LMFDB - Embedded newform 112.3.c.c.97.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,3,Mod(97,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("112.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	112.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$3.05177896084$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.4
Root			$1.84776i$ of defining polynomial
Character	$\chi$	$=$	112.97
Dual form			112.3.c.c.97.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.69552i q^{3} -0.634051i q^{5} +(1.82843 + 6.75699i) q^{7} -4.65685 q^{9} +O(q^{10})$	$q+3.69552i q^{3} -0.634051i q^{5} +(1.82843 + 6.75699i) q^{7} -4.65685 q^{9} -13.3137 q^{11} +21.5391i q^{13} +2.34315 q^{15} -6.12293i q^{17} -19.7457i q^{19} +(-24.9706 + 6.75699i) q^{21} +28.6274 q^{23} +24.5980 q^{25} +16.0502i q^{27} +37.3137 q^{29} -20.9050i q^{31} -49.2011i q^{33} +(4.28427 - 1.15932i) q^{35} -23.9411 q^{37} -79.5980 q^{39} -70.1061i q^{41} +46.0000 q^{43} +2.95268i q^{45} -1.05053i q^{47} +(-42.3137 + 24.7093i) q^{49} +22.6274 q^{51} -15.3726 q^{53} +8.44157i q^{55} +72.9706 q^{57} -14.8909i q^{59} +86.7903i q^{61} +(-8.51472 - 31.4663i) q^{63} +13.6569 q^{65} -24.6274 q^{67} +105.793i q^{69} +45.0294 q^{71} +51.7373i q^{73} +90.9023i q^{75} +(-24.3431 - 89.9605i) q^{77} +33.7157 q^{79} -101.225 q^{81} -77.3883i q^{83} -3.88225 q^{85} +137.893i q^{87} +78.7652i q^{89} +(-145.539 + 39.3826i) q^{91} +77.2548 q^{93} -12.5198 q^{95} -90.1781i q^{97} +62.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{7} + 4 q^{9} - 8 q^{11} + 32 q^{15} - 32 q^{21} + 24 q^{23} - 60 q^{25} + 104 q^{29} - 96 q^{35} + 40 q^{37} - 160 q^{39} + 184 q^{43} - 124 q^{49} - 152 q^{53} + 224 q^{57} - 68 q^{63} + 32 q^{65} - 8 q^{67} + 248 q^{71} - 120 q^{77} + 248 q^{79} - 156 q^{81} + 256 q^{85} - 288 q^{91} + 128 q^{93} - 480 q^{95} + 248 q^{99}+O(q^{100}) $ 4 * q - 4 * q^7 + 4 * q^9 - 8 * q^11 + 32 * q^15 - 32 * q^21 + 24 * q^23 - 60 * q^25 + 104 * q^29 - 96 * q^35 + 40 * q^37 - 160 * q^39 + 184 * q^43 - 124 * q^49 - 152 * q^53 + 224 * q^57 - 68 * q^63 + 32 * q^65 - 8 * q^67 + 248 * q^71 - 120 * q^77 + 248 * q^79 - 156 * q^81 + 256 * q^85 - 288 * q^91 + 128 * q^93 - 480 * q^95 + 248 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			3.69552i			1.23184i	0.787809	+	0.615920i	$0.211215\pi$
$3$			3.69552i			1.23184i	−0.787809	+	0.615920i	$0.788785\pi$
$4$		0			0
$5$		−	0.634051i		−	0.126810i	−0.997988	−	0.0634051i	$-0.979804\pi$
$5$		−	0.634051i		−	0.126810i	0.997988	−	0.0634051i	$-0.0201960\pi$
$6$		0			0
$7$	1.82843	+	6.75699i	0.261204	+	0.965284i
$7$	1.82843	+	6.75699i	0.261204	+	0.965284i
$8$		0			0
$9$	−4.65685			−0.517428
$10$		0			0
$11$	−13.3137			−1.21034			−0.605169	−	0.796097i	$-0.706894\pi$
$11$	−13.3137			−1.21034			−0.605169	+	0.796097i	$0.706894\pi$
$12$		0			0
$13$			21.5391i			1.65685i	0.560100	+	0.828425i	$0.310763\pi$
$13$			21.5391i			1.65685i	−0.560100	+	0.828425i	$0.689237\pi$
$14$		0			0
$15$	2.34315			0.156210
$16$		0			0
$17$		−	6.12293i		−	0.360173i	−0.983651	−	0.180086i	$-0.942362\pi$
$17$		−	6.12293i		−	0.360173i	0.983651	−	0.180086i	$-0.0576377\pi$
$18$		0			0
$19$		−	19.7457i		−	1.03925i	−0.854395	−	0.519623i	$-0.826072\pi$
$19$		−	19.7457i		−	1.03925i	0.854395	−	0.519623i	$-0.173928\pi$
$20$		0			0
$21$	−24.9706	+	6.75699i	−1.18907	+	0.321761i
$22$		0			0
$23$	28.6274			1.24467			0.622335	−	0.782751i	$-0.286184\pi$
$23$	28.6274			1.24467			0.622335	+	0.782751i	$0.286184\pi$
$24$		0			0
$25$	24.5980			0.983919
$26$		0			0
$27$			16.0502i			0.594451i
$28$		0			0
$29$	37.3137			1.28668			0.643340	−	0.765581i	$-0.277548\pi$
$29$	37.3137			1.28668			0.643340	+	0.765581i	$0.277548\pi$
$30$		0			0
$31$		−	20.9050i		−	0.674355i	−0.941441	−	0.337178i	$-0.890528\pi$
$31$		−	20.9050i		−	0.674355i	0.941441	−	0.337178i	$-0.109472\pi$
$32$		0			0
$33$		−	49.2011i		−	1.49094i
$34$		0			0
$35$	4.28427	−	1.15932i	0.122408	−	0.0331233i
$36$		0			0
$37$	−23.9411			−0.647057			−0.323529	−	0.946218i	$-0.604869\pi$
$37$	−23.9411			−0.647057			−0.323529	+	0.946218i	$0.604869\pi$
$38$		0			0
$39$	−79.5980			−2.04097
$40$		0			0
$41$		−	70.1061i		−	1.70990i	−0.518707	−	0.854952i	$-0.673587\pi$
$41$		−	70.1061i		−	1.70990i	0.518707	−	0.854952i	$-0.326413\pi$
$42$		0			0
$43$	46.0000			1.06977			0.534884	−	0.844926i	$-0.320355\pi$
$43$	46.0000			1.06977			0.534884	+	0.844926i	$0.320355\pi$
$44$		0			0
$45$			2.95268i			0.0656151i
$46$		0			0
$47$		−	1.05053i		−	0.0223517i	−0.999938	−	0.0111758i	$-0.996443\pi$
$47$		−	1.05053i		−	0.0223517i	0.999938	−	0.0111758i	$-0.00355746\pi$
$48$		0			0
$49$	−42.3137	+	24.7093i	−0.863545	+	0.504272i
$50$		0			0
$51$	22.6274			0.443675
$52$		0			0
$53$	−15.3726			−0.290049			−0.145024	−	0.989428i	$-0.546326\pi$
$53$	−15.3726			−0.290049			−0.145024	+	0.989428i	$0.546326\pi$
$54$		0			0
$55$			8.44157i			0.153483i
$56$		0			0
$57$	72.9706			1.28019
$58$		0			0
$59$		−	14.8909i		−	0.252387i	−0.992006	−	0.126194i	$-0.959724\pi$
$59$		−	14.8909i		−	0.252387i	0.992006	−	0.126194i	$-0.0402761\pi$
$60$		0			0
$61$			86.7903i			1.42279i	0.702792	+	0.711396i	$0.251937\pi$
$61$			86.7903i			1.42279i	−0.702792	+	0.711396i	$0.748063\pi$
$62$		0			0
$63$	−8.51472	−	31.4663i	−0.135154	−	0.499465i
$64$		0			0
$65$	13.6569			0.210105
$66$		0			0
$67$	−24.6274			−0.367573			−0.183787	−	0.982966i	$-0.558836\pi$
$67$	−24.6274			−0.367573			−0.183787	+	0.982966i	$0.558836\pi$
$68$		0			0
$69$			105.793i			1.53323i
$70$		0			0
$71$	45.0294			0.634217			0.317109	−	0.948389i	$-0.397288\pi$
$71$	45.0294			0.634217			0.317109	+	0.948389i	$0.397288\pi$
$72$		0			0
