1.25899i	−0.777004	+	0.629495i	$0.783262\pi$
$42$		0			0
$43$		−	31.5544i		−	0.733823i	−0.930256	−	0.366911i	$-0.880415\pi$
$43$		−	31.5544i		−	0.733823i	0.930256	−	0.366911i	$-0.119585\pi$
$44$		0			0
$45$			5.38413i			0.119647i
$46$		0			0
$47$	−40.1206			−0.853629			−0.426815	−	0.904339i	$-0.640364\pi$
$47$	−40.1206			−0.853629			−0.426815	+	0.904339i	$0.640364\pi$
$48$		0			0
$49$	1.29721			0.0264736
$50$		0			0
$51$	−11.8165			−0.231695
$52$		0			0
$53$		−	23.3327i		−	0.440239i	−0.975473	−	0.220120i	$-0.929355\pi$
$53$		−	23.3327i		−	0.440239i	0.975473	−	0.220120i	$-0.0706448\pi$
$54$		0			0
$55$			0.140409i			0.00255288i
$56$		0			0
$57$	−23.4225			−0.410921
$58$		0			0
$59$	−1.25854			−0.0213312			−0.0106656	−	0.999943i	$-0.503395\pi$
$59$	−1.25854			−0.0213312			−0.0106656	+	0.999943i	$0.503395\pi$
$60$		0			0
$61$	46.4689			0.761786			0.380893	−	0.924619i	$-0.375617\pi$
$61$	46.4689			0.761786			0.380893	+	0.924619i	$0.375617\pi$
$62$		0			0
$63$	−21.4076			−0.339804
$64$		0			0
$65$		−	23.1268i		−	0.355798i
$66$		0			0
$67$	47.0822			0.702720			0.351360	−	0.936240i	$-0.385719\pi$
$67$	47.0822			0.702720			0.351360	+	0.936240i	$0.385719\pi$
$68$		0			0
$69$	−123.276			−1.78661
$70$		0			0
$71$	−36.3976			−0.512642			−0.256321	−	0.966592i	$-0.582510\pi$
$71$	−36.3976			−0.512642			−0.256321	+	0.966592i	$0.582510\pi$
$72$		0			0
$73$		−	63.9488i		−	0.876010i	−0.898972	−	0.438005i	$-0.855685\pi$
$73$		−	63.9488i		−	0.876010i	0.898972	−	0.438005i	$-0.144315\pi$
$74$		0			0
$75$	−75.6397			−1.00853
$76$		0			0
$77$	−0.558273			−0.00725030
$78$		0			0
$79$		−	23.0688i		−	0.292010i	−0.989284	−	0.146005i	$-0.953358\pi$
$79$		−	23.0688i		−	0.292010i	0.989284	−	0.146005i	$-0.0466416\pi$
$80$		0			0
$81$	−99.0553			−1.22290
$82$		0			0
$83$			4.20676i			0.0506838i	0.999679	+	0.0253419i	$0.00806745\pi$
$83$			4.20676i			0.0506838i	−0.999679	+	0.0253419i	$0.991933\pi$
$84$		0			0
$85$			6.07968i			0.0715257i
$86$		0			0
$87$			156.553i			1.79946i
$88$		0			0
$89$		−	17.6508i		−	0.198323i	−0.995071	−	0.0991617i	$-0.968384\pi$
$89$		−	17.6508i		−	0.198323i	0.995071	−	0.0991617i	$-0.0316161\pi$
$90$		0			0
$91$	91.9537			1.01048
$92$		0			0
$93$		−	160.723i		−	1.72821i
$94$		0			0
$95$			12.0511i			0.126854i
$96$		0			0
$97$			70.6669i			0.728525i	0.931296	+	0.364262i	$0.118679\pi$
$97$			70.6669i			0.728525i	−0.931296	+	0.364262i	$0.881321\pi$
$98$		0			0
$99$	0.237614			0.00240014
$100$		0			0
$101$	33.1279			0.327999			0.163999	−	0.986460i	$-0.447560\pi$
$101$	33.1279			0.327999			0.163999	+	0.986460i	$0.447560\pi$
$102$		0			0
$103$	178.233			1.73042			0.865209	−	0.501411i	$-0.167186\pi$
$103$	178.233			1.73042			0.865209	+	0.501411i	$0.167186\pi$
$104$		0			0
$105$			43.8548i			0.417664i
$106$		0			0
$107$	132.352			1.23694			0.618469	−	0.785809i	$-0.287753\pi$
$107$	132.352			1.23694			0.618469	+	0.785809i	$0.287753\pi$
$108$		0			0
$109$			192.347i			1.76465i	0.470643	+	0.882324i	$0.344022\pi$
$109$			192.347i			1.76465i	−0.470643	+	0.882324i	$0.655978\pi$
$110$		0			0
$111$		−	215.888i		−	1.94494i
$112$		0			0
$113$	−180.188			−1.59458			−0.797291	−	0.603596i	$-0.793734\pi$
$113$	−180.188			−1.59458			−0.797291	+	0.603596i	$0.793734\pi$
$114$		0			0
$115$			63.4269i			0.551538i
$116$		0			0
$117$	−39.1376			−0.334509
$118$		0			0
$119$	−24.1732			−0.203136
$120$		0			0
$121$	−120.994			−0.999949
$122$		0			0
$123$		−	178.950i		−	1.45488i
$124$		0			0
$125$			83.5095i			0.668076i
$126$		0			0
$127$	−32.4774			−0.255728			−0.127864	−	0.991792i	$-0.540812\pi$
$127$	−32.4774			−0.255728			−0.127864	+	0.991792i	$0.540812\pi$
$128$		0			0
$129$			109.392i			0.848000i
$130$		0			0
$131$		−	216.144i		−	1.64995i	−0.565168	−	0.824976i	$-0.691189\pi$
$131$		−	216.144i		−	1.64995i	0.565168	−	0.824976i	$-0.308811\pi$
$132$		0			0
$133$	−47.9159			−0.360270
$134$		0			0
$135$			36.9873i			0.273980i
$136$		0			0
$137$	−80.6473			−0.588667			−0.294333	−	0.955703i	$-0.595098\pi$
$137$	−80.6473			−0.588667			−0.294333	+	0.955703i	$0.595098\pi$
$138$		0			0
$139$			134.493i			0.967576i	0.875185	+	0.483788i	$0.160739\pi$
$139$			134.493i			0.967576i	−0.875185	+	0.483788i	$0.839261\pi$
$140$		0			0
$141$	139.089			0.986447
$142$		0			0
$143$	−1.02064			−0.00713733
$144$		0			0
$145$	80.5478			0.555502
$146$		0			0
$147$	−4.49713			−0.0305927
$148$		0			0
$149$	−176.779			−1.18643			−0.593217	−	0.805043i	$-0.702142\pi$
$149$	−176.779			−1.18643			−0.593217	+	0.805043i	$0.702142\pi$
$150$		0			0
$151$		−	286.056i		−	1.89441i	−0.320630	−	0.947205i	$-0.603894\pi$
$151$		−	286.056i		−	1.89441i	0.320630	−	0.947205i	$-0.396106\pi$
$152$		0			0
$153$	10.2887			0.0672461
$154$		0			0
$155$	−82.6935			−0.533507
$156$		0			0
