LMFDB - Embedded newform 228.2.d.b.113.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [228,2,Mod(113,228)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(228, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 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("228.113"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 228 = 2^{2} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	228.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4] 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:	$1.82058916609$
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} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.3
Root			$1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	228.113
Dual form			228.2.d.b.113.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.58114 - 0.707107i) q^{3} -2.23607i q^{5} -1.00000 q^{7} +(2.00000 - 2.23607i) q^{9} -2.23607i q^{11} +4.24264i q^{13} +(-1.58114 - 3.53553i) q^{15} -2.23607i q^{17} +(1.00000 + 4.24264i) q^{19} +(-1.58114 + 0.707107i) q^{21} +4.47214i q^{23} +(1.58114 - 4.94975i) q^{27} +4.24264i q^{31} +(-1.58114 - 3.53553i) q^{33} +2.23607i q^{35} -4.24264i q^{37} +(3.00000 + 6.70820i) q^{39} +9.48683 q^{41} -7.00000 q^{43} +(-5.00000 - 4.47214i) q^{45} +11.1803i q^{47} -6.00000 q^{49} +(-1.58114 - 3.53553i) q^{51} -9.48683 q^{53} -5.00000 q^{55} +(4.58114 + 6.00110i) q^{57} -9.48683 q^{59} +11.0000 q^{61} +(-2.00000 + 2.23607i) q^{63} +9.48683 q^{65} +8.48528i q^{67} +(3.16228 + 7.07107i) q^{69} -9.48683 q^{71} +5.00000 q^{73} +2.23607i q^{77} +(-1.00000 - 8.94427i) q^{81} -8.94427i q^{83} -5.00000 q^{85} -9.48683 q^{89} -4.24264i q^{91} +(3.00000 + 6.70820i) q^{93} +(9.48683 - 2.23607i) q^{95} -12.7279i q^{97} +(-5.00000 - 4.47214i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} + 8 q^{9} + 4 q^{19} + 12 q^{39} - 28 q^{43} - 20 q^{45} - 24 q^{49} - 20 q^{55} + 12 q^{57} + 44 q^{61} - 8 q^{63} + 20 q^{73} - 4 q^{81} - 20 q^{85} + 12 q^{93} - 20 q^{99}+O(q^{100}) $ 4 * q - 4 * q^7 + 8 * q^9 + 4 * q^19 + 12 * q^39 - 28 * q^43 - 20 * q^45 - 24 * q^49 - 20 * q^55 + 12 * q^57 + 44 * q^61 - 8 * q^63 + 20 * q^73 - 4 * q^81 - 20 * q^85 + 12 * q^93 - 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/228\mathbb{Z}\right)^\times$.

$n$	$77$	$97$	$115$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	1.58114	−	0.707107i	0.912871	−	0.408248i
$3$	1.58114	−	0.707107i	0.912871	−	0.408248i
$4$		0			0
$5$		−	2.23607i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$5$		−	2.23607i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$6$		0			0
$7$	−1.00000			−0.377964			−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000			−0.377964			−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$	2.00000	−	2.23607i	0.666667	−	0.745356i
$10$		0			0
$11$		−	2.23607i		−	0.674200i	−0.941469	−	0.337100i	$-0.890554\pi$
$11$		−	2.23607i		−	0.674200i	0.941469	−	0.337100i	$-0.109446\pi$
$12$		0			0
$13$			4.24264i			1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$			4.24264i			1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$		0			0
$15$	−1.58114	−	3.53553i	−0.408248	−	0.912871i
$16$		0			0
$17$		−	2.23607i		−	0.542326i	−0.962533	−	0.271163i	$-0.912592\pi$
$17$		−	2.23607i		−	0.542326i	0.962533	−	0.271163i	$-0.0874083\pi$
$18$		0			0
$19$	1.00000	+	4.24264i	0.229416	+	0.973329i
$19$	1.00000	+	4.24264i	0.229416	+	0.973329i
$20$		0			0
$21$	−1.58114	+	0.707107i	−0.345033	+	0.154303i
$22$		0			0
$23$			4.47214i			0.932505i	0.884652	+	0.466252i	$0.154396\pi$
$23$			4.47214i			0.932505i	−0.884652	+	0.466252i	$0.845604\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	1.58114	−	4.94975i	0.304290	−	0.952579i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$			4.24264i			0.762001i	0.924575	+	0.381000i	$0.124420\pi$
$31$			4.24264i			0.762001i	−0.924575	+	0.381000i	$0.875580\pi$
$32$		0			0
$33$	−1.58114	−	3.53553i	−0.275241	−	0.615457i
$34$		0			0
$35$			2.23607i			0.377964i
$36$		0			0
$37$		−	4.24264i		−	0.697486i	−0.937218	−	0.348743i	$-0.886609\pi$
$37$		−	4.24264i		−	0.697486i	0.937218	−	0.348743i	$-0.113391\pi$
$38$		0			0
$39$	3.00000	+	6.70820i	0.480384	+	1.07417i
$40$		0			0
$41$	9.48683			1.48159			0.740797	−	0.671729i	$-0.234448\pi$
$41$	9.48683			1.48159			0.740797	+	0.671729i	$0.234448\pi$
$42$		0			0
$43$	−7.00000			−1.06749			−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000			−1.06749			−0.533745	+	0.845645i	$0.679216\pi$
$44$		0			0
$45$	−5.00000	−	4.47214i	−0.745356	−	0.666667i
$46$		0			0
$47$			11.1803i			1.63082i	0.578884	+	0.815410i	$0.303489\pi$
$47$			11.1803i			1.63082i	−0.578884	+	0.815410i	$0.696511\pi$
$48$		0			0
$49$	−6.00000			−0.857143
$50$		0			0
$51$	−1.58114	−	3.53553i	−0.221404	−	0.495074i
$52$		0			0
$53$	−9.48683			−1.30312			−0.651558	−	0.758599i	$-0.725884\pi$
$53$	−9.48683			−1.30312			−0.651558	+	0.758599i	$0.725884\pi$
$54$		0			0
$55$	−5.00000			−0.674200
$56$		0			0
$57$	4.58114	+	6.00110i	0.606787	+	0.794865i
$58$		0			0
$59$	−9.48683			−1.23508			−0.617540	−	0.786539i	$-0.711871\pi$
$59$	−9.48683			−1.23508			−0.617540	+	0.786539i	$0.711871\pi$
$60$		0			0
$61$	11.0000			1.40841			0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000			1.40841			0.704203	+	0.709999i	$0.251305\pi$
$62$		0			0
$63$	−2.00000	+	2.23607i	−0.251976	+	0.281718i
$64$		0			0
$65$	9.48683			1.17670
$66$		0			0
$67$			8.48528i			1.03664i	0.855186	+	0.518321i	$0.173443\pi$
$67$			8.48528i			1.03664i	−0.855186	+	0.518321i	$0.826557\pi$
$68$		0			0