$73$			51.7373i			0.708730i	0.935107	+	0.354365i	$0.115303\pi$
$73$			51.7373i			0.708730i	−0.935107	+	0.354365i	$0.884697\pi$
$74$		0			0
$75$			90.9023i			1.21203i
$76$		0			0
$77$	−24.3431	−	89.9605i	−0.316145	−	1.16832i
$78$		0			0
$79$	33.7157			0.426781			0.213391	−	0.976967i	$-0.431549\pi$
$79$	33.7157			0.426781			0.213391	+	0.976967i	$0.431549\pi$
$80$		0			0
$81$	−101.225			−1.24970
$82$		0			0
$83$		−	77.3883i		−	0.932389i	−0.884682	−	0.466195i	$-0.845625\pi$
$83$		−	77.3883i		−	0.932389i	0.884682	−	0.466195i	$-0.154375\pi$
$84$		0			0
$85$	−3.88225			−0.0456735
$86$		0			0
$87$			137.893i			1.58498i
$88$		0			0
$89$			78.7652i			0.885002i	0.896768	+	0.442501i	$0.145909\pi$
$89$			78.7652i			0.885002i	−0.896768	+	0.442501i	$0.854091\pi$
$90$		0			0
$91$	−145.539	+	39.3826i	−1.59933	+	0.432776i
$92$		0			0
$93$	77.2548			0.830697
$94$		0			0
$95$	−12.5198			−0.131787
$96$		0			0
$97$		−	90.1781i		−	0.929671i	−0.885397	−	0.464836i	$-0.846113\pi$
$97$		−	90.1781i		−	0.929671i	0.885397	−	0.464836i	$-0.153887\pi$
$98$		0			0
$99$	62.0000			0.626263
$100$		0			0
$101$		−	129.651i		−	1.28367i	−0.766842	−	0.641836i	$-0.778173\pi$
$101$		−	129.651i		−	1.28367i	0.766842	−	0.641836i	$-0.221827\pi$
$102$		0			0
$103$			34.4190i			0.334165i	0.985943	+	0.167082i	$0.0534346\pi$
$103$			34.4190i			0.334165i	−0.985943	+	0.167082i	$0.946565\pi$
$104$		0			0
$105$	4.28427	+	15.8326i	0.0408026	+	0.150787i
$106$		0			0
$107$	−149.882			−1.40077			−0.700384	−	0.713766i	$-0.746988\pi$
$107$	−149.882			−1.40077			−0.700384	+	0.713766i	$0.746988\pi$
$108$		0			0
$109$	−67.2548			−0.617017			−0.308508	−	0.951222i	$-0.599830\pi$
$109$	−67.2548			−0.617017			−0.308508	+	0.951222i	$0.599830\pi$
$110$		0			0
$111$		−	88.4749i		−	0.797071i
$112$		0			0
$113$	70.2843			0.621985			0.310992	−	0.950412i	$-0.399339\pi$
$113$	70.2843			0.621985			0.310992	+	0.950412i	$0.399339\pi$
$114$		0			0
$115$		−	18.1512i		−	0.157837i
$116$		0			0
$117$		−	100.304i		−	0.857301i
$118$		0			0
$119$	41.3726	−	11.1953i	0.347669	−	0.0940785i
$120$		0			0
$121$	56.2548			0.464916
$122$		0			0
$123$	259.078			2.10633
$124$		0			0
$125$		−	31.4476i		−	0.251581i
$126$		0			0
$127$	93.7645			0.738303			0.369152	−	0.929369i	$-0.379648\pi$
$127$	93.7645			0.738303			0.369152	+	0.929369i	$0.379648\pi$
$128$		0			0
$129$			169.994i			1.31778i
$130$		0			0
$131$		−	146.226i		−	1.11623i	−0.829763	−	0.558116i	$-0.811525\pi$
$131$		−	146.226i		−	1.11623i	0.829763	−	0.558116i	$-0.188475\pi$
$132$		0			0
$133$	133.421	−	36.1036i	1.00317	−	0.271455i
$134$		0			0
$135$	10.1766			0.0753824
$136$		0			0
$137$	−149.765			−1.09317			−0.546586	−	0.837403i	$-0.684073\pi$
$137$	−149.765			−1.09317			−0.546586	+	0.837403i	$0.684073\pi$
$138$		0			0
$139$			120.249i			0.865100i	0.901610	+	0.432550i	$0.142386\pi$
$139$			120.249i			0.865100i	−0.901610	+	0.432550i	$0.857614\pi$
$140$		0			0
$141$	3.88225			0.0275337
$142$		0			0
$143$		−	286.765i		−	2.00535i
$144$		0			0
$145$		−	23.6588i		−	0.163164i
$146$		0			0
$147$	−91.3137	−	156.371i	−0.621182	−	1.06375i
$148$		0			0
$149$	33.7645			0.226607			0.113304	−	0.993560i	$-0.463857\pi$
$149$	33.7645			0.226607			0.113304	+	0.993560i	$0.463857\pi$
$150$		0			0
$151$	−144.627			−0.957797			−0.478899	−	0.877870i	$-0.658964\pi$
$151$	−144.627			−0.957797			−0.478899	+	0.877870i	$0.658964\pi$
$152$		0			0
$153$			28.5136i			0.186363i
$154$		0			0
$155$	−13.2548			−0.0855151
$156$		0			0
$157$		−	128.818i		−	0.820496i	−0.911974	−	0.410248i	$-0.865442\pi$
$157$		−	128.818i		−	0.820496i	0.911974	−	0.410248i	$-0.134558\pi$
$158$		0			0
$159$		−	56.8097i		−	0.357293i
$160$		0			0
$161$	52.3431	+	193.435i	0.325113	+	1.20146i
$162$		0			0
$163$	−57.4315			−0.352340			−0.176170	−	0.984360i	$-0.556371\pi$
$163$	−57.4315			−0.352340			−0.176170	+	0.984360i	$0.556371\pi$
$164$		0			0
$165$	−31.1960			−0.189066
$166$		0			0
$167$			229.122i			1.37199i	0.727607	+	0.685994i	$0.240633\pi$
$167$			229.122i			1.37199i	−0.727607	+	0.685994i	$0.759367\pi$
$168$		0			0
$169$	−294.931			−1.74515
$170$		0			0
$171$			91.9528i			0.537736i
$172$		0			0
$173$		−	123.310i		−	0.712776i	−0.934338	−	0.356388i	$-0.884008\pi$
$173$		−	123.310i		−	0.712776i	0.934338	−	0.356388i	$-0.115992\pi$
$174$		0			0
$175$	44.9756	+	166.208i	0.257004	+	0.949761i
$176$		0			0
$177$	55.0294			0.310901
$178$		0			0
$179$	271.941			1.51922			0.759612	−	0.650376i	$-0.225389\pi$
$179$	271.941			1.51922			0.759612	+	0.650376i	$0.225389\pi$
$180$		0			0
$181$		−	30.6333i		−	0.169245i	−0.996413	−	0.0846225i	$-0.973032\pi$
$181$		−	30.6333i		−	0.169245i	0.996413	−	0.0846225i	$-0.0269684\pi$
$182$		0			0
$183$	−320.735			−1.75265
$184$		0			0
$185$			15.1799i			0.0820534i
$186$		0			0
$187$			81.5190i			0.435930i
$188$		0			0
$189$	−108.451	+	29.3466i	−0.573814	+	0.155273i
$190$		0			0
$191$	69.5980			0.364387			0.182194	−	0.983263i	$-0.441680\pi$
$191$	69.5980			0.364387			0.182194	+	0.983263i	$0.441680\pi$
$192$		0			0
$193$	279.421			1.44778			0.723890	−	0.689916i	$-0.242353\pi$
$193$	279.421			1.44778			0.723890	+	0.689916i	$0.242353\pi$
$194$		0			0
$195$			50.4692i			0.258816i
$196$		0			0
$197$	102.451			0.520055			0.260027	−	0.965601i	$-0.416268\pi$
$197$	102.451			0.520055			0.260027	+	0.965601i	$0.416268\pi$
$198$		0			0
$199$			161.950i			0.813820i	0.913468	+	0.406910i	$0.133394\pi$
$199$			161.950i			0.813820i	−0.913468	+	0.406910i	$0.866606\pi$
$200$		0			0
$201$		−	91.0111i		−	0.452791i
$202$		0			0
$203$	68.2254	+	252.128i	0.336086	+	1.24201i
$204$		0			0
$205$	−44.4508			−0.216833
$206$		0			0
$207$	−133.314			−0.644028