$157$		−	198.317i		−	1.26317i	−0.775309	−	0.631583i	$-0.782406\pi$
$157$		−	198.317i		−	1.26317i	0.775309	−	0.631583i	$-0.217594\pi$
$158$		0			0
$159$			80.8892i			0.508737i
$160$		0			0
$161$	−252.189			−1.56639
$162$		0			0
$163$		−	126.130i		−	0.773801i	−0.922121	−	0.386901i	$-0.873546\pi$
$163$		−	126.130i		−	0.773801i	0.922121	−	0.386901i	$-0.126454\pi$
$164$		0			0
$165$		−	0.486765i		−	0.00295009i
$166$		0			0
$167$			238.572i			1.42857i	0.699853	+	0.714287i	$0.253249\pi$
$167$			238.572i			1.42857i	−0.699853	+	0.714287i	$0.746751\pi$
$168$		0			0
$169$	−0.889560			−0.00526367
$170$		0			0
$171$	20.3941			0.119264
$172$		0			0
$173$			21.9466i			0.126859i	0.997986	+	0.0634296i	$0.0202038\pi$
$173$			21.9466i			0.126859i	−0.997986	+	0.0634296i	$0.979796\pi$
$174$		0			0
$175$	−154.738			−0.884215
$176$		0			0
$177$	4.36309			0.0246502
$178$		0			0
$179$	−98.6091			−0.550889			−0.275444	−	0.961317i	$-0.588825\pi$
$179$	−98.6091			−0.550889			−0.275444	+	0.961317i	$0.588825\pi$
$180$		0			0
$181$	−244.501			−1.35084			−0.675418	−	0.737435i	$-0.736037\pi$
$181$	−244.501			−1.35084			−0.675418	+	0.737435i	$0.736037\pi$
$182$		0			0
$183$	−161.097			−0.880314
$184$		0			0
$185$	−111.076			−0.600412
$186$		0			0
$187$	0.268310			0.00143481
$188$		0			0
$189$	−147.064			−0.778114
$190$		0			0
$191$	−54.1417			−0.283465			−0.141732	−	0.989905i	$-0.545267\pi$
$191$	−54.1417			−0.283465			−0.141732	+	0.989905i	$0.545267\pi$
$192$		0			0
$193$	−179.806			−0.931638			−0.465819	−	0.884880i	$-0.654240\pi$
$193$	−179.806			−0.931638			−0.465819	+	0.884880i	$0.654240\pi$
$194$		0			0
$195$			80.1756i			0.411157i
$196$		0			0
$197$	−263.058			−1.33532			−0.667661	−	0.744465i	$-0.732704\pi$
$197$	−263.058			−1.33532			−0.667661	+	0.744465i	$0.732704\pi$
$198$		0			0
$199$			21.5418i			0.108250i	0.998534	+	0.0541252i	$0.0172370\pi$
$199$			21.5418i			0.108250i	−0.998534	+	0.0541252i	$0.982763\pi$
$200$		0			0
$201$	−163.224			−0.812057
$202$		0			0
$203$			320.263i			1.57765i
$204$		0			0
$205$	−92.0715			−0.449129
$206$		0			0
$207$	107.337			0.518538
$208$		0			0
$209$	0.531842			0.00254470
$210$		0			0
$211$	−33.3902			−0.158247			−0.0791236	−	0.996865i	$-0.525212\pi$
$211$	−33.3902			−0.158247			−0.0791236	+	0.996865i	$0.525212\pi$
$212$		0			0
$213$	126.182			0.592405
$214$		0			0
$215$	56.2832			0.261782
$216$		0			0
$217$		−	328.794i		−	1.51518i
$218$		0			0
$219$			221.696i			1.01231i
$220$		0			0
$221$	−44.1936			−0.199971
$222$		0			0
$223$	51.8306			0.232424			0.116212	−	0.993224i	$-0.462925\pi$
$223$	51.8306			0.232424			0.116212	+	0.993224i	$0.462925\pi$
$224$		0			0
$225$	65.8599			0.292710
$226$		0			0
$227$			103.848i			0.457479i	0.973488	+	0.228739i	$0.0734603\pi$
$227$			103.848i			0.457479i	−0.973488	+	0.228739i	$0.926540\pi$
$228$		0			0
$229$		−	138.564i		−	0.605084i	−0.953136	−	0.302542i	$-0.902165\pi$
$229$		−	138.564i		−	0.605084i	0.953136	−	0.302542i	$-0.0978353\pi$
$230$		0			0
$231$	1.93541			0.00837839
$232$		0			0
$233$			93.6572i			0.401962i	0.979595	+	0.200981i	$0.0644130\pi$
$233$			93.6572i			0.401962i	−0.979595	+	0.200981i	$0.935587\pi$
$234$		0			0
$235$		−	71.5626i		−	0.304522i
$236$		0			0
$237$			79.9744i			0.337445i
$238$		0			0
$239$			24.4963i			0.102495i	0.998686	+	0.0512475i	$0.0163197\pi$
$239$			24.4963i			0.102495i	−0.998686	+	0.0512475i	$0.983680\pi$
$240$		0			0
$241$	−358.192			−1.48627			−0.743136	−	0.669140i	$-0.766662\pi$
$241$	−358.192			−1.48627			−0.743136	+	0.669140i	$0.766662\pi$
$242$		0			0
$243$	156.775			0.645165
$244$		0			0
$245$			2.31382i			0.00944415i
$246$		0			0
$247$	−87.6002			−0.354657
$248$		0			0
$249$		−	14.5839i		−	0.0585699i
$250$		0			0
$251$	66.5884			0.265292			0.132646	−	0.991163i	$-0.457653\pi$
$251$	66.5884			0.265292			0.132646	+	0.991163i	$0.457653\pi$
$252$		0			0
$253$	2.79917			0.0110639
$254$		0			0
$255$		−	21.0769i		−	0.0826545i
$256$		0			0
$257$	247.535			0.963169			0.481585	−	0.876400i	$-0.340061\pi$
$257$	247.535			0.963169			0.481585	+	0.876400i	$0.340061\pi$
$258$		0			0
$259$		−	441.646i		−	1.70520i
$260$		0			0
$261$		−	136.311i		−	0.522265i
$262$		0			0
$263$		−	119.407i		−	0.454017i	−0.973893	−	0.227009i	$-0.927105\pi$
$263$		−	119.407i		−	0.454017i	0.973893	−	0.227009i	$-0.0728946\pi$
$264$		0			0
$265$	41.6182			0.157050
$266$		0			0
$267$			61.1913i			0.229181i
$268$		0			0
$269$	−103.270			−0.383902			−0.191951	−	0.981405i	$-0.561481\pi$
$269$	−103.270			−0.383902			−0.191951	+	0.981405i	$0.561481\pi$
$270$		0			0
$271$	329.451			1.21568			0.607842	−	0.794058i	$-0.292035\pi$
$271$	329.451			1.21568			0.607842	+	0.794058i	$0.292035\pi$
$272$		0			0
$273$	−318.783			−1.16770
$274$		0			0