$69$	3.16228	+	7.07107i	0.380693	+	0.851257i
$70$		0			0
$71$	−9.48683			−1.12588			−0.562940	−	0.826498i	$-0.690330\pi$
$71$	−9.48683			−1.12588			−0.562940	+	0.826498i	$0.690330\pi$
$72$		0			0
$73$	5.00000			0.585206			0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000			0.585206			0.292603	+	0.956234i	$0.405479\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.23607i			0.254824i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−1.00000	−	8.94427i	−0.111111	−	0.993808i
$82$		0			0
$83$		−	8.94427i		−	0.981761i	−0.871227	−	0.490881i	$-0.836675\pi$
$83$		−	8.94427i		−	0.981761i	0.871227	−	0.490881i	$-0.163325\pi$
$84$		0			0
$85$	−5.00000			−0.542326
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.48683			−1.00560			−0.502801	−	0.864402i	$-0.667697\pi$
$89$	−9.48683			−1.00560			−0.502801	+	0.864402i	$0.667697\pi$
$90$		0			0
$91$		−	4.24264i		−	0.444750i
$92$		0			0
$93$	3.00000	+	6.70820i	0.311086	+	0.695608i
$94$		0			0
$95$	9.48683	−	2.23607i	0.973329	−	0.229416i
$96$		0			0
$97$		−	12.7279i		−	1.29232i	−0.763200	−	0.646162i	$-0.776373\pi$
$97$		−	12.7279i		−	1.29232i	0.763200	−	0.646162i	$-0.223627\pi$
$98$		0			0
$99$	−5.00000	−	4.47214i	−0.502519	−	0.449467i
$100$		0			0
$101$		−	8.94427i		−	0.889988i	−0.895533	−	0.444994i	$-0.853206\pi$
$101$		−	8.94427i		−	0.889988i	0.895533	−	0.444994i	$-0.146794\pi$
$102$		0			0
$103$			8.48528i			0.836080i	0.908429	+	0.418040i	$0.137283\pi$
$103$			8.48528i			0.836080i	−0.908429	+	0.418040i	$0.862717\pi$
$104$		0			0
$105$	1.58114	+	3.53553i	0.154303	+	0.345033i
$106$		0			0
$107$	18.9737			1.83425			0.917127	−	0.398596i	$-0.130502\pi$
$107$	18.9737			1.83425			0.917127	+	0.398596i	$0.130502\pi$
$108$		0			0
$109$		−	8.48528i		−	0.812743i	−0.913708	−	0.406371i	$-0.866794\pi$
$109$		−	8.48528i		−	0.812743i	0.913708	−	0.406371i	$-0.133206\pi$
$110$		0			0
$111$	−3.00000	−	6.70820i	−0.284747	−	0.636715i
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	10.0000			0.932505
$116$		0			0
$117$	9.48683	+	8.48528i	0.877058	+	0.784465i
$118$		0			0
$119$			2.23607i			0.204980i
$120$		0			0
$121$	6.00000			0.545455
$122$		0			0
$123$	15.0000	−	6.70820i	1.35250	−	0.604858i
$124$		0			0
$125$		−	11.1803i		−	1.00000i
$126$		0			0
$127$		−	16.9706i		−	1.50589i	−0.658081	−	0.752947i	$-0.728632\pi$
$127$		−	16.9706i		−	1.50589i	0.658081	−	0.752947i	$-0.271368\pi$
$128$		0			0
$129$	−11.0680	+	4.94975i	−0.974481	+	0.435801i
$130$		0			0
$131$		−	15.6525i		−	1.36756i	−0.729687	−	0.683782i	$-0.760334\pi$
$131$		−	15.6525i		−	1.36756i	0.729687	−	0.683782i	$-0.239666\pi$
$132$		0			0
$133$	−1.00000	−	4.24264i	−0.0867110	−	0.367884i
$134$		0			0
$135$	−11.0680	−	3.53553i	−0.952579	−	0.304290i
$136$		0			0
$137$			11.1803i			0.955201i	0.878577	+	0.477600i	$0.158493\pi$
$137$			11.1803i			0.955201i	−0.878577	+	0.477600i	$0.841507\pi$
$138$		0			0
$139$	11.0000			0.933008			0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000			0.933008			0.466504	+	0.884519i	$0.345513\pi$
$140$		0			0
$141$	7.90569	+	17.6777i	0.665780	+	1.48873i
$142$		0			0
$143$	9.48683			0.793329
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−9.48683	+	4.24264i	−0.782461	+	0.349927i
$148$		0			0
$149$			11.1803i			0.915929i	0.888970	+	0.457965i	$0.151421\pi$
$149$			11.1803i			0.915929i	−0.888970	+	0.457965i	$0.848579\pi$
$150$		0			0
$151$			4.24264i			0.345261i	0.984987	+	0.172631i	$0.0552267\pi$
$151$			4.24264i			0.345261i	−0.984987	+	0.172631i	$0.944773\pi$
$152$		0			0
$153$	−5.00000	−	4.47214i	−0.404226	−	0.361551i
$154$		0			0
$155$	9.48683			0.762001
$156$		0			0
$157$	−4.00000			−0.319235			−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000			−0.319235			−0.159617	+	0.987179i	$0.551026\pi$
$158$		0			0
$159$	−15.0000	+	6.70820i	−1.18958	+	0.531995i
$160$		0			0
$161$		−	4.47214i		−	0.352454i
$162$		0			0
$163$	2.00000			0.156652			0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000			0.156652			0.0783260	+	0.996928i	$0.475042\pi$
$164$		0			0
$165$	−7.90569	+	3.53553i	−0.615457	+	0.275241i
$166$		0			0
$167$	−18.9737			−1.46823			−0.734113	−	0.679027i	$-0.762402\pi$
$167$	−18.9737			−1.46823			−0.734113	+	0.679027i	$0.762402\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$	11.4868	+	6.24921i	0.878420	+	0.477889i
$172$		0			0
$173$	−18.9737			−1.44254			−0.721271	−	0.692653i	$-0.756442\pi$
$173$	−18.9737			−1.44254			−0.721271	+	0.692653i	$0.756442\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−15.0000	+	6.70820i	−1.12747	+	0.504219i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		−	25.4558i		−	1.89212i	−0.323994	−	0.946059i	$-0.605026\pi$
$181$		−	25.4558i		−	1.89212i	0.323994	−	0.946059i	$-0.394974\pi$
$182$		0			0
$183$	17.3925	−	7.77817i	1.28569	−	0.574979i
$184$		0			0
$185$	−9.48683			−0.697486
$186$		0			0
$187$	−5.00000			−0.365636
$188$		0			0
$189$	−1.58114	+	4.94975i	−0.115011	+	0.360041i
$190$		0			0
$191$		−	2.23607i		−	0.161796i	−0.996722	−	0.0808981i	$-0.974221\pi$
$191$		−	2.23607i		−	0.161796i	0.996722	−	0.0808981i	$-0.0257788\pi$
$192$		0			0
$193$			25.4558i			1.83235i	0.400776	+	0.916176i	$0.368740\pi$
$193$			25.4558i			1.83235i	−0.400776	+	0.916176i	$0.631260\pi$
$194$		0			0
$195$	15.0000	−	6.70820i	1.07417	−	0.480384i
$196$		0			0
$197$			17.8885i			1.27451i	0.770655	+	0.637253i	$0.219929\pi$
$197$			17.8885i			1.27451i	−0.770655	+	0.637253i	$0.780071\pi$
$198$		0			0
$199$	−13.0000			−0.921546			−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000			−0.921546			−0.460773	+	0.887518i	$0.652428\pi$