$208$		0			0
$209$			262.888i			1.25784i
$210$		0			0
$211$	51.0193			0.241798			0.120899	−	0.992665i	$-0.461422\pi$
$211$	51.0193			0.241798			0.120899	+	0.992665i	$0.461422\pi$
$212$		0			0
$213$			166.407i			0.781254i
$214$		0			0
$215$		−	29.1663i		−	0.135657i
$216$		0			0
$217$	141.255	−	38.2233i	0.650944	−	0.176144i
$218$		0			0
$219$	−191.196			−0.873041
$220$		0			0
$221$	131.882			0.596752
$222$		0			0
$223$		−	221.296i		−	0.992358i	−0.868220	−	0.496179i	$-0.834736\pi$
$223$		−	221.296i		−	0.992358i	0.868220	−	0.496179i	$-0.165264\pi$
$224$		0			0
$225$	−114.549			−0.509108
$226$		0			0
$227$			101.047i			0.445141i	0.974917	+	0.222571i	$0.0714448\pi$
$227$			101.047i			0.445141i	−0.974917	+	0.222571i	$0.928555\pi$
$228$		0			0
$229$			170.193i			0.743200i	0.928393	+	0.371600i	$0.121191\pi$
$229$			170.193i			0.743200i	−0.928393	+	0.371600i	$0.878809\pi$
$230$		0			0
$231$	332.451	−	89.9605i	1.43918	−	0.389440i
$232$		0			0
$233$	−451.019			−1.93571			−0.967853	−	0.251518i	$-0.919070\pi$
$233$	−451.019			−1.93571			−0.967853	+	0.251518i	$0.919070\pi$
$234$		0			0
$235$	−0.666089			−0.00283442
$236$		0			0
$237$			124.597i			0.525726i
$238$		0			0
$239$	−219.137			−0.916892			−0.458446	−	0.888722i	$-0.651594\pi$
$239$	−219.137			−0.916892			−0.458446	+	0.888722i	$0.651594\pi$
$240$		0			0
$241$		−	325.386i		−	1.35015i	−0.737750	−	0.675074i	$-0.764112\pi$
$241$		−	325.386i		−	1.35015i	0.737750	−	0.675074i	$-0.235888\pi$
$242$		0			0
$243$		−	229.629i		−	0.944974i
$244$		0			0
$245$	15.6670	+	26.8290i	0.0639468	+	0.109506i
$246$		0			0
$247$	425.304			1.72188
$248$		0			0
$249$	285.990			1.14855
$250$		0			0
$251$		−	260.063i		−	1.03611i	−0.855348	−	0.518054i	$-0.826657\pi$
$251$		−	260.063i		−	1.03611i	0.855348	−	0.518054i	$-0.173343\pi$
$252$		0			0
$253$	−381.137			−1.50647
$254$		0			0
$255$		−	14.3469i		−	0.0562625i
$256$		0			0
$257$			184.558i			0.718126i	0.933313	+	0.359063i	$0.116904\pi$
$257$			184.558i			0.718126i	−0.933313	+	0.359063i	$0.883096\pi$
$258$		0			0
$259$	−43.7746	−	161.770i	−0.169014	−	0.624594i
$260$		0			0
$261$	−173.765			−0.665764
$262$		0			0
$263$	−246.853			−0.938604			−0.469302	−	0.883038i	$-0.655494\pi$
$263$	−246.853			−0.938604			−0.469302	+	0.883038i	$0.655494\pi$
$264$		0			0
$265$			9.74700i			0.0367811i
$266$		0			0
$267$	−291.078			−1.09018
$268$		0			0
$269$			105.159i			0.390926i	0.980711	+	0.195463i	$0.0626210\pi$
$269$			105.159i			0.390926i	−0.980711	+	0.195463i	$0.937379\pi$
$270$		0			0
$271$			71.8093i			0.264979i	0.991184	+	0.132489i	$0.0422971\pi$
$271$			71.8093i			0.264979i	−0.991184	+	0.132489i	$0.957703\pi$
$272$		0			0
$273$	−145.539	−	537.842i	−0.533110	−	1.97012i
$274$		0			0
$275$	−327.490			−1.19087
$276$		0			0
$277$	402.098			1.45162			0.725808	−	0.687898i	$-0.241466\pi$
$277$	402.098			1.45162			0.725808	+	0.687898i	$0.241466\pi$
$278$		0			0
$279$			97.3516i			0.348930i
$280$		0			0
$281$	321.529			1.14423			0.572116	−	0.820173i	$-0.306123\pi$
$281$	321.529			1.14423			0.572116	+	0.820173i	$0.306123\pi$
$282$		0			0
$283$		−	485.561i		−	1.71576i	−0.513847	−	0.857882i	$-0.671780\pi$
$283$		−	485.561i		−	1.71576i	0.513847	−	0.857882i	$-0.328220\pi$
$284$		0			0
$285$		−	46.2670i		−	0.162340i
$286$		0			0
$287$	473.706	−	128.184i	1.65054	−	0.446634i
$288$		0			0
$289$	251.510			0.870276
$290$		0			0
$291$	333.255			1.14521
$292$		0			0
$293$		−	480.218i		−	1.63897i	−0.573100	−	0.819485i	$-0.694259\pi$
$293$		−	480.218i		−	1.63897i	0.573100	−	0.819485i	$-0.305741\pi$
$294$		0			0
$295$	−9.44156			−0.0320053
$296$		0			0
$297$		−	213.687i		−	0.719486i
$298$		0			0
$299$			616.608i			2.06223i
$300$		0			0
$301$	84.1076	+	310.821i	0.279427	+	1.03263i
$302$		0			0
$303$	479.127			1.58128
$304$		0			0
$305$	55.0294			0.180424
$306$		0			0
$307$		−	13.1876i		−	0.0429564i	−0.999769	−	0.0214782i	$-0.993163\pi$
$307$		−	13.1876i		−	0.0429564i	0.999769	−	0.0214782i	$-0.00683725\pi$
$308$		0			0
$309$	−127.196			−0.411637
$310$		0			0
$311$			393.211i			1.26434i	0.774829	+	0.632171i	$0.217836\pi$
$311$			393.211i			1.26434i	−0.774829	+	0.632171i	$0.782164\pi$
$312$		0			0
$313$		−	221.694i		−	0.708287i	−0.935191	−	0.354143i	$-0.884772\pi$
$313$		−	221.694i		−	0.708287i	0.935191	−	0.354143i	$-0.115228\pi$
$314$		0			0
$315$	−19.9512	+	5.39876i	−0.0633372	+	0.0171389i
$316$		0			0
$317$	−94.5685			−0.298323			−0.149162	−	0.988813i	$-0.547658\pi$
$317$	−94.5685			−0.298323			−0.149162	+	0.988813i	$0.547658\pi$
$318$		0			0
$319$	−496.784			−1.55732
$320$		0			0
$321$		−	553.893i		−	1.72552i
$322$		0			0
$323$	−120.902			−0.374308
$324$		0			0
$325$			529.817i			1.63021i
$326$		0			0
$327$		−	248.541i		−	0.760066i
$328$		0			0
$329$	7.09841	−	1.92082i	0.0215757	−	0.00583835i
$330$		0			0
$331$	190.333			0.575024			0.287512	−	0.957777i	$-0.407172\pi$
$331$	190.333			0.575024			0.287512	+	0.957777i	$0.407172\pi$
$332$		0			0
$333$	111.490			0.334806
$334$		0			0
$335$			15.6150i			0.0466120i
$336$		0			0
$337$	−430.735			−1.27815			−0.639073	−	0.769146i	$-0.720682\pi$
$337$	−430.735			−1.27815			−0.639073	+	0.769146i	$0.720682\pi$
$338$		0			0
$339$			259.737i			0.766185i
$340$		0			0
$341$			278.323i			0.816197i
$342$		0			0
$343$	−244.328	−	240.734i	−0.712327	−	0.701848i
$344$		0			0
$345$	67.0782			0.194430
$346$		0			0
$347$	251.255			0.724077			0.362039	−	0.932163i	$-0.382081\pi$
$347$	251.255			0.724077			0.362039	+	0.932163i	$0.382081\pi$
$348$		0			0
$349$		−	78.3487i		−	0.224495i	−0.993680	−	0.112247i	$-0.964195\pi$
$349$		−	78.3487i		−	0.224495i	0.993680	−	0.112247i	$-0.0358049\pi$
$350$		0			0