$275$	1.71751			0.00624548
$276$		0			0
$277$	278.030			1.00372			0.501860	−	0.864949i	$-0.332649\pi$
$277$	278.030			1.00372			0.501860	+	0.864949i	$0.332649\pi$
$278$		0			0
$279$			139.942i			0.501586i
$280$		0			0
$281$	−56.8539			−0.202327			−0.101164	−	0.994870i	$-0.532257\pi$
$281$	−56.8539			−0.202327			−0.101164	+	0.994870i	$0.532257\pi$
$282$		0			0
$283$			236.992i			0.837428i	0.908118	+	0.418714i	$0.137519\pi$
$283$			236.992i			0.837428i	−0.908118	+	0.418714i	$0.862481\pi$
$284$		0			0
$285$		−	41.7785i		−	0.146591i
$286$		0			0
$287$		−	366.082i		−	1.27555i
$288$		0			0
$289$	−277.382			−0.959800
$290$		0			0
$291$		−	244.986i		−	0.841877i
$292$		0			0
$293$		−	11.2885i		−	0.0385273i	−0.999814	−	0.0192636i	$-0.993868\pi$
$293$		−	11.2885i		−	0.0385273i	0.999814	−	0.0192636i	$-0.00613218\pi$
$294$		0			0
$295$		−	2.24485i		−	0.00760966i
$296$		0			0
$297$	1.63233			0.00549606
$298$		0			0
$299$	−461.054			−1.54199
$300$		0			0
$301$			223.785i			0.743473i
$302$		0			0
$303$	−114.847			−0.379033
$304$		0			0
$305$			82.8861i			0.271758i
$306$		0			0
$307$			499.213i			1.62610i	0.582193	+	0.813051i	$0.302195\pi$
$307$			499.213i			1.62610i	−0.582193	+	0.813051i	$0.697805\pi$
$308$		0			0
$309$	−617.894			−1.99966
$310$		0			0
$311$			569.619i			1.83157i	0.401665	+	0.915787i	$0.368432\pi$
$311$			569.619i			1.83157i	−0.401665	+	0.915787i	$0.631568\pi$
$312$		0			0
$313$		−	151.249i		−	0.483225i	−0.970373	−	0.241612i	$-0.922324\pi$
$313$		−	151.249i		−	0.483225i	0.970373	−	0.241612i	$-0.0776763\pi$
$314$		0			0
$315$		−	38.1845i		−	0.121221i
$316$		0			0
$317$			150.833i			0.475814i	0.971288	+	0.237907i	$0.0764613\pi$
$317$			150.833i			0.475814i	−0.971288	+	0.237907i	$0.923539\pi$
$318$		0			0
$319$		−	3.55475i		−	0.0111434i
$320$		0			0
$321$	−458.836			−1.42940
$322$		0			0
$323$	23.0287			0.0712963
$324$		0			0
$325$	−282.892			−0.870438
$326$		0			0
$327$		−	666.823i		−	2.03921i
$328$		0			0
$329$	284.537			0.864855
$330$		0			0
$331$			372.844i			1.12642i	0.826315	+	0.563209i	$0.190433\pi$
$331$			372.844i			1.12642i	−0.826315	+	0.563209i	$0.809567\pi$
$332$		0			0
$333$			187.975i			0.564488i
$334$		0			0
$335$			83.9800i			0.250687i
$336$		0			0
$337$	−397.922			−1.18078			−0.590389	−	0.807119i	$-0.701026\pi$
$337$	−397.922			−1.18078			−0.590389	+	0.807119i	$0.701026\pi$
$338$		0			0
$339$	624.670			1.84269
$340$		0			0
$341$			3.64945i			0.0107022i
$342$		0			0
$343$	338.311			0.986329
$344$		0			0
$345$		−	219.887i		−	0.637353i
$346$		0			0
$347$	−614.345			−1.77045			−0.885223	−	0.465166i	$-0.845995\pi$
$347$	−614.345			−1.77045			−0.885223	+	0.465166i	$0.845995\pi$
$348$		0			0
$349$	240.217			0.688300			0.344150	−	0.938915i	$-0.388167\pi$
$349$	240.217			0.688300			0.344150	+	0.938915i	$0.388167\pi$
$350$		0			0
$351$	−268.863			−0.765990
$352$		0			0
$353$		−	433.927i		−	1.22925i	−0.788818	−	0.614627i	$-0.789306\pi$
$353$		−	433.927i		−	1.22925i	0.788818	−	0.614627i	$-0.210694\pi$
$354$		0			0
$355$		−	64.9219i		−	0.182879i
$356$		0			0
$357$	83.8030			0.234742
$358$		0			0
$359$		−	166.263i		−	0.463128i	−0.972820	−	0.231564i	$-0.925616\pi$
$359$		−	166.263i		−	0.463128i	0.972820	−	0.231564i	$-0.0743842\pi$
$360$		0			0
$361$	−315.353			−0.873553
$362$		0			0
$363$	419.458			1.15553
$364$		0			0
$365$	114.065			0.312506
$366$		0			0
$367$		−	15.8241i		−	0.0431174i	−0.999768	−	0.0215587i	$-0.993137\pi$
$367$		−	15.8241i		−	0.0431174i	0.999768	−	0.0215587i	$-0.00686288\pi$
$368$		0			0
$369$			155.813i			0.422257i
$370$		0			0
$371$			165.477i			0.446029i
$372$		0			0
$373$			563.745i			1.51138i	0.654929	+	0.755691i	$0.272699\pi$
$373$			563.745i			1.51138i	−0.654929	+	0.755691i	$0.727301\pi$
$374$		0			0
$375$		−	289.509i		−	0.772024i
$376$		0			0
$377$			585.507i			1.55307i
$378$		0			0
$379$	−547.320			−1.44412			−0.722058	−	0.691833i	$-0.756804\pi$
$379$	−547.320			−1.44412			−0.722058	+	0.691833i	$0.756804\pi$
$380$		0			0
$381$	112.592			0.295517
$382$		0			0
$383$	228.334			0.596173			0.298086	−	0.954539i	$-0.403652\pi$
$383$	228.334			0.596173			0.298086	+	0.954539i	$0.403652\pi$
$384$		0			0
$385$		−	0.995785i		−	0.00258645i
$386$		0			0
$387$		−	95.2481i		−	0.246119i
$388$		0			0
$389$		−	419.932i		−	1.07952i	−0.841820	−	0.539759i	$-0.818516\pi$
$389$		−	419.932i		−	1.07952i	0.841820	−	0.539759i	$-0.181484\pi$
$390$		0			0
$391$	121.204			0.309984
$392$		0			0
$393$			749.322i			1.90667i
$394$		0			0
$395$	41.1475			0.104171
$396$		0			0
$397$			456.405i			1.14963i	0.818282	+	0.574817i	$0.194927\pi$
$397$			456.405i			1.14963i	−0.818282	+	0.574817i	$0.805073\pi$
$398$		0			0
$399$	166.114			0.416325
$400$		0			0
$401$	−271.042			−0.675916			−0.337958	−	0.941161i	$-0.609736\pi$