$200$		0			0
$201$	6.00000	+	13.4164i	0.423207	+	0.946320i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		−	21.2132i		−	1.48159i
$206$		0			0
$207$	10.0000	+	8.94427i	0.695048	+	0.621670i
$208$		0			0
$209$	9.48683	−	2.23607i	0.656218	−	0.154672i
$210$		0			0
$211$			4.24264i			0.292075i	0.989279	+	0.146038i	$0.0466521\pi$
$211$			4.24264i			0.292075i	−0.989279	+	0.146038i	$0.953348\pi$
$212$		0			0
$213$	−15.0000	+	6.70820i	−1.02778	+	0.459639i
$214$		0			0
$215$			15.6525i			1.06749i
$216$		0			0
$217$		−	4.24264i		−	0.288009i
$218$		0			0
$219$	7.90569	−	3.53553i	0.534217	−	0.238909i
$220$		0			0
$221$	9.48683			0.638153
$222$		0			0
$223$		−	4.24264i		−	0.284108i	−0.989859	−	0.142054i	$-0.954629\pi$
$223$		−	4.24264i		−	0.284108i	0.989859	−	0.142054i	$-0.0453707\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.48683			−0.629663			−0.314832	−	0.949148i	$-0.601948\pi$
$227$	−9.48683			−0.629663			−0.314832	+	0.949148i	$0.601948\pi$
$228$		0			0
$229$	−13.0000			−0.859064			−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000			−0.859064			−0.429532	+	0.903052i	$0.641321\pi$
$230$		0			0
$231$	1.58114	+	3.53553i	0.104031	+	0.232621i
$232$		0			0
$233$			24.5967i			1.61139i	0.592333	+	0.805693i	$0.298207\pi$
$233$			24.5967i			1.61139i	−0.592333	+	0.805693i	$0.701793\pi$
$234$		0			0
$235$	25.0000			1.63082
$236$		0			0
$237$		0			0
$238$		0			0
$239$			11.1803i			0.723196i	0.932334	+	0.361598i	$0.117769\pi$
$239$			11.1803i			0.723196i	−0.932334	+	0.361598i	$0.882231\pi$
$240$		0			0
$241$		−	4.24264i		−	0.273293i	−0.990620	−	0.136646i	$-0.956368\pi$
$241$		−	4.24264i		−	0.273293i	0.990620	−	0.136646i	$-0.0436324\pi$
$242$		0			0
$243$	−7.90569	−	13.4350i	−0.507151	−	0.861858i
$244$		0			0
$245$			13.4164i			0.857143i
$246$		0			0
$247$	−18.0000	+	4.24264i	−1.14531	+	0.269953i
$248$		0			0
$249$	−6.32456	−	14.1421i	−0.400802	−	0.896221i
$250$		0			0
$251$		−	29.0689i		−	1.83481i	−0.397953	−	0.917406i	$-0.630279\pi$
$251$		−	29.0689i		−	1.83481i	0.397953	−	0.917406i	$-0.369721\pi$
$252$		0			0
$253$	10.0000			0.628695
$254$		0			0
$255$	−7.90569	+	3.53553i	−0.495074	+	0.221404i
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$			4.24264i			0.263625i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	15.6525i		−	0.965173i	−0.875848	−	0.482587i	$-0.839697\pi$
$263$		−	15.6525i		−	0.965173i	0.875848	−	0.482587i	$-0.160303\pi$
$264$		0			0
$265$			21.2132i			1.30312i
$266$		0			0
$267$	−15.0000	+	6.70820i	−0.917985	+	0.410535i
$268$		0			0
$269$	28.4605			1.73527			0.867634	−	0.497204i	$-0.165640\pi$
$269$	28.4605			1.73527			0.867634	+	0.497204i	$0.165640\pi$
$270$		0			0
$271$	2.00000			0.121491			0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000			0.121491			0.0607457	+	0.998153i	$0.480652\pi$
$272$		0			0
$273$	−3.00000	−	6.70820i	−0.181568	−	0.405999i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.00000			−0.0600842			−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000			−0.0600842			−0.0300421	+	0.999549i	$0.509564\pi$
$278$		0			0
$279$	9.48683	+	8.48528i	0.567962	+	0.508001i
$280$		0			0
$281$	−18.9737			−1.13187			−0.565937	−	0.824448i	$-0.691485\pi$
$281$	−18.9737			−1.13187			−0.565937	+	0.824448i	$0.691485\pi$
$282$		0			0
$283$	−13.0000			−0.772770			−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000			−0.772770			−0.386385	+	0.922338i	$0.626276\pi$
$284$		0			0
$285$	13.4189	−	10.2437i	0.794865	−	0.606787i
$286$		0			0
$287$	−9.48683			−0.559990
$288$		0			0
$289$	12.0000			0.705882
$290$		0			0
$291$	−9.00000	−	20.1246i	−0.527589	−	1.17973i
$292$		0			0
$293$	18.9737			1.10845			0.554227	−	0.832366i	$-0.313014\pi$
$293$	18.9737			1.10845			0.554227	+	0.832366i	$0.313014\pi$
$294$		0			0
$295$			21.2132i			1.23508i
$296$		0			0
$297$	−11.0680	−	3.53553i	−0.642229	−	0.205152i
$298$		0			0
$299$	−18.9737			−1.09728
$300$		0			0
$301$	7.00000			0.403473
$302$		0			0
$303$	−6.32456	−	14.1421i	−0.363336	−	0.812444i
$304$		0			0
$305$		−	24.5967i		−	1.40841i
$306$		0			0
$307$			8.48528i			0.484281i	0.970241	+	0.242140i	$0.0778494\pi$
$307$			8.48528i			0.484281i	−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$	6.00000	+	13.4164i	0.341328	+	0.763233i
$310$		0			0
$311$		−	2.23607i		−	0.126796i	−0.997988	−	0.0633979i	$-0.979806\pi$
$311$		−	2.23607i		−	0.126796i	0.997988	−	0.0633979i	$-0.0201937\pi$
$312$		0			0
$313$	−28.0000			−1.58265			−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000			−1.58265			−0.791327	+	0.611393i	$0.790609\pi$
$314$		0			0
$315$	5.00000	+	4.47214i	0.281718	+	0.251976i
$316$		0			0
$317$	−9.48683			−0.532834			−0.266417	−	0.963858i	$-0.585840\pi$
$317$	−9.48683			−0.532834			−0.266417	+	0.963858i	$0.585840\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	30.0000	−	13.4164i	1.67444	−	0.748831i
$322$		0			0
$323$	9.48683	−	2.23607i	0.527862	−	0.124418i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−6.00000	−	13.4164i	−0.331801	−	0.741929i
$328$		0			0
$329$		−	11.1803i		−	0.616392i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−9.48683	−	8.48528i	−0.519875	−	0.464991i
$334$		0			0
$335$	18.9737			1.03664
$336$		0			0
$337$			12.7279i			0.693334i	0.937988	+	0.346667i	$0.112687\pi$
$337$			12.7279i			0.693334i	−0.937988	+	0.346667i	$0.887313\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	9.48683			0.513741
$342$		0			0
$343$	13.0000			0.701934
$344$		0			0