$351$	−345.706			−0.984916
$352$		0			0
$353$			43.9111i			0.124394i	0.998064	+	0.0621970i	$0.0198107\pi$
$353$			43.9111i			0.124394i	−0.998064	+	0.0621970i	$0.980189\pi$
$354$		0			0
$355$		−	28.5509i		−	0.0804252i
$356$		0			0
$357$	41.3726	+	152.893i	0.115890	+	0.428272i
$358$		0			0
$359$	−546.431			−1.52209			−0.761045	−	0.648699i	$-0.775314\pi$
$359$	−546.431			−1.52209			−0.761045	+	0.648699i	$0.775314\pi$
$360$		0			0
$361$	−28.8924			−0.0800342
$362$		0			0
$363$			207.891i			0.572702i
$364$		0			0
$365$	32.8040			0.0898741
$366$		0			0
$367$		−	481.505i		−	1.31200i	−0.754759	−	0.656002i	$-0.772246\pi$
$367$		−	481.505i		−	1.31200i	0.754759	−	0.656002i	$-0.227754\pi$
$368$		0			0
$369$			326.474i			0.884753i
$370$		0			0
$371$	−28.1076	−	103.872i	−0.0757619	−	0.279979i
$372$		0			0
$373$	−432.843			−1.16044			−0.580218	−	0.814461i	$-0.697033\pi$
$373$	−432.843			−1.16044			−0.580218	+	0.814461i	$0.697033\pi$
$374$		0			0
$375$	116.215			0.309907
$376$		0			0
$377$			803.702i			2.13184i
$378$		0			0
$379$	512.607			1.35253			0.676263	−	0.736660i	$-0.263598\pi$
$379$	512.607			1.35253			0.676263	+	0.736660i	$0.263598\pi$
$380$		0			0
$381$			346.508i			0.909471i
$382$		0			0
$383$			651.897i			1.70208i	0.525100	+	0.851040i	$0.324028\pi$
$383$			651.897i			1.70208i	−0.525100	+	0.851040i	$0.675972\pi$
$384$		0			0
$385$	−57.0395	+	15.4348i	−0.148155	+	0.0400904i
$386$		0			0
$387$	−214.215			−0.553528
$388$		0			0
$389$	−523.352			−1.34538			−0.672689	−	0.739925i	$-0.734861\pi$
$389$	−523.352			−1.34538			−0.672689	+	0.739925i	$0.734861\pi$
$390$		0			0
$391$		−	175.284i		−	0.448296i
$392$		0			0
$393$	540.382			1.37502
$394$		0			0
$395$		−	21.3775i		−	0.0541202i
$396$		0			0
$397$			15.8513i			0.0399276i	0.999801	+	0.0199638i	$0.00635510\pi$
$397$			15.8513i			0.0399276i	−0.999801	+	0.0199638i	$0.993645\pi$
$398$		0			0
$399$	133.421	+	493.061i	0.334389	+	1.23574i
$400$		0			0
$401$	11.7746			0.0293631			0.0146816	−	0.999892i	$-0.495327\pi$
$401$	11.7746			0.0293631			0.0146816	+	0.999892i	$0.495327\pi$
$402$		0			0
$403$	450.274			1.11731
$404$		0			0
$405$			64.1820i			0.158474i
$406$		0			0
$407$	318.745			0.783158
$408$		0			0
$409$		−	492.483i		−	1.20411i	−0.798453	−	0.602057i	$-0.794348\pi$
$409$		−	492.483i		−	1.20411i	0.798453	−	0.602057i	$-0.205652\pi$
$410$		0			0
$411$		−	553.457i		−	1.34661i
$412$		0			0
$413$	100.617	−	27.2268i	0.243625	−	0.0659246i
$414$		0			0
$415$	−49.0681			−0.118236
$416$		0			0
$417$	−444.382			−1.06566
$418$		0			0
$419$			276.946i			0.660970i	0.943811	+	0.330485i	$0.107212\pi$
$419$			276.946i			0.660970i	−0.943811	+	0.330485i	$0.892788\pi$
$420$		0			0
$421$	14.3532			0.0340932			0.0170466	−	0.999855i	$-0.494574\pi$
$421$	14.3532			0.0340932			0.0170466	+	0.999855i	$0.494574\pi$
$422$		0			0
$423$			4.89216i			0.0115654i
$424$		0			0
$425$		−	150.612i		−	0.354381i
$426$		0			0
$427$	−586.441	+	158.690i	−1.37340	+	0.371639i
$428$		0			0
$429$	1059.74			2.47027
$430$		0			0
$431$	−250.471			−0.581139			−0.290570	−	0.956854i	$-0.593845\pi$
$431$	−250.471			−0.581139			−0.290570	+	0.956854i	$0.593845\pi$
$432$		0			0
$433$			656.534i			1.51625i	0.652112	+	0.758123i	$0.273883\pi$
$433$			656.534i			1.51625i	−0.652112	+	0.758123i	$0.726117\pi$
$434$		0			0
$435$	87.4315			0.200992
$436$		0			0
$437$		−	565.268i		−	1.29352i
$438$		0			0
$439$		−	356.398i		−	0.811841i	−0.913908	−	0.405921i	$-0.866951\pi$
$439$		−	356.398i		−	0.811841i	0.913908	−	0.405921i	$-0.133049\pi$
$440$		0			0
$441$	197.049	−	115.068i	0.446823	−	0.260924i
$442$		0			0
$443$	−253.785			−0.572877			−0.286439	−	0.958099i	$-0.592471\pi$
$443$	−253.785			−0.572877			−0.286439	+	0.958099i	$0.592471\pi$
$444$		0			0
$445$	49.9411			0.112227
$446$		0			0
$447$			124.777i			0.279144i
$448$		0			0
$449$	−107.921			−0.240358			−0.120179	−	0.992752i	$-0.538347\pi$
$449$	−107.921			−0.240358			−0.120179	+	0.992752i	$0.538347\pi$
$450$		0			0
$451$			933.372i			2.06956i
$452$		0			0
$453$		−	534.473i		−	1.17985i
$454$		0			0
$455$	24.9706	+	92.2792i	0.0548804	+	0.202811i
$456$		0			0
$457$	143.990			0.315076			0.157538	−	0.987513i	$-0.449644\pi$
$457$	143.990			0.315076			0.157538	+	0.987513i	$0.449644\pi$
$458$		0			0
$459$	98.2742			0.214105
$460$		0			0
$461$		−	380.728i		−	0.825875i	−0.910759	−	0.412938i	$-0.864503\pi$
$461$		−	380.728i		−	0.825875i	0.910759	−	0.412938i	$-0.135497\pi$
$462$		0			0
$463$	227.990			0.492419			0.246209	−	0.969217i	$-0.420815\pi$
$463$	227.990			0.492419			0.246209	+	0.969217i	$0.420815\pi$
$464$		0			0
$465$		−	48.9835i		−	0.105341i
$466$		0			0
$467$		−	502.917i		−	1.07691i	−0.842654	−	0.538455i	$-0.819008\pi$
$467$		−	502.917i		−	1.07691i	0.842654	−	0.538455i	$-0.180992\pi$
$468$		0			0
$469$	−45.0294	−	166.407i	−0.0960116	−	0.354813i
$470$		0			0
$471$	476.049			1.01072
$472$		0			0
$473$	−612.431			−1.29478
$474$		0			0
$475$		−	485.704i		−	1.02254i
$476$		0			0
$477$	71.5879			0.150079
$478$		0			0
$479$		−	212.637i		−	0.443918i	−0.975056	−	0.221959i	$-0.928755\pi$
$479$		−	212.637i		−	0.443918i	0.975056	−	0.221959i	$-0.0712451\pi$
$480$		0			0
$481$		−	515.669i		−	1.07208i
$482$		0			0
$483$	−714.843	+	193.435i	−1.48001	+	0.400487i
$484$		0			0
$485$	−57.1775			−0.117892
$486$		0			0
$487$	121.411			0.249304			0.124652	−	0.992200i	$-0.460218\pi$
$487$	121.411			0.249304			0.124652	+	0.992200i	$0.460218\pi$
$488$		0			0
$489$		−	212.239i		−	0.434027i
$490$		0			0
$491$	130.353			0.265485			0.132743	−	0.991151i	$-0.457622\pi$
$491$	130.353			0.265485			0.132743	+	0.991151i	$0.457622\pi$
$492$		0			0