$401$	−271.042			−0.675916			−0.337958	+	0.941161i	$0.609736\pi$
$402$		0			0
$403$		−	601.104i		−	1.49157i
$404$		0			0
$405$		−	176.684i		−	0.436256i
$406$		0			0
$407$			4.90204i			0.0120443i
$408$		0			0
$409$	−501.217			−1.22547			−0.612734	−	0.790289i	$-0.709930\pi$
$409$	−501.217			−1.22547			−0.612734	+	0.790289i	$0.709930\pi$
$410$		0			0
$411$	279.586			0.680259
$412$		0			0
$413$	8.92565			0.0216118
$414$		0			0
$415$	−7.50355			−0.0180808
$416$		0			0
$417$		−	466.257i		−	1.11812i
$418$		0			0
$419$		−	389.164i		−	0.928791i	−0.885628	−	0.464396i	$-0.846272\pi$
$419$		−	389.164i		−	0.928791i	0.885628	−	0.464396i	$-0.153728\pi$
$420$		0			0
$421$	738.853			1.75500			0.877498	−	0.479581i	$-0.159211\pi$
$421$	738.853			1.75500			0.877498	+	0.479581i	$0.159211\pi$
$422$		0			0
$423$	−121.106			−0.286301
$424$		0			0
$425$	74.3680			0.174983
$426$		0			0
$427$	−329.560			−0.771804
$428$		0			0
$429$	3.53833			0.00824785
$430$		0			0
$431$		−	315.719i		−	0.732527i	−0.930511	−	0.366264i	$-0.880637\pi$
$431$		−	315.719i		−	0.732527i	0.930511	−	0.366264i	$-0.119363\pi$
$432$		0			0
$433$			62.2338i			0.143727i	0.997414	+	0.0718636i	$0.0228946\pi$
$433$			62.2338i			0.143727i	−0.997414	+	0.0718636i	$0.977105\pi$
$434$		0			0
$435$	−279.241			−0.641934
$436$		0			0
$437$	240.249			0.549769
$438$		0			0
$439$	−326.306			−0.743295			−0.371647	−	0.928374i	$-0.621207\pi$
$439$	−326.306			−0.743295			−0.371647	+	0.928374i	$0.621207\pi$
$440$		0			0
$441$	3.91567			0.00887908
$442$		0			0
$443$		−	22.4440i		−	0.0506636i	−0.999679	−	0.0253318i	$-0.991936\pi$
$443$		−	22.4440i		−	0.0506636i	0.999679	−	0.0253318i	$-0.00806422\pi$
$444$		0			0
$445$	31.4835			0.0707494
$446$		0			0
$447$	612.852			1.37103
$448$		0			0
$449$	640.430			1.42635			0.713174	−	0.700988i	$-0.247257\pi$
$449$	640.430			1.42635			0.713174	+	0.700988i	$0.247257\pi$
$450$		0			0
$451$			4.06332i			0.00900958i
$452$		0			0
$453$			991.691i			2.18916i
$454$		0			0
$455$			164.017i			0.360476i
$456$		0			0
$457$		−	608.551i		−	1.33162i	−0.746121	−	0.665811i	$-0.768086\pi$
$457$		−	608.551i		−	1.33162i	0.746121	−	0.665811i	$-0.231914\pi$
$458$		0			0
$459$	70.6797			0.153986
$460$		0			0
$461$	−268.061			−0.581477			−0.290739	−	0.956802i	$-0.593901\pi$
$461$	−268.061			−0.581477			−0.290739	+	0.956802i	$0.593901\pi$
$462$		0			0
$463$		−	519.238i		−	1.12146i	−0.827997	−	0.560732i	$-0.810520\pi$
$463$		−	519.238i		−	1.12146i	0.827997	−	0.560732i	$-0.189480\pi$
$464$		0			0
$465$	286.680			0.616516
$466$		0			0
$467$	−214.808			−0.459975			−0.229987	−	0.973194i	$-0.573868\pi$
$467$	−214.808			−0.459975			−0.229987	+	0.973194i	$0.573868\pi$
$468$		0			0
$469$	−333.910			−0.711961
$470$		0			0
$471$			687.521i			1.45970i
$472$		0			0
$473$		−	2.48390i		−	0.00525138i
$474$		0			0
$475$	147.412			0.310340
$476$		0			0
$477$		−	70.4306i		−	0.147653i
$478$		0			0
$479$		−	521.722i		−	1.08919i	−0.838699	−	0.544595i	$-0.816683\pi$
$479$		−	521.722i		−	1.08919i	0.838699	−	0.544595i	$-0.183317\pi$
$480$		0			0
$481$		−	807.420i		−	1.67863i
$482$		0			0
$483$	874.283			1.81011
$484$		0			0
$485$	−126.048			−0.259892
$486$		0			0
$487$			415.337i			0.852848i	0.904523	+	0.426424i	$0.140227\pi$
$487$			415.337i			0.852848i	−0.904523	+	0.426424i	$0.859773\pi$
$488$		0			0
$489$			437.263i			0.894199i
$490$		0			0
$491$	263.760			0.537189			0.268595	−	0.963253i	$-0.413441\pi$
$491$	263.760			0.537189			0.268595	+	0.963253i	$0.413441\pi$
$492$		0			0
$493$		−	153.921i		−	0.312212i
$494$		0			0
$495$			0.423829i			0.000856220i
$496$		0			0
$497$	258.133			0.519383
$498$		0			0
$499$			620.671i			1.24383i	0.783085	+	0.621914i	$0.213645\pi$
$499$			620.671i			1.24383i	−0.783085	+	0.621914i	$0.786355\pi$
$500$		0			0
$501$		−	827.075i		−	1.65085i
$502$		0			0
$503$	−305.342			−0.607042			−0.303521	−	0.952825i	$-0.598162\pi$
$503$	−305.342			−0.607042			−0.303521	+	0.952825i	$0.598162\pi$
$504$		0			0
$505$			59.0898i			0.117010i
$506$		0			0
$507$	3.08391			0.00608266
$508$		0			0
$509$	210.207			0.412981			0.206491	−	0.978449i	$-0.433796\pi$
$509$	210.207			0.412981			0.206491	+	0.978449i	$0.433796\pi$
$510$		0			0
$511$			453.528i			0.887530i
$512$		0			0
$513$	140.101			0.273101
$514$		0			0
$515$			317.912i			0.617305i
$516$		0			0
$517$	−3.15822			−0.00610874
$518$		0			0
$519$		−	76.0841i		−	0.146598i
$520$		0			0
$521$	−18.0976			−0.0347364			−0.0173682	−	0.999849i	$-0.505529\pi$
$521$	−18.0976			−0.0347364			−0.0173682	+	0.999849i	$0.505529\pi$
$522$		0			0
$523$			230.738i			0.441182i	0.975366	+	0.220591i	$0.0707986\pi$
$523$			230.738i			0.441182i	−0.975366	+	0.220591i	$0.929201\pi$
$524$		0			0
$525$	536.441			1.02179
$526$		0			0
