LMFDB - Embedded newform 912.2.f.f.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(113,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("912.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.f (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:	$7.28235666434$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.1
Root			$-0.707107 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	912.113
Dual form			912.2.f.f.113.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 1.58114i) q^{3} +2.23607i q^{5} -1.00000 q^{7} +(-2.00000 + 2.23607i) q^{9} +O(q^{10})$	\(q+(-0.707107 - 1.58114i) q^{3} +2.23607i q^{5} -1.00000 q^{7} +(-2.00000 + 2.23607i) q^{9} +2.23607i q^{11} -3.16228i q^{13} +(3.53553 - 1.58114i) q^{15} -6.70820i q^{17} +(-3.00000 - 3.16228i) q^{19} +(0.707107 + 1.58114i) q^{21} -4.47214i q^{23} +(4.94975 + 1.58114i) q^{27} +5.65685 q^{29} -3.16228i q^{31} +(3.53553 - 1.58114i) q^{33} -2.23607i q^{35} -9.48683i q^{37} +(-5.00000 + 2.23607i) q^{39} -9.89949 q^{41} +5.00000 q^{43} +(-5.00000 - 4.47214i) q^{45} -2.23607i q^{47} -6.00000 q^{49} +(-10.6066 + 4.74342i) q^{51} +4.24264 q^{53} -5.00000 q^{55} +(-2.87868 + 6.97948i) q^{57} -1.41421 q^{59} -1.00000 q^{61} +(2.00000 - 2.23607i) q^{63} +7.07107 q^{65} +6.32456i q^{67} +(-7.07107 + 3.16228i) q^{69} +4.24264 q^{71} -3.00000 q^{73} -2.23607i q^{77} -12.6491i q^{79} +(-1.00000 - 8.94427i) q^{81} +8.94427i q^{83} +15.0000 q^{85} +(-4.00000 - 8.94427i) q^{87} -12.7279 q^{89} +3.16228i q^{91} +(-5.00000 + 2.23607i) q^{93} +(7.07107 - 6.70820i) q^{95} -3.16228i q^{97} +(-5.00000 - 4.47214i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} - 8 q^{9}+O(q^{10}) $ 4 * q - 4 * q^7 - 8 * q^9	$ 4 q - 4 q^{7} - 8 q^{9} - 12 q^{19} - 20 q^{39} + 20 q^{43} - 20 q^{45} - 24 q^{49} - 20 q^{55} - 20 q^{57} - 4 q^{61} + 8 q^{63} - 12 q^{73} - 4 q^{81} + 60 q^{85} - 16 q^{87} - 20 q^{93} - 20 q^{99}+O(q^{100}) $ 4 * q - 4 * q^7 - 8 * q^9 - 12 * q^19 - 20 * q^39 + 20 * q^43 - 20 * q^45 - 24 * q^49 - 20 * q^55 - 20 * q^57 - 4 * q^61 + 8 * q^63 - 12 * q^73 - 4 * q^81 + 60 * q^85 - 16 * q^87 - 20 * q^93 - 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.707107	−	1.58114i	−0.408248	−	0.912871i
$3$	−0.707107	−	1.58114i	−0.408248	−	0.912871i
$4$		0			0
$5$			2.23607i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$5$			2.23607i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$6$		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	$0.109446\pi$
$11$			2.23607i			0.674200i	−0.941469	+	0.337100i	$0.890554\pi$
$12$		0			0
$13$		−	3.16228i		−	0.877058i	−0.898717	−	0.438529i	$-0.855500\pi$
$13$		−	3.16228i		−	0.877058i	0.898717	−	0.438529i	$-0.144500\pi$
$14$		0			0
$15$	3.53553	−	1.58114i	0.912871	−	0.408248i
$16$		0			0
$17$		−	6.70820i		−	1.62698i	−0.581580	−	0.813489i	$-0.697565\pi$
$17$		−	6.70820i		−	1.62698i	0.581580	−	0.813489i	$-0.302435\pi$
$18$		0			0
$19$	−3.00000	−	3.16228i	−0.688247	−	0.725476i
$19$	−3.00000	−	3.16228i	−0.688247	−	0.725476i
$20$		0			0
$21$	0.707107	+	1.58114i	0.154303	+	0.345033i
$22$		0			0
$23$		−	4.47214i		−	0.932505i	−0.884652	−	0.466252i	$-0.845604\pi$
$23$		−	4.47214i		−	0.932505i	0.884652	−	0.466252i	$-0.154396\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.94975	+	1.58114i	0.952579	+	0.304290i
$28$		0			0
$29$	5.65685			1.05045			0.525226	−	0.850963i	$-0.323981\pi$