$493$		−	228.469i		−	0.463427i
$494$		0			0
$495$		−	39.3111i		−	0.0794164i
$496$		0			0
$497$	82.3330	+	304.263i	0.165660	+	0.612200i
$498$		0			0
$499$	−157.980			−0.316593			−0.158296	−	0.987392i	$-0.550600\pi$
$499$	−157.980			−0.316593			−0.158296	+	0.987392i	$0.550600\pi$
$500$		0			0
$501$	−846.725			−1.69007
$502$		0			0
$503$		−	323.502i		−	0.643146i	−0.946885	−	0.321573i	$-0.895788\pi$
$503$		−	323.502i		−	0.643146i	0.946885	−	0.321573i	$-0.104212\pi$
$504$		0			0
$505$	−82.2052			−0.162783
$506$		0			0
$507$		−	1089.92i		−	2.14975i
$508$		0			0
$509$		−	505.723i		−	0.993563i	−0.867876	−	0.496781i	$-0.834515\pi$
$509$		−	505.723i		−	0.993563i	0.867876	−	0.496781i	$-0.165485\pi$
$510$		0			0
$511$	−349.588	+	94.5978i	−0.684125	+	0.185123i
$512$		0			0
$513$	316.922			0.617781
$514$		0			0
$515$	21.8234			0.0423755
$516$		0			0
$517$			13.9864i			0.0270531i
$518$		0			0
$519$	455.696			0.878026
$520$		0			0
$521$			827.833i			1.58893i	0.607309	+	0.794466i	$0.292249\pi$
$521$			827.833i			1.58893i	−0.607309	+	0.794466i	$0.707751\pi$
$522$		0			0
$523$			114.306i			0.218559i	0.994011	+	0.109279i	$0.0348543\pi$
$523$			114.306i			0.218559i	−0.994011	+	0.109279i	$0.965146\pi$
$524$		0			0
$525$	−614.225	+	166.208i	−1.16995	+	0.316587i
$526$		0			0
$527$	−128.000			−0.242884
$528$		0			0
$529$	290.529			0.549204
$530$		0			0
$531$			69.3446i			0.130592i
$532$		0			0
$533$	1510.02			2.83306
$534$		0			0
$535$			95.0329i			0.177632i
$536$		0			0
$537$			1004.96i			1.87144i
$538$		0			0
$539$	563.352	−	328.973i	1.04518	−	0.610339i
$540$		0			0
$541$	529.098			0.978001			0.489000	−	0.872284i	$-0.337362\pi$
$541$	529.098			0.978001			0.489000	+	0.872284i	$0.337362\pi$
$542$		0			0
$543$	113.206			0.208483
$544$		0			0
$545$			42.6430i			0.0782440i
$546$		0			0
$547$	281.314			0.514285			0.257142	−	0.966374i	$-0.417219\pi$
$547$	281.314			0.514285			0.257142	+	0.966374i	$0.417219\pi$
$548$		0			0
$549$		−	404.170i		−	0.736193i
$550$		0			0
$551$		−	736.785i		−	1.33718i
$552$		0			0
$553$	61.6468	+	227.817i	0.111477	+	0.411965i
$554$		0			0
$555$	−56.0975			−0.101077
$556$		0			0
$557$	−198.471			−0.356321			−0.178161	−	0.984001i	$-0.557015\pi$
$557$	−198.471			−0.356321			−0.178161	+	0.984001i	$0.557015\pi$
$558$		0			0
$559$			990.797i			1.77244i
$560$		0			0
$561$	−301.255			−0.536996
$562$		0			0
$563$		−	98.2933i		−	0.174588i	−0.996183	−	0.0872942i	$-0.972178\pi$
$563$		−	98.2933i		−	0.174588i	0.996183	−	0.0872942i	$-0.0278220\pi$
$564$		0			0
$565$		−	44.5638i		−	0.0788740i
$566$		0			0
$567$	−185.083	−	683.979i	−0.326426	−	1.20631i
$568$		0			0
$569$	−709.029			−1.24610			−0.623049	−	0.782183i	$-0.714106\pi$
$569$	−709.029			−1.24610			−0.623049	+	0.782183i	$0.714106\pi$
$570$		0			0
$571$	−462.118			−0.809313			−0.404657	−	0.914469i	$-0.632609\pi$
$571$	−462.118			−0.809313			−0.404657	+	0.914469i	$0.632609\pi$
$572$		0			0
$573$			257.201i			0.448867i
$574$		0			0
$575$	704.177			1.22465
$576$		0			0
$577$			586.571i			1.01659i	0.861184	+	0.508294i	$0.169724\pi$
$577$			586.571i			1.01659i	−0.861184	+	0.508294i	$0.830276\pi$
$578$		0			0
$579$			1032.61i			1.78343i
$580$		0			0
$581$	522.912	−	141.499i	0.900020	−	0.243544i
$582$		0			0
$583$	204.666			0.351057
$584$		0			0
$585$	−63.5980			−0.108714
$586$		0			0
$587$			441.613i			0.752322i	0.926554	+	0.376161i	$0.122756\pi$
$587$			441.613i			0.752322i	−0.926554	+	0.376161i	$0.877244\pi$
$588$		0			0
$589$	−412.784			−0.700821
$590$		0			0
$591$			378.609i			0.640624i
$592$		0			0
$593$		−	87.0265i		−	0.146756i	−0.997304	−	0.0733782i	$-0.976622\pi$
$593$		−	87.0265i		−	0.146756i	0.997304	−	0.0733782i	$-0.0233780\pi$
$594$		0			0
$595$	−7.09841	−	26.2323i	−0.0119301	−	0.0440879i
$596$		0			0
$597$	−598.489			−1.00249
$598$		0			0
$599$	−484.108			−0.808193			−0.404097	−	0.914716i	$-0.632414\pi$
$599$	−484.108			−0.808193			−0.404097	+	0.914716i	$0.632414\pi$
$600$		0			0
$601$		−	904.895i		−	1.50565i	−0.658221	−	0.752825i	$-0.728691\pi$
$601$		−	904.895i		−	1.50565i	0.658221	−	0.752825i	$-0.271309\pi$
$602$		0			0
$603$	114.686			0.190193
$604$		0			0
$605$		−	35.6684i		−	0.0589561i
$606$		0			0
$607$			940.980i			1.55021i	0.631830	+	0.775107i	$0.282304\pi$
$607$			940.980i			1.55021i	−0.631830	+	0.775107i	$0.717696\pi$
$608$		0			0
$609$	−931.744	+	252.128i	−1.52996	+	0.414004i
$610$		0			0
$611$	22.6274			0.0370334
$612$		0			0
$613$	240.627			0.392541			0.196270	−	0.980550i	$-0.437117\pi$
$613$	240.627			0.392541			0.196270	+	0.980550i	$0.437117\pi$
$614$		0			0
$615$		−	164.269i		−	0.267104i
$616$		0			0
$617$	781.716			1.26696			0.633481	−	0.773758i	$-0.281625\pi$
$617$	781.716			1.26696			0.633481	+	0.773758i	$0.281625\pi$
$618$		0			0
$619$		−	9.99870i		−	0.0161530i	−0.999967	−	0.00807649i	$-0.997429\pi$
$619$		−	9.99870i		−	0.0161530i	0.999967	−	0.00807649i	$-0.00257085\pi$
$620$		0			0
$621$			459.475i			0.739895i
$622$		0			0
$623$	−532.215	+	144.016i	−0.854278	+	0.231166i
$624$		0			0
$625$	595.010			0.952016
$626$		0			0
$627$	−971.509			−1.54946
$628$		0			0
$629$			146.590i			0.233052i
$630$		0			0
$631$	−908.538			−1.43984			−0.719919	−	0.694058i	$-0.755821\pi$
$631$	−908.538			−1.43984			−0.719919	+	0.694058i	$0.755821\pi$
$632$		0			0
$633$			188.543i			0.297856i
$634$		0			0
$635$		−	59.4514i		−	0.0936243i
$636$		0			0
$637$	−532.215	−	911.397i	−0.835503	−	1.43077i
$638$		0			0
$639$	−209.696			−0.328162
$640$		0			0
$641$	193.460			0.301810			0.150905	−	0.988548i	$-0.451781\pi$
$641$	193.460			0.301810			0.150905	+	0.988548i	$0.451781\pi$
$642$		0			0
$643$			362.412i			0.563627i	0.959469	+	0.281814i	$0.0909360\pi$