$527$			158.021i			0.299850i
$528$		0			0
$529$	735.470			1.39030
$530$		0			0
$531$	−3.79896			−0.00715435
$532$		0			0
$533$		−	669.274i		−	1.25567i
$534$		0			0
$535$			236.075i			0.441263i
$536$		0			0
$537$	341.856			0.636603
$538$		0			0
$539$	0.102114			0.000189451
$540$		0			0
$541$	−665.526			−1.23018			−0.615088	−	0.788458i	$-0.710880\pi$
$541$	−665.526			−1.23018			−0.615088	+	0.788458i	$0.710880\pi$
$542$		0			0
$543$	847.631			1.56102
$544$		0			0
$545$	−343.086			−0.629516
$546$		0			0
$547$		−	631.348i		−	1.15420i	−0.816673	−	0.577100i	$-0.804184\pi$
$547$		−	631.348i		−	1.15420i	0.816673	−	0.577100i	$-0.195816\pi$
$548$		0			0
$549$	140.268			0.255498
$550$		0			0
$551$		−	305.100i		−	0.553721i
$552$		0			0
$553$			163.605i			0.295850i
$554$		0			0
$555$	385.077			0.693832
$556$		0			0
$557$	766.871			1.37679			0.688394	−	0.725337i	$-0.258316\pi$
$557$	766.871			1.37679			0.688394	+	0.725337i	$0.258316\pi$
$558$		0			0
$559$			409.126i			0.731889i
$560$		0			0
$561$	−0.930170			−0.00165806
$562$		0			0
$563$	562.959	−	6.75533i	0.999928	−	0.0119988i
$563$	562.959	−	6.75533i	0.999928	−	0.0119988i
$564$		0			0
$565$		−	321.399i		−	0.568847i
$566$		0			0
$567$	702.505			1.23899
$568$		0			0
$569$		−	811.664i		−	1.42647i	−0.700923	−	0.713237i	$-0.747228\pi$
$569$		−	811.664i		−	1.42647i	0.700923	−	0.713237i	$-0.252772\pi$
$570$		0			0
$571$			774.929i			1.35714i	0.734534	+	0.678572i	$0.237401\pi$
$571$			774.929i			1.35714i	−0.734534	+	0.678572i	$0.762599\pi$
$572$		0			0
$573$	187.697			0.327569
$574$		0			0
$575$	775.851			1.34931
$576$		0			0
$577$		−	235.802i		−	0.408669i	−0.978901	−	0.204335i	$-0.934497\pi$
$577$		−	235.802i		−	0.408669i	0.978901	−	0.204335i	$-0.0655031\pi$
$578$		0			0
$579$	623.348			1.07659
$580$		0			0
$581$		−	29.8346i		−	0.0513504i
$582$		0			0
$583$		−	1.83671i		−	0.00315044i
$584$		0			0
$585$		−	69.8093i		−	0.119332i
$586$		0			0
$587$			221.534i			0.377400i	0.982035	+	0.188700i	$0.0604273\pi$
$587$			221.534i			0.377400i	−0.982035	+	0.188700i	$0.939573\pi$
$588$		0			0
$589$			313.228i			0.531796i
$590$		0			0
$591$	911.965			1.54309
$592$		0			0
$593$	−608.513			−1.02616			−0.513080	−	0.858341i	$-0.671496\pi$
$593$	−608.513			−1.02616			−0.513080	+	0.858341i	$0.671496\pi$
$594$		0			0
$595$		−	43.1174i		−	0.0724662i
$596$		0			0
$597$		−	74.6807i		−	0.125093i
$598$		0			0
$599$	−485.059			−0.809782			−0.404891	−	0.914365i	$-0.632690\pi$
$599$	−485.059			−0.809782			−0.404891	+	0.914365i	$0.632690\pi$
$600$		0			0
$601$		−	460.254i		−	0.765813i	−0.923787	−	0.382907i	$-0.874923\pi$
$601$		−	460.254i		−	0.765813i	0.923787	−	0.382907i	$-0.125077\pi$
$602$		0			0
$603$	142.120			0.235687
$604$		0			0
$605$		−	215.815i		−	0.356719i
$606$		0			0
$607$	−1.60019			−0.00263623			−0.00131812	−	0.999999i	$-0.500420\pi$
$607$	−1.60019			−0.00263623			−0.00131812	+	0.999999i	$0.500420\pi$
$608$		0			0
$609$		−	1110.28i		−	1.82312i
$610$		0			0
$611$	520.193			0.851380
$612$		0			0
$613$			563.377i			0.919049i	0.888165	+	0.459524i	$0.151980\pi$
$613$			563.377i			0.919049i	−0.888165	+	0.459524i	$0.848020\pi$
$614$		0			0
$615$	319.191			0.519010
$616$		0			0
$617$			287.155i			0.465405i	0.972548	+	0.232703i	$0.0747568\pi$
$617$			287.155i			0.465405i	−0.972548	+	0.232703i	$0.925243\pi$
$618$		0			0
$619$			980.394i			1.58383i	0.610628	+	0.791917i	$0.290917\pi$
$619$			980.394i			1.58383i	−0.610628	+	0.791917i	$0.709083\pi$
$620$		0			0
$621$	737.373			1.18740
$622$		0			0
$623$			125.180i			0.200931i
$624$		0			0
$625$	396.506			0.634410
$626$		0			0
$627$	−1.84378			−0.00294063
$628$		0			0
$629$			212.258i			0.337453i
$630$		0			0
$631$	−891.574			−1.41295			−0.706477	−	0.707736i	$-0.749717\pi$
$631$	−891.574			−1.41295			−0.706477	+	0.707736i	$0.749717\pi$
$632$		0			0
$633$	115.756			0.182869
$634$		0			0
$635$		−	57.9296i		−	0.0912277i
$636$		0			0
$637$	−16.8193			−0.0264039
$638$		0			0
$639$	−109.867			−0.171937
$640$		0			0
$641$			1012.21i			1.57911i	0.613681	+	0.789554i	$0.289688\pi$
$641$			1012.21i			1.57911i	−0.613681	+	0.789554i	$0.710312\pi$
$642$		0			0
$643$		−	617.389i		−	0.960170i	−0.877222	−	0.480085i	$-0.840606\pi$
$643$		−	617.389i		−	0.960170i	0.877222	−	0.480085i	$-0.159394\pi$
$644$		0			0
$645$	−195.121			−0.302514
$646$		0			0
$647$	211.427			0.326780			0.163390	−	0.986562i	$-0.447757\pi$
$647$	211.427			0.326780			0.163390	+	0.986562i	$0.447757\pi$
$648$		0			0
$649$	−0.0990701			−0.000152650
$650$		0			0
$651$			1139.86i			1.75093i
$652$		0			0
$653$	968.072			1.48250			0.741250	−	0.671229i	$-0.234233\pi$
$653$	968.072			1.48250			0.741250	+	0.671229i	$0.234233\pi$
$654$		0			0
$655$	385.533			0.588600
$656$		0			0
$657$		−	193.032i		−	0.293808i