$345$	15.8114	−	7.07107i	0.851257	−	0.380693i
$346$		0			0
$347$		−	2.23607i		−	0.120038i	−0.998197	−	0.0600192i	$-0.980884\pi$
$347$		−	2.23607i		−	0.120038i	0.998197	−	0.0600192i	$-0.0191162\pi$
$348$		0			0
$349$	29.0000			1.55233			0.776167	−	0.630527i	$-0.217161\pi$
$349$	29.0000			1.55233			0.776167	+	0.630527i	$0.217161\pi$
$350$		0			0
$351$	21.0000	+	6.70820i	1.12090	+	0.358057i
$352$		0			0
$353$		−	8.94427i		−	0.476056i	−0.971258	−	0.238028i	$-0.923499\pi$
$353$		−	8.94427i		−	0.476056i	0.971258	−	0.238028i	$-0.0765009\pi$
$354$		0			0
$355$			21.2132i			1.12588i
$356$		0			0
$357$	1.58114	+	3.53553i	0.0836827	+	0.187120i
$358$		0			0
$359$			11.1803i			0.590076i	0.955486	+	0.295038i	$0.0953323\pi$
$359$			11.1803i			0.590076i	−0.955486	+	0.295038i	$0.904668\pi$
$360$		0			0
$361$	−17.0000	+	8.48528i	−0.894737	+	0.446594i
$362$		0			0
$363$	9.48683	−	4.24264i	0.497930	−	0.222681i
$364$		0			0
$365$		−	11.1803i		−	0.585206i
$366$		0			0
$367$	26.0000			1.35719			0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000			1.35719			0.678594	+	0.734513i	$0.262589\pi$
$368$		0			0
$369$	18.9737	−	21.2132i	0.987730	−	1.10432i
$370$		0			0
$371$	9.48683			0.492532
$372$		0			0
$373$			16.9706i			0.878702i	0.898315	+	0.439351i	$0.144792\pi$
$373$			16.9706i			0.878702i	−0.898315	+	0.439351i	$0.855208\pi$
$374$		0			0
$375$	−7.90569	−	17.6777i	−0.408248	−	0.912871i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	33.9411i		−	1.74344i	−0.490006	−	0.871719i	$-0.663005\pi$
$379$		−	33.9411i		−	1.74344i	0.490006	−	0.871719i	$-0.336995\pi$
$380$		0			0
$381$	−12.0000	−	26.8328i	−0.614779	−	1.37469i
$382$		0			0
$383$	−9.48683			−0.484755			−0.242377	−	0.970182i	$-0.577927\pi$
$383$	−9.48683			−0.484755			−0.242377	+	0.970182i	$0.577927\pi$
$384$		0			0
$385$	5.00000			0.254824
$386$		0			0
$387$	−14.0000	+	15.6525i	−0.711660	+	0.795660i
$388$		0			0
$389$		−	15.6525i		−	0.793612i	−0.917902	−	0.396806i	$-0.870119\pi$
$389$		−	15.6525i		−	0.793612i	0.917902	−	0.396806i	$-0.129881\pi$
$390$		0			0
$391$	10.0000			0.505722
$392$		0			0
$393$	−11.0680	−	24.7487i	−0.558305	−	1.24841i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−19.0000			−0.953583			−0.476791	−	0.879017i	$-0.658200\pi$
$397$	−19.0000			−0.953583			−0.476791	+	0.879017i	$0.658200\pi$
$398$		0			0
$399$	−4.58114	−	6.00110i	−0.229344	−	0.300431i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−18.0000			−0.896644
$404$		0			0
$405$	−20.0000	+	2.23607i	−0.993808	+	0.111111i
$406$		0			0
$407$	−9.48683			−0.470245
$408$		0			0
$409$			29.6985i			1.46850i	0.678882	+	0.734248i	$0.262465\pi$
$409$			29.6985i			1.46850i	−0.678882	+	0.734248i	$0.737535\pi$
$410$		0			0
$411$	7.90569	+	17.6777i	0.389959	+	0.871975i
$412$		0			0
$413$	9.48683			0.466817
$414$		0			0
$415$	−20.0000			−0.981761
$416$		0			0
$417$	17.3925	−	7.77817i	0.851716	−	0.380899i
$418$		0			0
$419$			4.47214i			0.218478i	0.994016	+	0.109239i	$0.0348414\pi$
$419$			4.47214i			0.218478i	−0.994016	+	0.109239i	$0.965159\pi$
$420$		0			0
$421$		−	21.2132i		−	1.03387i	−0.856025	−	0.516934i	$-0.827073\pi$
$421$		−	21.2132i		−	1.03387i	0.856025	−	0.516934i	$-0.172927\pi$
$422$		0			0
$423$	25.0000	+	22.3607i	1.21554	+	1.08721i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−11.0000			−0.532327
$428$		0			0
$429$	15.0000	−	6.70820i	0.724207	−	0.323875i
$430$		0			0
$431$	9.48683			0.456965			0.228482	−	0.973548i	$-0.426624\pi$
$431$	9.48683			0.456965			0.228482	+	0.973548i	$0.426624\pi$
$432$		0			0
$433$			25.4558i			1.22333i	0.791117	+	0.611665i	$0.209500\pi$
$433$			25.4558i			1.22333i	−0.791117	+	0.611665i	$0.790500\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−18.9737	+	4.47214i	−0.907634	+	0.213931i
$438$		0			0
$439$		−	8.48528i		−	0.404980i	−0.979284	−	0.202490i	$-0.935097\pi$
$439$		−	8.48528i		−	0.404980i	0.979284	−	0.202490i	$-0.0649034\pi$
$440$		0			0
$441$	−12.0000	+	13.4164i	−0.571429	+	0.638877i
$442$		0			0
$443$			24.5967i			1.16863i	0.811528	+	0.584313i	$0.198636\pi$
$443$			24.5967i			1.16863i	−0.811528	+	0.584313i	$0.801364\pi$
$444$		0			0
$445$			21.2132i			1.00560i
$446$		0			0
$447$	7.90569	+	17.6777i	0.373927	+	0.836125i
$448$		0			0
$449$	−37.9473			−1.79085			−0.895423	−	0.445217i	$-0.853127\pi$
$449$	−37.9473			−1.79085			−0.895423	+	0.445217i	$0.853127\pi$
$450$		0			0
$451$		−	21.2132i		−	0.998891i
$452$		0			0
$453$	3.00000	+	6.70820i	0.140952	+	0.315179i
$454$		0			0
$455$	−9.48683			−0.444750
$456$		0			0
$457$	5.00000			0.233890			0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000			0.233890			0.116945	+	0.993138i	$0.462690\pi$
$458$		0			0
$459$	−11.0680	−	3.53553i	−0.516609	−	0.165025i
$460$		0			0
$461$		−	29.0689i		−	1.35387i	−0.736041	−	0.676936i	$-0.763307\pi$
$461$		−	29.0689i		−	1.35387i	0.736041	−	0.676936i	$-0.236693\pi$
$462$		0			0
$463$	5.00000			0.232370			0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000			0.232370			0.116185	+	0.993228i	$0.462933\pi$
$464$		0			0
$465$	15.0000	−	6.70820i	0.695608	−	0.311086i
$466$		0			0
$467$			38.0132i			1.75904i	0.475863	+	0.879520i	$0.342136\pi$
$467$			38.0132i			1.75904i	−0.475863	+	0.879520i	$0.657864\pi$
$468$		0			0
$469$		−	8.48528i		−	0.391814i
$470$		0			0
$471$	−6.32456	+	2.82843i	−0.291420	+	0.130327i
$472$		0			0
$473$			15.6525i			0.719702i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−18.9737	+	21.2132i	−0.868744	+	0.971286i
$478$		0			0
$479$			31.3050i			1.43036i	0.698940	+	0.715180i	$0.253655\pi$