$643$			362.412i			0.563627i	−0.959469	+	0.281814i	$0.909064\pi$
$644$		0			0
$645$	107.785			0.167108
$646$		0			0
$647$			359.625i			0.555834i	0.960605	+	0.277917i	$0.0896440\pi$
$647$			359.625i			0.555834i	−0.960605	+	0.277917i	$0.910356\pi$
$648$		0			0
$649$			198.253i			0.305474i
$650$		0			0
$651$	141.255	+	522.010i	0.216981	+	0.801858i
$652$		0			0
$653$	−715.685			−1.09600			−0.547998	−	0.836480i	$-0.684610\pi$
$653$	−715.685			−1.09600			−0.547998	+	0.836480i	$0.684610\pi$
$654$		0			0
$655$	−92.7149			−0.141549
$656$		0			0
$657$		−	240.933i		−	0.366717i
$658$		0			0
$659$	390.431			0.592459			0.296230	−	0.955117i	$-0.404271\pi$
$659$	390.431			0.592459			0.296230	+	0.955117i	$0.404271\pi$
$660$		0			0
$661$			350.474i			0.530218i	0.964218	+	0.265109i	$0.0854080\pi$
$661$			350.474i			0.530218i	−0.964218	+	0.265109i	$0.914592\pi$
$662$		0			0
$663$			487.373i			0.735103i
$664$		0			0
$665$	−22.8915	−	84.5959i	−0.0344233	−	0.127212i
$666$		0			0
$667$	1068.20			1.60149
$668$		0			0
$669$	817.803			1.22243
$670$		0			0
$671$		−	1155.50i		−	1.72206i
$672$		0			0
$673$	487.214			0.723944			0.361972	−	0.932189i	$-0.382104\pi$
$673$	487.214			0.723944			0.361972	+	0.932189i	$0.382104\pi$
$674$		0			0
$675$			394.802i			0.584892i
$676$		0			0
$677$			451.742i			0.667271i	0.942702	+	0.333635i	$0.108275\pi$
$677$			451.742i			0.667271i	−0.942702	+	0.333635i	$0.891725\pi$
$678$		0			0
$679$	609.332	−	164.884i	0.897396	−	0.242834i
$680$		0			0
$681$	−373.421			−0.548343
$682$		0			0
$683$	224.745			0.329056			0.164528	−	0.986372i	$-0.447390\pi$
$683$	224.745			0.329056			0.164528	+	0.986372i	$0.447390\pi$
$684$		0			0
$685$			94.9583i			0.138625i
$686$		0			0
$687$	−628.950			−0.915503
$688$		0			0
$689$		−	331.111i		−	0.480567i
$690$		0			0
$691$			56.0855i			0.0811657i	0.999176	+	0.0405828i	$0.0129215\pi$
$691$			56.0855i			0.0811657i	−0.999176	+	0.0405828i	$0.987079\pi$
$692$		0			0
$693$	113.362	+	418.933i	0.163582	+	0.604521i
$694$		0			0
$695$	76.2439			0.109703
$696$		0			0
$697$	−429.255			−0.615861
$698$		0			0
$699$		−	1666.75i		−	2.38448i
$700$		0			0
$701$	−769.647			−1.09793			−0.548963	−	0.835846i	$-0.684977\pi$
$701$	−769.647			−1.09793			−0.548963	+	0.835846i	$0.684977\pi$
$702$		0			0
$703$			472.734i			0.672452i
$704$		0			0
$705$		−	2.46154i		−	0.00349155i
$706$		0			0
$707$	876.049	−	237.057i	1.23911	−	0.335300i
$708$		0			0
$709$	465.960			0.657207			0.328603	−	0.944468i	$-0.393422\pi$
$709$	465.960			0.657207			0.328603	+	0.944468i	$0.393422\pi$
$710$		0			0
$711$	−157.009			−0.220829
$712$		0			0
$713$		−	598.456i		−	0.839350i
$714$		0			0
$715$	−181.823			−0.254298
$716$		0			0
$717$		−	809.825i		−	1.12946i
$718$		0			0
$719$		−	1051.88i		−	1.46298i	−0.681852	−	0.731490i	$-0.738825\pi$
$719$		−	1051.88i		−	1.46298i	0.681852	−	0.731490i	$-0.261175\pi$
$720$		0			0
$721$	−232.569	+	62.9326i	−0.322564	+	0.0872852i
$722$		0			0
$723$	1202.47			1.66317
$724$		0			0
$725$	917.842			1.26599
$726$		0			0
$727$		−	161.080i		−	0.221568i	−0.993845	−	0.110784i	$-0.964664\pi$
$727$		−	161.080i		−	0.221568i	0.993845	−	0.110784i	$-0.0353361\pi$
$728$		0			0
$729$	−62.4315			−0.0856399
$730$		0			0
$731$		−	281.655i		−	0.385301i
$732$		0			0
$733$			1438.70i			1.96275i	0.192097	+	0.981376i	$0.438471\pi$
$733$			1438.70i			1.96275i	−0.192097	+	0.981376i	$0.561529\pi$
$734$		0			0
$735$	−99.1472	+	57.8975i	−0.134894	+	0.0787721i
$736$		0			0
$737$	327.882			0.444888
$738$		0			0
$739$	−1137.39			−1.53909			−0.769547	−	0.638590i	$-0.779518\pi$
$739$	−1137.39			−1.53909			−0.769547	+	0.638590i	$0.779518\pi$
$740$		0			0
$741$			1571.72i			2.12108i
$742$		0			0
$743$	745.882			1.00388			0.501940	−	0.864903i	$-0.332620\pi$
$743$	745.882			1.00388			0.501940	+	0.864903i	$0.332620\pi$
$744$		0			0
$745$		−	21.4084i		−	0.0287361i
$746$		0			0
$747$			360.386i			0.482445i
$748$		0			0
$749$	−274.049	−	1012.75i	−0.365886	−	1.35214i
$750$		0			0
$751$	153.882			0.204903			0.102452	−	0.994738i	$-0.467331\pi$
$751$	153.882			0.204903			0.102452	+	0.994738i	$0.467331\pi$
$752$		0			0
$753$	961.068			1.27632
$754$		0			0
$755$			91.7011i			0.121458i
$756$		0			0
$757$	−119.137			−0.157381			−0.0786903	−	0.996899i	$-0.525074\pi$
$757$	−119.137			−0.157381			−0.0786903	+	0.996899i	$0.525074\pi$
$758$		0			0
$759$		−	1408.50i		−	1.85573i
$760$		0			0
$761$		−	1385.20i		−	1.82024i	−0.414348	−	0.910119i	$-0.635990\pi$
$761$		−	1385.20i		−	1.82024i	0.414348	−	0.910119i	$-0.364010\pi$
$762$		0			0
$763$	−122.971	−	454.440i	−0.161167	−	0.595596i
$764$		0			0
$765$	18.0791			0.0236328
$766$		0			0
$767$	320.735			0.418168
$768$		0			0
$769$		−	55.9020i		−	0.0726945i	−0.999339	−	0.0363472i	$-0.988428\pi$
$769$		−	55.9020i		−	0.0726945i	0.999339	−	0.0363472i	$-0.0115722\pi$
$770$		0			0
$771$	−682.039			−0.884616
$772$		0			0
$773$		−	179.759i		−	0.232548i	−0.993217	−	0.116274i	$-0.962905\pi$
$773$		−	179.759i		−	0.232548i	0.993217	−	0.116274i	$-0.0370950\pi$
$774$		0			0
$775$		−	514.221i		−	0.663511i
$776$		0			0
$777$	597.823	−	161.770i	0.769399	−	0.208198i
$778$		0			0
$779$	−1384.29			−1.77701
$780$		0			0
$781$	−599.509			−0.767617
$782$		0			0
$783$			598.892i			0.764868i
$784$		0			0
$785$	−81.6771			−0.104047
$786$		0			0
$787$			208.363i			0.264756i	0.991199	+	0.132378i	$0.0422613\pi$
$787$			208.363i			0.264756i	−0.991199	+	0.132378i	$0.957739\pi$
$788$		0			0
$789$		−	912.249i		−	1.15621i
$790$		0			0
$791$	128.510	+	474.910i	0.162465	+	0.600392i
$792$		0			0
$793$	−1869.38			−2.35735
$794$		0			0
$795$	−36.0202			−0.0453084