$658$		0			0
$659$		−	94.9656i		−	0.144106i	−0.997401	−	0.0720528i	$-0.977045\pi$
$659$		−	94.9656i		−	0.144106i	0.997401	−	0.0720528i	$-0.0229550\pi$
$660$		0			0
$661$			750.561i			1.13549i	0.823204	+	0.567746i	$0.192185\pi$
$661$			750.561i			1.13549i	−0.823204	+	0.567746i	$0.807815\pi$
$662$		0			0
$663$	153.209			0.231085
$664$		0			0
$665$		−	85.4670i		−	0.128522i
$666$		0			0
$667$		−	1605.79i		−	2.40748i
$668$		0			0
$669$	−179.685			−0.268588
$670$		0			0
$671$	3.65795			0.00545149
$672$		0			0
$673$	499.931			0.742839			0.371420	−	0.928465i	$-0.378871\pi$
$673$	499.931			0.742839			0.371420	+	0.928465i	$0.378871\pi$
$674$		0			0
$675$	452.436			0.670276
$676$		0			0
$677$		−	689.660i		−	1.01870i	−0.860559	−	0.509350i	$-0.829886\pi$
$677$		−	689.660i		−	1.01870i	0.860559	−	0.509350i	$-0.170114\pi$
$678$		0			0
$679$		−	501.173i		−	0.738105i
$680$		0			0
$681$		−	360.017i		−	0.528659i
$682$		0			0
$683$	801.523			1.17353			0.586766	−	0.809756i	$-0.300401\pi$
$683$	801.523			1.17353			0.586766	+	0.809756i	$0.300401\pi$
$684$		0			0
$685$		−	143.850i		−	0.210000i
$686$		0			0
$687$			480.371i			0.699230i
$688$		0			0
$689$			302.526i			0.439079i
$690$		0			0
$691$		−	1315.37i		−	1.90358i	−0.306751	−	0.951790i	$-0.599242\pi$
$691$		−	1315.37i		−	1.90358i	0.306751	−	0.951790i	$-0.400758\pi$
$692$		0			0
$693$	−1.68517			−0.00243170
$694$		0			0
$695$	−239.894			−0.345171
$696$		0			0
$697$			175.942i			0.252427i
$698$		0			0
$699$		−	324.689i		−	0.464505i
$700$		0			0
$701$			770.892i			1.09970i	0.835262	+	0.549852i	$0.185316\pi$
$701$			770.892i			1.09970i	−0.835262	+	0.549852i	$0.814684\pi$
$702$		0			0
$703$			420.736i			0.598487i
$704$		0			0
$705$			248.092i			0.351903i
$706$		0			0
$707$	−234.945			−0.332312
$708$		0			0
$709$	−1231.20			−1.73653			−0.868265	−	0.496101i	$-0.834765\pi$
$709$	−1231.20			−1.73653			−0.868265	+	0.496101i	$0.834765\pi$
$710$		0			0
$711$		−	69.6340i		−	0.0979382i
$712$		0			0
$713$			1648.57i			2.31216i
$714$		0			0
$715$		−	1.82050i		−	0.00254616i
$716$		0			0
$717$		−	84.9233i		−	0.118442i
$718$		0			0
$719$	−525.218			−0.730484			−0.365242	−	0.930913i	$-0.619014\pi$
$719$	−525.218			−0.730484			−0.365242	+	0.930913i	$0.619014\pi$
$720$		0			0
$721$	−1264.04			−1.75317
$722$		0			0
$723$	1241.77			1.71752
$724$		0			0
$725$		−	985.278i		−	1.35900i
$726$		0			0
$727$			236.734i			0.325632i	0.986656	+	0.162816i	$0.0520577\pi$
$727$			236.734i			0.325632i	−0.986656	+	0.162816i	$0.947942\pi$
$728$		0			0
$729$	347.994			0.477358
$730$		0			0
$731$		−	107.553i		−	0.147131i
$732$		0			0
$733$	73.5579			0.100352			0.0501759	−	0.998740i	$-0.484022\pi$
$733$	73.5579			0.100352			0.0501759	+	0.998740i	$0.484022\pi$
$734$		0			0
$735$		−	8.02148i		−	0.0109136i
$736$		0			0
$737$	3.70622			0.00502880
$738$		0			0
$739$	271.733			0.367704			0.183852	−	0.982954i	$-0.441143\pi$
$739$	271.733			0.367704			0.183852	+	0.982954i	$0.441143\pi$
$740$		0			0
$741$	303.690			0.409838
$742$		0			0
$743$			1054.54i			1.41930i	0.704553	+	0.709651i	$0.251148\pi$
$743$			1054.54i			1.41930i	−0.704553	+	0.709651i	$0.748852\pi$
$744$		0			0
$745$		−	315.318i		−	0.423245i
$746$		0			0
$747$			12.6983i			0.0169990i
$748$		0			0
$749$	−938.650			−1.25320
$750$		0			0
$751$	−688.375			−0.916612			−0.458306	−	0.888795i	$-0.651544\pi$
$751$	−688.375			−0.916612			−0.458306	+	0.888795i	$0.651544\pi$
$752$		0			0
$753$	−230.847			−0.306570
$754$		0			0
$755$	510.234			0.675807
$756$		0			0
$757$	−254.236			−0.335847			−0.167924	−	0.985800i	$-0.553706\pi$
$757$	−254.236			−0.335847			−0.167924	+	0.985800i	$0.553706\pi$
$758$		0			0
$759$	−9.70409			−0.0127854
$760$		0			0
$761$			169.796i			0.223122i	0.993758	+	0.111561i	$0.0355850\pi$
$761$			169.796i			0.223122i	−0.993758	+	0.111561i	$0.964415\pi$
$762$		0			0
$763$		−	1364.13i		−	1.78785i
$764$		0			0
$765$			18.3518i			0.0239892i
$766$		0			0
$767$	16.3179			0.0212750
$768$		0			0
$769$			122.260i			0.158986i	0.996835	+	0.0794930i	$0.0253301\pi$
$769$			122.260i			0.158986i	−0.996835	+	0.0794930i	$0.974670\pi$
$770$		0			0
$771$	−858.147			−1.11303
$772$		0			0
$773$	1077.99			1.39456			0.697279	−	0.716799i	$-0.254394\pi$
$773$	1077.99			1.39456			0.697279	+	0.716799i	$0.254394\pi$
$774$		0			0
$775$			1011.52i			1.30519i
$776$		0			0
$777$			1531.09i			1.97051i
$778$		0			0
$779$			348.750i			0.447689i
$780$		0			0
$781$	−2.86515			−0.00366856
$782$		0			0
$783$		−	936.414i		−	1.19593i
$784$		0			0
$785$	353.736			0.450619
$786$		0			0
$787$		−	191.192i		−	0.242938i	−0.992595	−	0.121469i	$-0.961240\pi$
$787$		−	191.192i		−	0.242938i	0.992595	−	0.121469i	$-0.0387605\pi$
$788$		0			0
$789$			413.956i			0.524659i
$790$		0			0