$479$			31.3050i			1.43036i	−0.698940	+	0.715180i	$0.746345\pi$
$480$		0			0
$481$	18.0000			0.820729
$482$		0			0
$483$	−3.16228	−	7.07107i	−0.143889	−	0.321745i
$484$		0			0
$485$	−28.4605			−1.29232
$486$		0			0
$487$			12.7279i			0.576757i	0.957516	+	0.288379i	$0.0931162\pi$
$487$			12.7279i			0.576757i	−0.957516	+	0.288379i	$0.906884\pi$
$488$		0			0
$489$	3.16228	−	1.41421i	0.143003	−	0.0639529i
$490$		0			0
$491$			4.47214i			0.201825i	0.994895	+	0.100912i	$0.0321762\pi$
$491$			4.47214i			0.201825i	−0.994895	+	0.100912i	$0.967824\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−10.0000	+	11.1803i	−0.449467	+	0.502519i
$496$		0			0
$497$	9.48683			0.425543
$498$		0			0
$499$	−37.0000			−1.65635			−0.828174	−	0.560471i	$-0.810620\pi$
$499$	−37.0000			−1.65635			−0.828174	+	0.560471i	$0.810620\pi$
$500$		0			0
$501$	−30.0000	+	13.4164i	−1.34030	+	0.599401i
$502$		0			0
$503$		−	22.3607i		−	0.997013i	−0.866886	−	0.498507i	$-0.833882\pi$
$503$		−	22.3607i		−	0.997013i	0.866886	−	0.498507i	$-0.166118\pi$
$504$		0			0
$505$	−20.0000			−0.889988
$506$		0			0
$507$	−7.90569	+	3.53553i	−0.351104	+	0.157019i
$508$		0			0
$509$	28.4605			1.26149			0.630745	−	0.775990i	$-0.282750\pi$
$509$	28.4605			1.26149			0.630745	+	0.775990i	$0.282750\pi$
$510$		0			0
$511$	−5.00000			−0.221187
$512$		0			0
$513$	22.5811	+	1.75846i	0.996982	+	0.0776377i
$514$		0			0
$515$	18.9737			0.836080
$516$		0			0
$517$	25.0000			1.09950
$518$		0			0
$519$	−30.0000	+	13.4164i	−1.31685	+	0.588915i
$520$		0			0
$521$	18.9737			0.831251			0.415626	−	0.909536i	$-0.363563\pi$
$521$	18.9737			0.831251			0.415626	+	0.909536i	$0.363563\pi$
$522$		0			0
$523$		−	16.9706i		−	0.742071i	−0.928619	−	0.371035i	$-0.879003\pi$
$523$		−	16.9706i		−	0.742071i	0.928619	−	0.371035i	$-0.120997\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.48683			0.413253
$528$		0			0
$529$	3.00000			0.130435
$530$		0			0
$531$	−18.9737	+	21.2132i	−0.823387	+	0.920575i
$532$		0			0
$533$			40.2492i			1.74339i
$534$		0			0
$535$		−	42.4264i		−	1.83425i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			13.4164i			0.577886i
$540$		0			0
$541$	−1.00000			−0.0429934			−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000			−0.0429934			−0.0214967	+	0.999769i	$0.506843\pi$
$542$		0			0
$543$	−18.0000	−	40.2492i	−0.772454	−	1.72726i
$544$		0			0
$545$	−18.9737			−0.812743
$546$		0			0
$547$		−	4.24264i		−	0.181402i	−0.995878	−	0.0907011i	$-0.971089\pi$
$547$		−	4.24264i		−	0.181402i	0.995878	−	0.0907011i	$-0.0289108\pi$
$548$		0			0
$549$	22.0000	−	24.5967i	0.938937	−	1.04976i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−15.0000	+	6.70820i	−0.636715	+	0.284747i
$556$		0			0
$557$		−	15.6525i		−	0.663217i	−0.943417	−	0.331608i	$-0.892409\pi$
$557$		−	15.6525i		−	0.663217i	0.943417	−	0.331608i	$-0.107591\pi$
$558$		0			0
$559$		−	29.6985i		−	1.25611i
$560$		0			0
$561$	−7.90569	+	3.53553i	−0.333779	+	0.149270i
$562$		0			0
$563$	9.48683			0.399822			0.199911	−	0.979814i	$-0.435935\pi$
$563$	9.48683			0.399822			0.199911	+	0.979814i	$0.435935\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.00000	+	8.94427i	0.0419961	+	0.375624i
$568$		0			0
$569$	28.4605			1.19313			0.596563	−	0.802566i	$-0.296533\pi$
$569$	28.4605			1.19313			0.596563	+	0.802566i	$0.296533\pi$
$570$		0			0
$571$	14.0000			0.585882			0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000			0.585882			0.292941	+	0.956131i	$0.405366\pi$
$572$		0			0
$573$	−1.58114	−	3.53553i	−0.0660530	−	0.147699i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	11.0000			0.457936			0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000			0.457936			0.228968	+	0.973434i	$0.426465\pi$
$578$		0			0
$579$	18.0000	+	40.2492i	0.748054	+	1.67270i
$580$		0			0
$581$			8.94427i			0.371071i
$582$		0			0
$583$			21.2132i			0.878561i
$584$		0			0
$585$	18.9737	−	21.2132i	0.784465	−	0.877058i
$586$		0			0
$587$		−	15.6525i		−	0.646047i	−0.946391	−	0.323023i	$-0.895301\pi$
$587$		−	15.6525i		−	0.646047i	0.946391	−	0.323023i	$-0.104699\pi$
$588$		0			0
$589$	−18.0000	+	4.24264i	−0.741677	+	0.174815i
$590$		0			0
$591$	12.6491	+	28.2843i	0.520315	+	1.16346i
$592$		0			0
$593$		−	22.3607i		−	0.918243i	−0.888373	−	0.459122i	$-0.848164\pi$
$593$		−	22.3607i		−	0.918243i	0.888373	−	0.459122i	$-0.151836\pi$
$594$		0			0
$595$	5.00000			0.204980
$596$		0			0
$597$	−20.5548	+	9.19239i	−0.841252	+	0.376219i
$598$		0			0
$599$	18.9737			0.775243			0.387621	−	0.921819i	$-0.373297\pi$
$599$	18.9737			0.775243			0.387621	+	0.921819i	$0.373297\pi$
$600$		0			0
$601$		−	38.1838i		−	1.55755i	−0.627304	−	0.778774i	$-0.715842\pi$
$601$		−	38.1838i		−	1.55755i	0.627304	−	0.778774i	$-0.284158\pi$
$602$		0			0
$603$	18.9737	+	16.9706i	0.772667	+	0.691095i
$604$		0			0
$605$		−	13.4164i		−	0.545455i
$606$		0			0
$607$		−	4.24264i		−	0.172203i	−0.996286	−	0.0861017i	$-0.972559\pi$
$607$		−	4.24264i		−	0.172203i	0.996286	−	0.0861017i	$-0.0274410\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−47.4342			−1.91898
$612$		0			0
$613$	5.00000			0.201948			0.100974	−	0.994889i	$-0.467804\pi$
$613$	5.00000			0.201948			0.100974	+	0.994889i	$0.467804\pi$
$614$		0			0
$615$	−15.0000	−	33.5410i	−0.604858	−	1.35250i
$616$		0			0
$617$			11.1803i			0.450104i	0.974347	+	0.225052i	$0.0722551\pi$
$617$			11.1803i			0.450104i	−0.974347	+	0.225052i	$0.927745\pi$
$618$		0			0
$619$	8.00000			0.321547			0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000			0.321547			0.160774	+	0.986991i	$0.448601\pi$