$796$		0			0
$797$			237.327i			0.297776i	0.988854	+	0.148888i	$0.0475694\pi$
$797$			237.327i			0.297776i	−0.988854	+	0.148888i	$0.952431\pi$
$798$		0			0
$799$	−6.43232			−0.00805047
$800$		0			0
$801$		−	366.798i		−	0.457925i
$802$		0			0
$803$		−	688.815i		−	0.857802i
$804$		0			0
$805$	122.648	−	33.1882i	0.152357	−	0.0412276i
$806$		0			0
$807$	−388.617			−0.481558
$808$		0			0
$809$	513.793			0.635097			0.317548	−	0.948242i	$-0.397140\pi$
$809$	513.793			0.635097			0.317548	+	0.948242i	$0.397140\pi$
$810$		0			0
$811$			452.373i			0.557797i	0.960321	+	0.278898i	$0.0899692\pi$
$811$			452.373i			0.557797i	−0.960321	+	0.278898i	$0.910031\pi$
$812$		0			0
$813$	−265.373			−0.326412
$814$		0			0
$815$			36.4145i			0.0446803i
$816$		0			0
$817$		−	908.302i		−	1.11175i
$818$		0			0
$819$	677.754	−	183.399i	0.827539	−	0.223930i
$820$		0			0
$821$	−1441.04			−1.75522			−0.877611	−	0.479373i	$-0.840864\pi$
$821$	−1441.04			−1.75522			−0.877611	+	0.479373i	$0.840864\pi$
$822$		0			0
$823$	1603.66			1.94855			0.974275	−	0.225362i	$-0.0723564\pi$
$823$	1603.66			1.94855			0.974275	+	0.225362i	$0.0723564\pi$
$824$		0			0
$825$		−	1210.25i		−	1.46697i
$826$		0			0
$827$	−933.549			−1.12884			−0.564419	−	0.825488i	$-0.690900\pi$
$827$	−933.549			−1.12884			−0.564419	+	0.825488i	$0.690900\pi$
$828$		0			0
$829$			418.809i			0.505198i	0.967571	+	0.252599i	$0.0812853\pi$
$829$			418.809i			0.505198i	−0.967571	+	0.252599i	$0.918715\pi$
$830$		0			0
$831$			1485.96i			1.78816i
$832$		0			0
$833$	151.294	+	259.084i	0.181625	+	0.311025i
$834$		0			0
$835$	145.275			0.173982
$836$		0			0
$837$	335.529			0.400871
$838$		0			0
$839$			995.689i			1.18676i	0.804924	+	0.593378i	$0.202206\pi$
$839$			995.689i			1.18676i	−0.804924	+	0.593378i	$0.797794\pi$
$840$		0			0
$841$	551.313			0.655544
$842$		0			0
$843$			1188.22i			1.40951i
$844$		0			0
$845$			187.001i			0.221303i
$846$		0			0
$847$	102.858	+	380.113i	0.121438	+	0.448776i
$848$		0			0
$849$	1794.40			2.11355
$850$		0			0
$851$	−685.373			−0.805373
$852$		0			0
$853$			315.117i			0.369422i	0.982793	+	0.184711i	$0.0591349\pi$
$853$			315.117i			0.369422i	−0.982793	+	0.184711i	$0.940865\pi$
$854$		0			0
$855$	58.3027			0.0681903
$856$		0			0
$857$			218.287i			0.254711i	0.991857	+	0.127355i	$0.0406489\pi$
$857$			218.287i			0.254711i	−0.991857	+	0.127355i	$0.959351\pi$
$858$		0			0
$859$		−	1441.76i		−	1.67842i	−0.543811	−	0.839208i	$-0.683019\pi$
$859$		−	1441.76i		−	1.67842i	0.543811	−	0.839208i	$-0.316981\pi$
$860$		0			0
$861$	473.706	+	1750.59i	0.550181	+	2.03320i
$862$		0			0
$863$	−874.873			−1.01376			−0.506879	−	0.862017i	$-0.669201\pi$
$863$	−874.873			−1.01376			−0.506879	+	0.862017i	$0.669201\pi$
$864$		0			0
$865$	−78.1850			−0.0903873
$866$		0			0
$867$			929.459i			1.07204i
$868$		0			0
$869$	−448.881			−0.516549
$870$		0			0
$871$		−	530.451i		−	0.609014i
$872$		0			0
$873$			419.946i			0.481038i
$874$		0			0
$875$	212.491	−	57.4997i	0.242847	−	0.0657139i
$876$		0			0
$877$	1702.65			1.94144			0.970722	−	0.240207i	$-0.0772153\pi$
$877$	1702.65			1.94144			0.970722	+	0.240207i	$0.0772153\pi$
$878$		0			0
$879$	1774.66			2.01895
$880$		0			0
$881$		−	1018.58i		−	1.15617i	−0.815978	−	0.578083i	$-0.803801\pi$
$881$		−	1018.58i		−	1.15617i	0.815978	−	0.578083i	$-0.196199\pi$
$882$		0			0
$883$	−1646.04			−1.86414			−0.932071	−	0.362276i	$-0.882000\pi$
$883$	−1646.04			−1.86414			−0.932071	+	0.362276i	$0.882000\pi$
$884$		0			0
$885$		−	34.8915i		−	0.0394254i
$886$		0			0
$887$			688.597i			0.776321i	0.921592	+	0.388161i	$0.126889\pi$
$887$			688.597i			0.776321i	−0.921592	+	0.388161i	$0.873111\pi$
$888$		0			0
$889$	171.442	+	633.565i	0.192848	+	0.712672i
$890$		0			0
$891$	1347.69			1.51255
$892$		0			0
$893$	−20.7434			−0.0232289
$894$		0			0
$895$		−	172.424i		−	0.192653i
$896$		0			0
$897$	−2278.68			−2.54034
$898$		0			0
$899$		−	780.043i		−	0.867679i
$900$		0			0
$901$			94.1253i			0.104468i
$902$		0			0
$903$	−1148.65	+	310.821i	−1.27203	+	0.344210i
$904$		0			0
$905$	−19.4231			−0.0214620
$906$		0			0
$907$	212.018			0.233758			0.116879	−	0.993146i	$-0.462711\pi$
$907$	212.018			0.233758			0.116879	+	0.993146i	$0.462711\pi$
$908$		0			0
$909$			603.765i			0.664208i
$910$		0			0
$911$	601.882			0.660683			0.330342	−	0.943861i	$-0.392836\pi$
$911$	601.882			0.660683			0.330342	+	0.943861i	$0.392836\pi$
$912$		0			0
$913$			1030.33i			1.12851i
$914$		0			0
$915$			203.362i			0.222254i
$916$		0			0
$917$	988.049	−	267.364i	1.07748	−	0.291564i
$918$		0			0
$919$	−1118.34			−1.21691			−0.608456	−	0.793588i	$-0.708211\pi$
$919$	−1118.34			−1.21691			−0.608456	+	0.793588i	$0.708211\pi$
$920$		0			0
$921$	48.7351			0.0529154
$922$		0			0
$923$			969.892i			1.05080i
$924$		0			0
$925$	−588.903			−0.636652
$926$		0			0
$927$		−	160.284i		−	0.172906i
$928$		0			0
$929$			401.907i			0.432623i	0.976324	+	0.216312i	$0.0694027\pi$
$929$			401.907i			0.432623i	−0.976324	+	0.216312i	$0.930597\pi$
$930$		0			0
$931$	487.902	+	835.513i	0.524063	+	0.897437i
$932$		0			0
$933$	−1453.12			−1.55747
$934$		0			0
$935$	51.6872			0.0552804
$936$		0			0
$937$			1597.84i			1.70527i	0.522507	+	0.852635i	$0.324997\pi$
$937$			1597.84i			1.70527i	−0.522507	+	0.852635i	$0.675003\pi$
$938$		0			0
$939$	819.273			0.872496
$940$		0			0
$941$			1537.71i			1.63413i	0.576547	+	0.817064i	$0.304400\pi$
$941$			1537.71i			1.63413i	−0.576547	+	0.817064i	$0.695600\pi$
$942$		0			0
$943$		−	2006.96i		−	2.12827i
$944$		0			0
$945$	18.6072	+	68.7633i	0.0196902	+	0.0727654i
$946$		0			0
$947$	320.274			0.338199			0.169099	−	0.985599i	$-0.445914\pi$