$791$	1277.90			1.61555
$792$		0			0
$793$	−602.504			−0.759778
$794$		0			0
$795$	−144.281			−0.181486
$796$		0			0
$797$		−	216.924i		−	0.272175i	−0.990697	−	0.136088i	$-0.956547\pi$
$797$		−	216.924i		−	0.272175i	0.990697	−	0.136088i	$-0.0434529\pi$
$798$		0			0
$799$	−136.751			−0.171152
$800$		0			0
$801$		−	53.2796i		−	0.0665163i
$802$		0			0
$803$		−	5.03393i		−	0.00626890i
$804$		0			0
$805$		−	449.827i		−	0.558791i
$806$		0			0
$807$	358.012			0.443634
$808$		0			0
$809$	−1022.75			−1.26422			−0.632110	−	0.774879i	$-0.717811\pi$
$809$	−1022.75			−1.26422			−0.632110	+	0.774879i	$0.717811\pi$
$810$		0			0
$811$	−336.769			−0.415251			−0.207626	−	0.978208i	$-0.566574\pi$
$811$	−336.769			−0.415251			−0.207626	+	0.978208i	$0.566574\pi$
$812$		0			0
$813$	−1142.13			−1.40484
$814$		0			0
$815$	224.976			0.276044
$816$		0			0
$817$		−	213.190i		−	0.260943i
$818$		0			0
$819$	277.566			0.338908
$820$		0			0
$821$	−498.182			−0.606799			−0.303400	−	0.952863i	$-0.598122\pi$
$821$	−498.182			−0.606799			−0.303400	+	0.952863i	$0.598122\pi$
$822$		0			0
$823$			77.7654i			0.0944902i	0.998883	+	0.0472451i	$0.0150442\pi$
$823$			77.7654i			0.0944902i	−0.998883	+	0.0472451i	$0.984956\pi$
$824$		0			0
$825$	−5.95422			−0.00721723
$826$		0			0
$827$		−	1512.80i		−	1.82926i	−0.404295	−	0.914628i	$-0.632483\pi$
$827$		−	1512.80i		−	1.82926i	0.404295	−	0.914628i	$-0.367517\pi$
$828$		0			0
$829$		−	1558.71i		−	1.88023i	−0.340853	−	0.940117i	$-0.610716\pi$
$829$		−	1558.71i		−	1.88023i	0.340853	−	0.940117i	$-0.389284\pi$
$830$		0			0
$831$	−963.869			−1.15989
$832$		0			0
$833$	4.42152			0.00530795
$834$		0			0
$835$	−425.538			−0.509626
$836$		0			0
$837$			961.359i			1.14858i
$838$		0			0
$839$	−458.595			−0.546598			−0.273299	−	0.961929i	$-0.588115\pi$
$839$	−458.595			−0.546598			−0.273299	+	0.961929i	$0.588115\pi$
$840$		0			0
$841$	−1198.24			−1.42479
$842$		0			0
$843$	197.100			0.233808
$844$		0			0
$845$		−	1.58670i		−	0.00187775i
$846$		0			0
$847$	858.094			1.01310
$848$		0			0
$849$		−	821.599i		−	0.967725i
$850$		0			0
$851$			2214.40i			2.60212i
$852$		0			0
$853$			855.927i			1.00343i	0.865033	+	0.501716i	$0.167298\pi$
$853$			855.927i			1.00343i	−0.865033	+	0.501716i	$0.832702\pi$
$854$		0			0
$855$			36.3767i			0.0425459i
$856$		0			0
$857$		−	371.457i		−	0.433438i	−0.976234	−	0.216719i	$-0.930464\pi$
$857$		−	371.457i		−	0.433438i	0.976234	−	0.216719i	$-0.0695356\pi$
$858$		0			0
$859$	−1042.70			−1.21386			−0.606928	−	0.794756i	$-0.707599\pi$
$859$	−1042.70			−1.21386			−0.606928	+	0.794756i	$0.707599\pi$
$860$		0			0
$861$			1269.12i			1.47401i
$862$		0			0
$863$	−1360.23			−1.57616			−0.788080	−	0.615573i	$-0.788925\pi$
$863$	−1360.23			−1.57616			−0.788080	+	0.615573i	$0.788925\pi$
$864$		0			0
$865$	−39.1460			−0.0452555
$866$		0			0
$867$	961.622			1.10914
$868$		0			0
$869$		−	1.81593i		−	0.00208968i
$870$		0			0
$871$	−610.456			−0.700868
$872$		0			0
$873$			213.311i			0.244342i
$874$		0			0
$875$		−	592.254i		−	0.676862i
$876$		0			0
$877$	939.441			1.07120			0.535599	−	0.844472i	$-0.320086\pi$
$877$	939.441			1.07120			0.535599	+	0.844472i	$0.320086\pi$
$878$		0			0
$879$			39.1347i			0.0445218i
$880$		0			0
$881$	−1576.80			−1.78979			−0.894894	−	0.446279i	$-0.852749\pi$
$881$	−1576.80			−1.78979			−0.894894	+	0.446279i	$0.852749\pi$
$882$		0			0
$883$			506.285i			0.573369i	0.958025	+	0.286684i	$0.0925531\pi$
$883$			506.285i			0.573369i	−0.958025	+	0.286684i	$0.907447\pi$
$884$		0			0
$885$			7.78239i			0.00879366i
$886$		0			0
$887$	−567.090			−0.639335			−0.319668	−	0.947530i	$-0.603571\pi$
$887$	−567.090			−0.639335			−0.319668	+	0.947530i	$0.603571\pi$
$888$		0			0
$889$	230.331			0.259091
$890$		0			0
$891$	−7.79745			−0.00875134
$892$		0			0
$893$	−271.066			−0.303545
$894$		0			0
$895$		−	175.888i		−	0.196523i
$896$		0			0
$897$	1598.37			1.78191
$898$		0			0
$899$	2093.57			2.32877
$900$		0			0
$901$		−	79.5292i		−	0.0882677i
$902$		0			0
$903$		−	775.814i		−	0.859152i
$904$		0			0
$905$		−	436.114i		−	0.481894i
$906$		0			0
$907$	592.683			0.653454			0.326727	−	0.945119i	$-0.394054\pi$
$907$	592.683			0.653454			0.326727	+	0.945119i	$0.394054\pi$
$908$		0			0
$909$	99.9978			0.110009
$910$		0			0
$911$		−	632.663i		−	0.694470i	−0.937778	−	0.347235i	$-0.887121\pi$
$911$		−	632.663i		−	0.694470i	0.937778	−	0.347235i	$-0.112879\pi$
$912$		0			0
$913$			0.331148i			0.000362703i
$914$		0			0
$915$		−	287.348i		−	0.314041i
$916$		0			0
$917$			1532.90i			1.67165i
$918$		0			0
$919$		−	278.808i		−	0.303382i	−0.988428	−	0.151691i	$-0.951528\pi$
$919$		−	278.808i		−	0.303382i	0.988428	−	0.151691i	$-0.0484719\pi$
$920$		0			0
$921$		−	1730.66i		−	1.87911i
$922$		0			0