$620$		0			0
$621$	22.1359	+	7.07107i	0.888285	+	0.283752i
$622$		0			0
$623$	9.48683			0.380082
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$	13.4189	−	10.2437i	0.535898	−	0.409095i
$628$		0			0
$629$	−9.48683			−0.378265
$630$		0			0
$631$	41.0000			1.63218			0.816092	−	0.577922i	$-0.196136\pi$
$631$	41.0000			1.63218			0.816092	+	0.577922i	$0.196136\pi$
$632$		0			0
$633$	3.00000	+	6.70820i	0.119239	+	0.266627i
$634$		0			0
$635$	−37.9473			−1.50589
$636$		0			0
$637$		−	25.4558i		−	1.00860i
$638$		0			0
$639$	−18.9737	+	21.2132i	−0.750587	+	0.839181i
$640$		0			0
$641$	28.4605			1.12412			0.562061	−	0.827096i	$-0.310009\pi$
$641$	28.4605			1.12412			0.562061	+	0.827096i	$0.310009\pi$
$642$		0			0
$643$	35.0000			1.38027			0.690133	−	0.723683i	$-0.257552\pi$
$643$	35.0000			1.38027			0.690133	+	0.723683i	$0.257552\pi$
$644$		0			0
$645$	11.0680	+	24.7487i	0.435801	+	0.974481i
$646$		0			0
$647$		−	15.6525i		−	0.615362i	−0.951490	−	0.307681i	$-0.900447\pi$
$647$		−	15.6525i		−	0.615362i	0.951490	−	0.307681i	$-0.0995530\pi$
$648$		0			0
$649$			21.2132i			0.832691i
$650$		0			0
$651$	−3.00000	−	6.70820i	−0.117579	−	0.262915i
$652$		0			0
$653$		−	29.0689i		−	1.13755i	−0.822492	−	0.568777i	$-0.807417\pi$
$653$		−	29.0689i		−	1.13755i	0.822492	−	0.568777i	$-0.192583\pi$
$654$		0			0
$655$	−35.0000			−1.36756
$656$		0			0
$657$	10.0000	−	11.1803i	0.390137	−	0.436187i
$658$		0			0
$659$	18.9737			0.739109			0.369555	−	0.929209i	$-0.379510\pi$
$659$	18.9737			0.739109			0.369555	+	0.929209i	$0.379510\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$	15.0000	−	6.70820i	0.582552	−	0.260525i
$664$		0			0
$665$	−9.48683	+	2.23607i	−0.367884	+	0.0867110i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−3.00000	−	6.70820i	−0.115987	−	0.259354i
$670$		0			0
$671$		−	24.5967i		−	0.949547i
$672$		0			0
$673$			12.7279i			0.490625i	0.969444	+	0.245313i	$0.0788906\pi$
$673$			12.7279i			0.490625i	−0.969444	+	0.245313i	$0.921109\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.9737			−0.729217			−0.364609	−	0.931161i	$-0.618797\pi$
$677$	−18.9737			−0.729217			−0.364609	+	0.931161i	$0.618797\pi$
$678$		0			0
$679$			12.7279i			0.488453i
$680$		0			0
$681$	−15.0000	+	6.70820i	−0.574801	+	0.257059i
$682$		0			0
$683$	−28.4605			−1.08901			−0.544505	−	0.838757i	$-0.683283\pi$
$683$	−28.4605			−1.08901			−0.544505	+	0.838757i	$0.683283\pi$
$684$		0			0
$685$	25.0000			0.955201
$686$		0			0
$687$	−20.5548	+	9.19239i	−0.784215	+	0.350711i
$688$		0			0
$689$		−	40.2492i		−	1.53337i
$690$		0			0
$691$	23.0000			0.874961			0.437481	−	0.899228i	$-0.355871\pi$
$691$	23.0000			0.874961			0.437481	+	0.899228i	$0.355871\pi$
$692$		0			0
$693$	5.00000	+	4.47214i	0.189934	+	0.169882i
$694$		0			0
$695$		−	24.5967i		−	0.933008i
$696$		0			0
$697$		−	21.2132i		−	0.803507i
$698$		0			0
$699$	17.3925	+	38.8909i	0.657846	+	1.47099i
$700$		0			0
$701$			17.8885i			0.675641i	0.941211	+	0.337820i	$0.109690\pi$
$701$			17.8885i			0.675641i	−0.941211	+	0.337820i	$0.890310\pi$
$702$		0			0
$703$	18.0000	−	4.24264i	0.678883	−	0.160014i
$704$		0			0
$705$	39.5285	−	17.6777i	1.48873	−	0.665780i
$706$		0			0
$707$			8.94427i			0.336384i
$708$		0			0
$709$	8.00000			0.300446			0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000			0.300446			0.150223	+	0.988652i	$0.452001\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−18.9737			−0.710569
$714$		0			0
$715$		−	21.2132i		−	0.793329i
$716$		0			0
$717$	7.90569	+	17.6777i	0.295244	+	0.660185i
$718$		0			0
$719$		−	29.0689i		−	1.08409i	−0.840351	−	0.542043i	$-0.817651\pi$
$719$		−	29.0689i		−	1.08409i	0.840351	−	0.542043i	$-0.182349\pi$
$720$		0			0
$721$		−	8.48528i		−	0.316008i
$722$		0			0
$723$	−3.00000	−	6.70820i	−0.111571	−	0.249481i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−31.0000			−1.14973			−0.574863	−	0.818250i	$-0.694945\pi$
$727$	−31.0000			−1.14973			−0.574863	+	0.818250i	$0.694945\pi$
$728$		0			0
$729$	−22.0000	−	15.6525i	−0.814815	−	0.579721i
$730$		0			0
$731$			15.6525i			0.578928i
$732$		0			0
$733$	−40.0000			−1.47743			−0.738717	−	0.674016i	$-0.764568\pi$
$733$	−40.0000			−1.47743			−0.738717	+	0.674016i	$0.764568\pi$
$734$		0			0
$735$	9.48683	+	21.2132i	0.349927	+	0.782461i
$736$		0			0
$737$	18.9737			0.698904
$738$		0			0
$739$	−19.0000			−0.698926			−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000			−0.698926			−0.349463	+	0.936950i	$0.613636\pi$
$740$		0			0
$741$	−25.4605	+	19.4361i	−0.935315	+	0.714004i
$742$		0			0
$743$	37.9473			1.39215			0.696076	−	0.717968i	$-0.254928\pi$
$743$	37.9473			1.39215			0.696076	+	0.717968i	$0.254928\pi$
$744$		0			0
$745$	25.0000			0.915929
$746$		0			0
$747$	−20.0000	−	17.8885i	−0.731762	−	0.654508i
$748$		0			0
$749$	−18.9737			−0.693283
$750$		0			0
$751$		−	42.4264i		−	1.54816i	−0.633087	−	0.774081i	$-0.718212\pi$
$751$		−	42.4264i		−	1.54816i	0.633087	−	0.774081i	$-0.281788\pi$
$752$		0			0
$753$	−20.5548	−	45.9619i	−0.749059	−	1.67495i
$754$		0			0
$755$	9.48683			0.345261
$756$		0			0
$757$	−1.00000			−0.0363456			−0.0181728	−	0.999835i	$-0.505785\pi$
$757$	−1.00000			−0.0363456			−0.0181728	+	0.999835i	$0.505785\pi$
$758$		0			0
$759$	15.8114	−	7.07107i	0.573917	−	0.256664i
$760$		0			0
$761$			38.0132i			1.37798i	0.724773	+	0.688988i	$0.241945\pi$
$761$			38.0132i			1.37798i	−0.724773	+	0.688988i	$0.758055\pi$
$762$		0			0
$763$			8.48528i			0.307188i
$764$		0			0
$765$	−10.0000	+	11.1803i	−0.361551	+	0.404226i