$29$	5.65685			1.05045			0.525226	+	0.850963i	$0.323981\pi$
$30$		0			0
$31$		−	3.16228i		−	0.567962i	−0.958830	−	0.283981i	$-0.908345\pi$
$31$		−	3.16228i		−	0.567962i	0.958830	−	0.283981i	$-0.0916552\pi$
$32$		0			0
$33$	3.53553	−	1.58114i	0.615457	−	0.275241i
$34$		0			0
$35$		−	2.23607i		−	0.377964i
$36$		0			0
$37$		−	9.48683i		−	1.55963i	−0.626013	−	0.779813i	$-0.715314\pi$
$37$		−	9.48683i		−	1.55963i	0.626013	−	0.779813i	$-0.284686\pi$
$38$		0			0
$39$	−5.00000	+	2.23607i	−0.800641	+	0.358057i
$40$		0			0
$41$	−9.89949			−1.54604			−0.773021	−	0.634381i	$-0.781255\pi$
$41$	−9.89949			−1.54604			−0.773021	+	0.634381i	$0.781255\pi$
$42$		0			0
$43$	5.00000			0.762493			0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000			0.762493			0.381246	+	0.924473i	$0.375495\pi$
$44$		0			0
$45$	−5.00000	−	4.47214i	−0.745356	−	0.666667i
$46$		0			0
$47$		−	2.23607i		−	0.326164i	−0.986613	−	0.163082i	$-0.947856\pi$
$47$		−	2.23607i		−	0.326164i	0.986613	−	0.163082i	$-0.0521435\pi$
$48$		0			0
$49$	−6.00000			−0.857143
$50$		0			0
$51$	−10.6066	+	4.74342i	−1.48522	+	0.664211i
$52$		0			0
$53$	4.24264			0.582772			0.291386	−	0.956606i	$-0.405884\pi$
$53$	4.24264			0.582772			0.291386	+	0.956606i	$0.405884\pi$
$54$		0			0
$55$	−5.00000			−0.674200
$56$		0			0
$57$	−2.87868	+	6.97948i	−0.381290	+	0.924455i
$58$		0			0
$59$	−1.41421			−0.184115			−0.0920575	−	0.995754i	$-0.529344\pi$
$59$	−1.41421			−0.184115			−0.0920575	+	0.995754i	$0.529344\pi$
$60$		0			0
$61$	−1.00000			−0.128037			−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000			−0.128037			−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$	2.00000	−	2.23607i	0.251976	−	0.281718i
$64$		0			0
$65$	7.07107			0.877058
$66$		0			0
$67$			6.32456i			0.772667i	0.922359	+	0.386334i	$0.126259\pi$
$67$			6.32456i			0.772667i	−0.922359	+	0.386334i	$0.873741\pi$
$68$		0			0
$69$	−7.07107	+	3.16228i	−0.851257	+	0.380693i
$70$		0			0
$71$	4.24264			0.503509			0.251754	−	0.967791i	$-0.418992\pi$
$71$	4.24264			0.503509			0.251754	+	0.967791i	$0.418992\pi$
$72$		0			0
$73$	−3.00000			−0.351123			−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000			−0.351123			−0.175562	+	0.984468i	$0.556174\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.23607i		−	0.254824i
$78$		0			0
$79$		−	12.6491i		−	1.42314i	−0.702617	−	0.711568i	$-0.747985\pi$
$79$		−	12.6491i		−	1.42314i	0.702617	−	0.711568i	$-0.252015\pi$
$80$		0			0
$81$	−1.00000	−	8.94427i	−0.111111	−	0.993808i
$82$		0			0
$83$			8.94427i			0.981761i	0.871227	+	0.490881i	$0.163325\pi$
$83$			8.94427i			0.981761i	−0.871227	+	0.490881i	$0.836675\pi$
$84$		0			0
$85$	15.0000			1.62698
$86$		0			0
$87$	−4.00000	−	8.94427i	−0.428845	−	0.958927i
$88$		0			0
$89$	−12.7279			−1.34916			−0.674579	−	0.738203i	$-0.735675\pi$
$89$	−12.7279			−1.34916			−0.674579	+	0.738203i	$0.735675\pi$
$90$		0			0
$91$			3.16228i			0.331497i
$92$		0			0
$93$	−5.00000	+	2.23607i	−0.518476	+	0.231869i
$94$		0			0
$95$	7.07107	−	6.70820i	0.725476	−	0.688247i
$96$		0			0
$97$		−	3.16228i		−	0.321081i	−0.987029	−	0.160540i	$-0.948676\pi$
$97$		−	3.16228i		−	0.321081i	0.987029	−	0.160540i	$-0.0513237\pi$
$98$		0			0
$99$	−5.00000	−	4.47214i	−0.502519	−	0.449467i
$100$		0			0
$101$			8.94427i			0.889988i	0.895533	+	0.444994i	$0.146794\pi$