$923$	471.921			0.511291
$924$		0			0
$925$			1358.71i			1.46887i
$926$		0			0
$927$	538.003			0.580370
$928$		0			0
$929$			1571.51i			1.69162i	0.533485	+	0.845809i	$0.320882\pi$
$929$			1571.51i			1.69162i	−0.533485	+	0.845809i	$0.679118\pi$
$930$		0			0
$931$	8.76431			0.00941386
$932$		0			0
$933$		−	1974.74i		−	2.11655i
$934$		0			0
$935$			0.478581i			0.000511851i
$936$		0			0
$937$		−	1723.64i		−	1.83953i	−0.392472	−	0.919764i	$-0.628380\pi$
$937$		−	1723.64i		−	1.83953i	0.392472	−	0.919764i	$-0.371620\pi$
$938$		0			0
$939$			524.348i			0.558411i
$940$		0			0
$941$		−	109.977i		−	0.116873i	−0.998291	−	0.0584363i	$-0.981389\pi$
$941$		−	109.977i		−	0.116873i	0.998291	−	0.0584363i	$-0.0186115\pi$
$942$		0			0
$943$			1835.53i			1.94648i
$944$		0			0
$945$		−	262.316i		−	0.277583i
$946$		0			0
$947$		−	472.288i		−	0.498720i	−0.968411	−	0.249360i	$-0.919780\pi$
$947$		−	472.288i		−	0.498720i	0.968411	−	0.249360i	$-0.0802203\pi$
$948$		0			0
$949$			829.143i			0.873702i
$950$		0			0
$951$		−	522.904i		−	0.549847i
$952$		0			0
$953$	585.822			0.614713			0.307357	−	0.951594i	$-0.400556\pi$
$953$	585.822			0.614713			0.307357	+	0.951594i	$0.400556\pi$
$954$		0			0
$955$		−	96.5720i		−	0.101123i
$956$		0			0
$957$			12.3235i			0.0128773i
$958$		0			0
$959$	571.955			0.596408
$960$		0			0
$961$	−1188.34			−1.23657
$962$		0			0
$963$	399.511			0.414861
$964$		0			0
$965$		−	320.718i		−	0.332350i
$966$		0			0
$967$	197.341			0.204075			0.102038	−	0.994781i	$-0.467464\pi$
$967$	197.341			0.204075			0.102038	+	0.994781i	$0.467464\pi$
$968$		0			0
$969$	−79.8354			−0.0823895
$970$		0			0
$971$			395.335i			0.407142i	0.979060	+	0.203571i	$0.0652548\pi$
$971$			395.335i			0.407142i	−0.979060	+	0.203571i	$0.934745\pi$
$972$		0			0
$973$		−	953.832i		−	0.980300i
$974$		0			0
$975$	980.725			1.00587
$976$		0			0
$977$			1286.98i			1.31728i	0.752459	+	0.658639i	$0.228868\pi$
$977$			1286.98i			1.31728i	−0.752459	+	0.658639i	$0.771132\pi$
$978$		0			0
$979$		−	1.38944i		−	0.00141924i
$980$		0			0
$981$			580.606i			0.591851i
$982$		0			0
$983$		−	816.858i		−	0.830985i	−0.909597	−	0.415492i	$-0.863609\pi$
$983$		−	816.858i		−	0.830985i	0.909597	−	0.415492i	$-0.136391\pi$
$984$		0			0
$985$		−	469.214i		−	0.476360i
$986$		0			0
$987$	−986.427			−0.999419
$988$		0			0
$989$		−	1122.05i		−	1.13453i
$990$		0			0
$991$	1160.32			1.17086			0.585431	−	0.810722i	$-0.300925\pi$
$991$	1160.32			1.17086			0.585431	+	0.810722i	$0.300925\pi$
$992$		0			0
$993$		−	1292.57i		−	1.30168i
$994$		0			0
$995$	−38.4239			−0.0386170
$996$		0			0
$997$	86.6082			0.0868688			0.0434344	−	0.999056i	$-0.486170\pi$
$997$	86.6082			0.0868688			0.0434344	+	0.999056i	$0.486170\pi$
$998$		0			0
$999$			1291.32i			1.29262i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2252.3.d.b.1125.16	yes	76
563.562	odd	2	inner	2252.3.d.b.1125.15	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2252.3.d.b.1125.15	✓	76	563.562	odd	2	inner
2252.3.d.b.1125.16	yes	76	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2252 = 2^{2} \cdot 563 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2252.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.46678 q^{3} +1.78369i q^{5} -7.09205 q^{7} +3.01854 q^{9} +0.0787181 q^{11} -12.9657 q^{13} -6.18365i q^{15} +3.40849 q^{17} +6.75628 q^{19} +24.5866 q^{21} +35.5594 q^{23} +21.8185 q^{25} +20.7364 q^{27} -45.1580i q^{29} +46.3610i q^{31} -0.272898 q^{33} -12.6500i q^{35} +62.2734i q^{37} +44.9493 q^{39} +51.6186i q^{41} -31.5544i q^{43} +5.38413i q^{45} -40.1206 q^{47} +1.29721 q^{49} -11.8165 q^{51} -23.3327i q^{53} +0.140409i q^{55} -23.4225 q^{57} -1.25854 q^{59} +46.4689 q^{61} -21.4076 q^{63} -23.1268i q^{65} +47.0822 q^{67} -123.276 q^{69} -36.3976 q^{71} -63.9488i q^{73} -75.6397 q^{75} -0.558273 q^{77} -23.0688i q^{79} -99.0553 q^{81} +4.20676i q^{83} +6.07968i q^{85} +156.553i q^{87} -17.6508i q^{89} +91.9537 q^{91} -160.723i q^{93} +12.0511i q^{95} +70.6669i q^{97} +0.237614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q + 4 q^{3} - 8 q^{7} + 128 q^{9} + 2 q^{11} - 6 q^{13} + 22 q^{17} + 12 q^{19} - 6 q^{21} + 24 q^{23} - 912 q^{25} + 22 q^{27} - 52 q^{33} - 70 q^{39} - 28 q^{47} - 340 q^{49} + 314 q^{51} - 98 q^{57}+ \cdots + 348 q^{99}+O(q^{100}) \) 76 * q + 4 * q^3 - 8 * q^7 + 128 * q^9 + 2 * q^11 - 6 * q^13 + 22 * q^17 + 12 * q^19 - 6 * q^21 + 24 * q^23 - 912 * q^25 + 22 * q^27 - 52 * q^33 - 70 * q^39 - 28 * q^47 - 340 * q^49 + 314 * q^51 - 98 * q^57 + 18 * q^59 - 162 * q^61 - 246 * q^63 - 178 * q^67 + 2 * q^69 + 72 * q^71 + 14 * q^75 - 498 * q^77 - 468 * q^81 + 132 * q^91 + 348 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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