$766$		0			0
$767$		−	40.2492i		−	1.45332i
$768$		0			0
$769$	23.0000			0.829401			0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000			0.829401			0.414701	+	0.909958i	$0.363886\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	28.4605			1.02365			0.511826	−	0.859089i	$-0.328969\pi$
$773$	28.4605			1.02365			0.511826	+	0.859089i	$0.328969\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	3.00000	+	6.70820i	0.107624	+	0.240655i
$778$		0			0
$779$	9.48683	+	40.2492i	0.339901	+	1.44208i
$780$		0			0
$781$			21.2132i			0.759068i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			8.94427i			0.319235i
$786$		0			0
$787$		−	33.9411i		−	1.20987i	−0.796275	−	0.604935i	$-0.793199\pi$
$787$		−	33.9411i		−	1.20987i	0.796275	−	0.604935i	$-0.206801\pi$
$788$		0			0
$789$	−11.0680	−	24.7487i	−0.394030	−	0.881078i
$790$		0			0
$791$		0			0
$792$		0			0
$793$			46.6690i			1.65727i
$794$		0			0
$795$	15.0000	+	33.5410i	0.531995	+	1.18958i
$796$		0			0
$797$	18.9737			0.672082			0.336041	−	0.941847i	$-0.390912\pi$
$797$	18.9737			0.672082			0.336041	+	0.941847i	$0.390912\pi$
$798$		0			0
$799$	25.0000			0.884436
$800$		0			0
$801$	−18.9737	+	21.2132i	−0.670402	+	0.749532i
$802$		0			0
$803$		−	11.1803i		−	0.394546i
$804$		0			0
$805$	−10.0000			−0.352454
$806$		0			0
$807$	45.0000	−	20.1246i	1.58408	−	0.708420i
$808$		0			0
$809$			11.1803i			0.393080i	0.980496	+	0.196540i	$0.0629705\pi$
$809$			11.1803i			0.393080i	−0.980496	+	0.196540i	$0.937029\pi$
$810$		0			0
$811$			4.24264i			0.148979i	0.997222	+	0.0744896i	$0.0237328\pi$
$811$			4.24264i			0.148979i	−0.997222	+	0.0744896i	$0.976267\pi$
$812$		0			0
$813$	3.16228	−	1.41421i	0.110906	−	0.0495986i
$814$		0			0
$815$		−	4.47214i		−	0.156652i
$816$		0			0
$817$	−7.00000	−	29.6985i	−0.244899	−	1.03902i
$818$		0			0
$819$	−9.48683	−	8.48528i	−0.331497	−	0.296500i
$820$		0			0
$821$			38.0132i			1.32667i	0.748323	+	0.663334i	$0.230859\pi$
$821$			38.0132i			1.32667i	−0.748323	+	0.663334i	$0.769141\pi$
$822$		0			0
$823$	47.0000			1.63832			0.819159	−	0.573567i	$-0.194441\pi$
$823$	47.0000			1.63832			0.819159	+	0.573567i	$0.194441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18.9737			0.659779			0.329890	−	0.944020i	$-0.392989\pi$
$827$	18.9737			0.659779			0.329890	+	0.944020i	$0.392989\pi$
$828$		0			0
$829$		−	42.4264i		−	1.47353i	−0.676149	−	0.736765i	$-0.736352\pi$
$829$		−	42.4264i		−	1.47353i	0.676149	−	0.736765i	$-0.263648\pi$
$830$		0			0
$831$	−1.58114	+	0.707107i	−0.0548491	+	0.0245293i
$832$		0			0
$833$			13.4164i			0.464851i
$834$		0			0
$835$			42.4264i			1.46823i
$836$		0			0
$837$	21.0000	+	6.70820i	0.725866	+	0.231869i
$838$		0			0
$839$	−56.9210			−1.96513			−0.982566	−	0.185917i	$-0.940475\pi$
$839$	−56.9210			−1.96513			−0.982566	+	0.185917i	$0.940475\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	−30.0000	+	13.4164i	−1.03325	+	0.462086i
$844$		0			0
$845$			11.1803i			0.384615i
$846$		0			0
$847$	−6.00000			−0.206162
$848$		0			0
$849$	−20.5548	+	9.19239i	−0.705439	+	0.315482i
$850$		0			0
$851$	18.9737			0.650409
$852$		0			0
$853$	−28.0000			−0.958702			−0.479351	−	0.877623i	$-0.659128\pi$
$853$	−28.0000			−0.958702			−0.479351	+	0.877623i	$0.659128\pi$
$854$		0			0
$855$	13.9737	−	25.6853i	0.477889	−	0.878420i
$856$		0			0
$857$	47.4342			1.62032			0.810160	−	0.586209i	$-0.199380\pi$
$857$	47.4342			1.62032			0.810160	+	0.586209i	$0.199380\pi$
$858$		0			0
$859$	−1.00000			−0.0341196			−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000			−0.0341196			−0.0170598	+	0.999854i	$0.505431\pi$
$860$		0			0
$861$	−15.0000	+	6.70820i	−0.511199	+	0.228615i
$862$		0			0
$863$	−28.4605			−0.968807			−0.484403	−	0.874845i	$-0.660963\pi$
$863$	−28.4605			−0.968807			−0.484403	+	0.874845i	$0.660963\pi$
$864$		0			0
$865$			42.4264i			1.44254i
$866$		0			0
$867$	18.9737	−	8.48528i	0.644379	−	0.288175i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−36.0000			−1.21981
$872$		0			0
$873$	−28.4605	−	25.4558i	−0.963242	−	0.861550i
$874$		0			0
$875$			11.1803i			0.377964i
$876$		0			0
$877$			8.48528i			0.286528i	0.989685	+	0.143264i	$0.0457597\pi$
$877$			8.48528i			0.286528i	−0.989685	+	0.143264i	$0.954240\pi$
$878$		0			0
$879$	30.0000	−	13.4164i	1.01187	−	0.452524i
$880$		0			0
$881$			38.0132i			1.28070i	0.768085	+	0.640348i	$0.221210\pi$
$881$			38.0132i			1.28070i	−0.768085	+	0.640348i	$0.778790\pi$
$882$		0			0
$883$	−43.0000			−1.44707			−0.723533	−	0.690290i	$-0.757483\pi$
$883$	−43.0000			−1.44707			−0.723533	+	0.690290i	$0.757483\pi$
$884$		0			0
$885$	15.0000	+	33.5410i	0.504219	+	1.12747i
$886$		0			0
$887$	28.4605			0.955610			0.477805	−	0.878466i	$-0.341433\pi$
$887$	28.4605			0.955610			0.477805	+	0.878466i	$0.341433\pi$
$888$		0			0
$889$			16.9706i			0.569174i
$890$		0			0
$891$	−20.0000	+	2.23607i	−0.670025	+	0.0749111i
$892$		0			0
$893$	−47.4342	+	11.1803i	−1.58732	+	0.374136i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−30.0000	+	13.4164i	−1.00167	+	0.447961i
$898$		0			0
$899$		0			0
$900$		0			0
$901$			21.2132i			0.706714i
$902$		0			0
$903$	11.0680	−	4.94975i	0.368319	−	0.164717i
$904$		0			0
$905$	−56.9210			−1.89212
$906$		0			0
$907$			4.24264i			0.140875i	0.997516	+	0.0704373i	$0.0224395\pi$
$907$			4.24264i			0.140875i	−0.997516	+	0.0704373i	$0.977561\pi$
$908$		0			0
$909$	−20.0000	−	17.8885i	−0.663358	−	0.593326i
$910$		0			0
$911$	−18.9737			−0.628626			−0.314313	−	0.949319i	$-0.601774\pi$