$947$	320.274			0.338199			0.169099	+	0.985599i	$0.445914\pi$
$948$		0			0
$949$	−1114.37			−1.17426
$950$		0			0
$951$		−	349.480i		−	0.367487i
$952$		0			0
$953$	−734.861			−0.771103			−0.385552	−	0.922686i	$-0.625989\pi$
$953$	−734.861			−0.771103			−0.385552	+	0.922686i	$0.625989\pi$
$954$		0			0
$955$		−	44.1286i		−	0.0462080i
$956$		0			0
$957$		−	1835.87i		−	1.91836i
$958$		0			0
$959$	−273.833	−	1011.96i	−0.285541	−	1.05522i
$960$		0			0
$961$	523.981			0.545245
$962$		0			0
$963$	697.980			0.724797
$964$		0			0
$965$		−	177.167i		−	0.183593i
$966$		0			0
$967$	−1446.98			−1.49636			−0.748179	−	0.663497i	$-0.769072\pi$
$967$	−1446.98			−1.49636			−0.748179	+	0.663497i	$0.769072\pi$
$968$		0			0
$969$		−	446.794i		−	0.461088i
$970$		0			0
$971$			966.084i			0.994937i	0.867482	+	0.497469i	$0.165737\pi$
$971$			966.084i			0.994937i	−0.867482	+	0.497469i	$0.834263\pi$
$972$		0			0
$973$	−812.520	+	219.866i	−0.835067	+	0.225967i
$974$		0			0
$975$	−1957.95			−2.00815
$976$		0			0
$977$	−609.606			−0.623957			−0.311979	−	0.950089i	$-0.600992\pi$
$977$	−609.606			−0.623957			−0.311979	+	0.950089i	$0.600992\pi$
$978$		0			0
$979$		−	1048.66i		−	1.07115i
$980$		0			0
$981$	313.196			0.319262
$982$		0			0
$983$			1277.27i			1.29936i	0.760209	+	0.649679i	$0.225097\pi$
$983$			1277.27i			1.29936i	−0.760209	+	0.649679i	$0.774903\pi$
$984$		0			0
$985$		−	64.9590i		−	0.0659482i
$986$		0			0
$987$	7.09841	+	26.2323i	0.00719191	+	0.0265778i
$988$		0			0
$989$	1316.86			1.33151
$990$		0			0
$991$	1669.99			1.68515			0.842577	−	0.538575i	$-0.181037\pi$
$991$	1669.99			1.68515			0.842577	+	0.538575i	$0.181037\pi$
$992$		0			0
$993$			703.379i			0.708338i
$994$		0			0
$995$	102.685			0.103201
$996$		0			0
$997$		−	789.952i		−	0.792329i	−0.918180	−	0.396164i	$-0.870341\pi$
$997$		−	789.952i		−	0.792329i	0.918180	−	0.396164i	$-0.129659\pi$
$998$		0			0
$999$		−	384.259i		−	0.384644i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.3.c.c.97.4		4
3.2	odd	2		1008.3.f.h.433.3		4
4.3	odd	2		56.3.c.a.41.1	✓	4
7.2	even	3		784.3.s.f.129.1		8
7.3	odd	6		784.3.s.f.705.1		8
7.4	even	3		784.3.s.f.705.4		8
7.5	odd	6		784.3.s.f.129.4		8
7.6	odd	2	inner	112.3.c.c.97.1		4
8.3	odd	2		448.3.c.f.321.4		4
8.5	even	2		448.3.c.e.321.1		4
12.11	even	2		504.3.f.a.433.3		4
20.3	even	4		1400.3.p.a.1049.8		8
20.7	even	4		1400.3.p.a.1049.1		8
20.19	odd	2		1400.3.f.a.601.4		4
21.20	even	2		1008.3.f.h.433.2		4
28.3	even	6		392.3.o.b.313.4		8
28.11	odd	6		392.3.o.b.313.1		8
28.19	even	6		392.3.o.b.129.1		8
28.23	odd	6		392.3.o.b.129.4		8
28.27	even	2		56.3.c.a.41.4	yes	4
56.13	odd	2		448.3.c.e.321.4		4
56.27	even	2		448.3.c.f.321.1		4
84.83	odd	2		504.3.f.a.433.2		4
140.27	odd	4		1400.3.p.a.1049.7		8
140.83	odd	4		1400.3.p.a.1049.2		8
140.139	even	2		1400.3.f.a.601.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.c.a.41.1	✓	4	4.3	odd	2
56.3.c.a.41.4	yes	4	28.27	even	2
112.3.c.c.97.1		4	7.6	odd	2	inner
112.3.c.c.97.4		4	1.1	even	1	trivial
392.3.o.b.129.1		8	28.19	even	6
392.3.o.b.129.4		8	28.23	odd	6
392.3.o.b.313.1		8	28.11	odd	6
392.3.o.b.313.4		8	28.3	even	6
448.3.c.e.321.1		4	8.5	even	2
448.3.c.e.321.4		4	56.13	odd	2
448.3.c.f.321.1		4	56.27	even	2
448.3.c.f.321.4		4	8.3	odd	2
504.3.f.a.433.2		4	84.83	odd	2
504.3.f.a.433.3		4	12.11	even	2
784.3.s.f.129.1		8	7.2	even	3
784.3.s.f.129.4		8	7.5	odd	6
784.3.s.f.705.1		8	7.3	odd	6
784.3.s.f.705.4		8	7.4	even	3
1008.3.f.h.433.2		4	21.20	even	2
1008.3.f.h.433.3		4	3.2	odd	2
1400.3.f.a.601.1		4	140.139	even	2
1400.3.f.a.601.4		4	20.19	odd	2
1400.3.p.a.1049.1		8	20.7	even	4
1400.3.p.a.1049.2		8	140.83	odd	4
1400.3.p.a.1049.7		8	140.27	odd	4
1400.3.p.a.1049.8		8	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	112.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.69552i q^{3} -0.634051i q^{5} +(1.82843 + 6.75699i) q^{7} -4.65685 q^{9} +O(q^{10})\)	\(q+3.69552i q^{3} -0.634051i q^{5} +(1.82843 + 6.75699i) q^{7} -4.65685 q^{9} -13.3137 q^{11} +21.5391i q^{13} +2.34315 q^{15} -6.12293i q^{17} -19.7457i q^{19} +(-24.9706 + 6.75699i) q^{21} +28.6274 q^{23} +24.5980 q^{25} +16.0502i q^{27} +37.3137 q^{29} -20.9050i q^{31} -49.2011i q^{33} +(4.28427 - 1.15932i) q^{35} -23.9411 q^{37} -79.5980 q^{39} -70.1061i q^{41} +46.0000 q^{43} +2.95268i q^{45} -1.05053i q^{47} +(-42.3137 + 24.7093i) q^{49} +22.6274 q^{51} -15.3726 q^{53} +8.44157i q^{55} +72.9706 q^{57} -14.8909i q^{59} +86.7903i q^{61} +(-8.51472 - 31.4663i) q^{63} +13.6569 q^{65} -24.6274 q^{67} +105.793i q^{69} +45.0294 q^{71} +51.7373i q^{73} +90.9023i q^{75} +(-24.3431 - 89.9605i) q^{77} +33.7157 q^{79} -101.225 q^{81} -77.3883i q^{83} -3.88225 q^{85} +137.893i q^{87} +78.7652i q^{89} +(-145.539 + 39.3826i) q^{91} +77.2548 q^{93} -12.5198 q^{95} -90.1781i q^{97} +62.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{7} + 4 q^{9} - 8 q^{11} + 32 q^{15} - 32 q^{21} + 24 q^{23} - 60 q^{25} + 104 q^{29} - 96 q^{35} + 40 q^{37} - 160 q^{39} + 184 q^{43} - 124 q^{49} - 152 q^{53} + 224 q^{57} - 68 q^{63} + 32 q^{65} - 8 q^{67} + 248 q^{71} - 120 q^{77} + 248 q^{79} - 156 q^{81} + 256 q^{85} - 288 q^{91} + 128 q^{93} - 480 q^{95} + 248 q^{99}+O(q^{100}) \) 4 * q - 4 * q^7 + 4 * q^9 - 8 * q^11 + 32 * q^15 - 32 * q^21 + 24 * q^23 - 60 * q^25 + 104 * q^29 - 96 * q^35 + 40 * q^37 - 160 * q^39 + 184 * q^43 - 124 * q^49 - 152 * q^53 + 224 * q^57 - 68 * q^63 + 32 * q^65 - 8 * q^67 + 248 * q^71 - 120 * q^77 + 248 * q^79 - 156 * q^81 + 256 * q^85 - 288 * q^91 + 128 * q^93 - 480 * q^95 + 248 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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