$911$	−18.9737			−0.628626			−0.314313	+	0.949319i	$0.601774\pi$
$912$		0			0
$913$	−20.0000			−0.661903
$914$		0			0
$915$	−17.3925	−	38.8909i	−0.574979	−	1.28569i
$916$		0			0
$917$			15.6525i			0.516890i
$918$		0			0
$919$	2.00000			0.0659739			0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000			0.0659739			0.0329870	+	0.999456i	$0.489498\pi$
$920$		0			0
$921$	6.00000	+	13.4164i	0.197707	+	0.442086i
$922$		0			0
$923$		−	40.2492i		−	1.32482i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	18.9737	+	16.9706i	0.623177	+	0.557386i
$928$		0			0
$929$		−	8.94427i		−	0.293452i	−0.989177	−	0.146726i	$-0.953126\pi$
$929$		−	8.94427i		−	0.293452i	0.989177	−	0.146726i	$-0.0468736\pi$
$930$		0			0
$931$	−6.00000	−	25.4558i	−0.196642	−	0.834282i
$932$		0			0
$933$	−1.58114	−	3.53553i	−0.0517642	−	0.115748i
$934$		0			0
$935$			11.1803i			0.365636i
$936$		0			0
$937$	−19.0000			−0.620703			−0.310351	−	0.950622i	$-0.600447\pi$
$937$	−19.0000			−0.620703			−0.310351	+	0.950622i	$0.600447\pi$
$938$		0			0
$939$	−44.2719	+	19.7990i	−1.44476	+	0.646116i
$940$		0			0
$941$	−37.9473			−1.23705			−0.618524	−	0.785766i	$-0.712269\pi$
$941$	−37.9473			−1.23705			−0.618524	+	0.785766i	$0.712269\pi$
$942$		0			0
$943$			42.4264i			1.38159i
$944$		0			0
$945$	11.0680	+	3.53553i	0.360041	+	0.115011i
$946$		0			0
$947$			31.3050i			1.01727i	0.860981	+	0.508637i	$0.169851\pi$
$947$			31.3050i			1.01727i	−0.860981	+	0.508637i	$0.830149\pi$
$948$		0			0
$949$			21.2132i			0.688610i
$950$		0			0
$951$	−15.0000	+	6.70820i	−0.486408	+	0.217528i
$952$		0			0
$953$	−18.9737			−0.614617			−0.307309	−	0.951610i	$-0.599428\pi$
$953$	−18.9737			−0.614617			−0.307309	+	0.951610i	$0.599428\pi$
$954$		0			0
$955$	−5.00000			−0.161796
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	11.1803i		−	0.361032i
$960$		0			0
$961$	13.0000			0.419355
$962$		0			0
$963$	37.9473	−	42.4264i	1.22284	−	1.36717i
$964$		0			0
$965$	56.9210			1.83235
$966$		0			0
$967$	50.0000			1.60789			0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000			1.60789			0.803946	+	0.594703i	$0.202730\pi$
$968$		0			0
$969$	13.4189	−	10.2437i	0.431076	−	0.329076i
$970$		0			0
$971$	37.9473			1.21779			0.608894	−	0.793252i	$-0.291614\pi$
$971$	37.9473			1.21779			0.608894	+	0.793252i	$0.291614\pi$
$972$		0			0
$973$	−11.0000			−0.352644
$974$		0			0
$975$		0			0
$976$		0			0
$977$	9.48683			0.303511			0.151755	−	0.988418i	$-0.451507\pi$
$977$	9.48683			0.303511			0.151755	+	0.988418i	$0.451507\pi$
$978$		0			0
$979$			21.2132i			0.677977i
$980$		0			0
$981$	−18.9737	−	16.9706i	−0.605783	−	0.541828i
$982$		0			0
$983$	18.9737			0.605166			0.302583	−	0.953123i	$-0.402151\pi$
$983$	18.9737			0.605166			0.302583	+	0.953123i	$0.402151\pi$
$984$		0			0
$985$	40.0000			1.27451
$986$		0			0
$987$	−7.90569	−	17.6777i	−0.251641	−	0.562686i
$988$		0			0
$989$		−	31.3050i		−	0.995440i
$990$		0			0
$991$		−	4.24264i		−	0.134772i	−0.997727	−	0.0673860i	$-0.978534\pi$
$991$		−	4.24264i		−	0.134772i	0.997727	−	0.0673860i	$-0.0214659\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			29.0689i			0.921546i
$996$		0			0
$997$	5.00000			0.158352			0.0791758	−	0.996861i	$-0.474771\pi$
$997$	5.00000			0.158352			0.0791758	+	0.996861i	$0.474771\pi$
$998$		0			0
$999$	−21.0000	−	6.70820i	−0.664411	−	0.212238i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	228.2.d.b.113.3	yes	4
3.2	odd	2	inner	228.2.d.b.113.1	✓	4
4.3	odd	2		912.2.f.g.113.2		4
12.11	even	2		912.2.f.g.113.4		4
19.18	odd	2	inner	228.2.d.b.113.2	yes	4
57.56	even	2	inner	228.2.d.b.113.4	yes	4
76.75	even	2		912.2.f.g.113.3		4
228.227	odd	2		912.2.f.g.113.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.d.b.113.1	✓	4	3.2	odd	2	inner
228.2.d.b.113.2	yes	4	19.18	odd	2	inner
228.2.d.b.113.3	yes	4	1.1	even	1	trivial
228.2.d.b.113.4	yes	4	57.56	even	2	inner
912.2.f.g.113.1		4	228.227	odd	2
912.2.f.g.113.2		4	4.3	odd	2
912.2.f.g.113.3		4	76.75	even	2
912.2.f.g.113.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	228.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.58114 - 0.707107i) q^{3} -2.23607i q^{5} -1.00000 q^{7} +(2.00000 - 2.23607i) q^{9} -2.23607i q^{11} +4.24264i q^{13} +(-1.58114 - 3.53553i) q^{15} -2.23607i q^{17} +(1.00000 + 4.24264i) q^{19} +(-1.58114 + 0.707107i) q^{21} +4.47214i q^{23} +(1.58114 - 4.94975i) q^{27} +4.24264i q^{31} +(-1.58114 - 3.53553i) q^{33} +2.23607i q^{35} -4.24264i q^{37} +(3.00000 + 6.70820i) q^{39} +9.48683 q^{41} -7.00000 q^{43} +(-5.00000 - 4.47214i) q^{45} +11.1803i q^{47} -6.00000 q^{49} +(-1.58114 - 3.53553i) q^{51} -9.48683 q^{53} -5.00000 q^{55} +(4.58114 + 6.00110i) q^{57} -9.48683 q^{59} +11.0000 q^{61} +(-2.00000 + 2.23607i) q^{63} +9.48683 q^{65} +8.48528i q^{67} +(3.16228 + 7.07107i) q^{69} -9.48683 q^{71} +5.00000 q^{73} +2.23607i q^{77} +(-1.00000 - 8.94427i) q^{81} -8.94427i q^{83} -5.00000 q^{85} -9.48683 q^{89} -4.24264i q^{91} +(3.00000 + 6.70820i) q^{93} +(9.48683 - 2.23607i) q^{95} -12.7279i q^{97} +(-5.00000 - 4.47214i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} + 8 q^{9} + 4 q^{19} + 12 q^{39} - 28 q^{43} - 20 q^{45} - 24 q^{49} - 20 q^{55} + 12 q^{57} + 44 q^{61} - 8 q^{63} + 20 q^{73} - 4 q^{81} - 20 q^{85} + 12 q^{93} - 20 q^{99}+O(q^{100}) \) 4 * q - 4 * q^7 + 8 * q^9 + 4 * q^19 + 12 * q^39 - 28 * q^43 - 20 * q^45 - 24 * q^49 - 20 * q^55 + 12 * q^57 + 44 * q^61 - 8 * q^63 + 20 * q^73 - 4 * q^81 - 20 * q^85 + 12 * q^93 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more