$101$			8.94427i			0.889988i	−0.895533	+	0.444994i	$0.853206\pi$
$102$		0			0
$103$		−	6.32456i		−	0.623177i	−0.950217	−	0.311588i	$-0.899139\pi$
$103$		−	6.32456i		−	0.623177i	0.950217	−	0.311588i	$-0.100861\pi$
$104$		0			0
$105$	−3.53553	+	1.58114i	−0.345033	+	0.154303i
$106$		0			0
$107$	−8.48528			−0.820303			−0.410152	−	0.912017i	$-0.634524\pi$
$107$	−8.48528			−0.820303			−0.410152	+	0.912017i	$0.634524\pi$
$108$		0			0
$109$		−	6.32456i		−	0.605783i	−0.953025	−	0.302891i	$-0.902048\pi$
$109$		−	6.32456i		−	0.605783i	0.953025	−	0.302891i	$-0.0979519\pi$
$110$		0			0
$111$	−15.0000	+	6.70820i	−1.42374	+	0.636715i
$112$		0			0
$113$	5.65685			0.532152			0.266076	−	0.963952i	$-0.414273\pi$
$113$	5.65685			0.532152			0.266076	+	0.963952i	$0.414273\pi$
$114$		0			0
$115$	10.0000			0.932505
$116$		0			0
$117$	7.07107	+	6.32456i	0.653720	+	0.584705i
$118$		0			0
$119$			6.70820i			0.614940i
$120$		0			0
$121$	6.00000			0.545455
$122$		0			0
$123$	7.00000	+	15.6525i	0.631169	+	1.41134i
$124$		0			0
$125$			11.1803i			1.00000i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	−3.53553	−	7.90569i	−0.311286	−	0.696058i
$130$		0			0
$131$		−	2.23607i		−	0.195366i	−0.995218	−	0.0976831i	$-0.968857\pi$
$131$		−	2.23607i		−	0.195366i	0.995218	−	0.0976831i	$-0.0311432\pi$
$132$		0			0
$133$	3.00000	+	3.16228i	0.260133	+	0.274204i
$134$		0			0
$135$	−3.53553	+	11.0680i	−0.304290	+	0.952579i
$136$		0			0
$137$		−	2.23607i		−	0.191040i	−0.995428	−	0.0955201i	$-0.969549\pi$
$137$		−	2.23607i		−	0.191040i	0.995428	−	0.0955201i	$-0.0304514\pi$
$138$		0			0
$139$	7.00000			0.593732			0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000			0.593732			0.296866	+	0.954919i	$0.404058\pi$
$140$		0			0
$141$	−3.53553	+	1.58114i	−0.297746	+	0.133156i
$142$		0			0
$143$	7.07107			0.591312
$144$		0			0
$145$			12.6491i			1.05045i
$146$		0			0
$147$	4.24264	+	9.48683i	0.349927	+	0.782461i
$148$		0			0
$149$		−	11.1803i		−	0.915929i	−0.888970	−	0.457965i	$-0.848579\pi$
$149$		−	11.1803i		−	0.915929i	0.888970	−	0.457965i	$-0.151421\pi$
$150$		0			0
$151$		−	15.8114i		−	1.28671i	−0.765567	−	0.643356i	$-0.777541\pi$
$151$		−	15.8114i		−	1.28671i	0.765567	−	0.643356i	$-0.222459\pi$
$152$		0			0
$153$	15.0000	+	13.4164i	1.21268	+	1.08465i
$154$		0			0
$155$	7.07107			0.567962
$156$		0			0
$157$	4.00000			0.319235			0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000			0.319235			0.159617	+	0.987179i	$0.448974\pi$
$158$		0			0
$159$	−3.00000	−	6.70820i	−0.237915	−	0.531995i
$160$		0			0
$161$			4.47214i			0.352454i
$162$		0			0
$163$	18.0000			1.40987			0.704934	−	0.709273i	$-0.250976\pi$
$163$	18.0000			1.40987			0.704934	+	0.709273i	$0.250976\pi$
$164$		0			0
$165$	3.53553	+	7.90569i	0.275241	+	0.615457i
$166$		0			0
$167$	−19.7990			−1.53209			−0.766046	−	0.642786i	$-0.777779\pi$
$167$	−19.7990			−1.53209			−0.766046	+	0.642786i	$0.777779\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	13.0711	−	0.383649i	0.999570	−	0.0293383i
$172$		0			0
$173$	−2.82843			−0.215041			−0.107521	−	0.994203i	$-0.534291\pi$
$173$	−2.82843			−0.215041			−0.107521	+	0.994203i	$0.534291\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	1.00000	+	2.23607i	0.0751646	+	0.168073i
$178$		0			0
$179$	5.65685			0.422813			0.211407	−	0.977398i	$-0.432196\pi$
$179$	5.65685			0.422813			0.211407	+	0.977398i	$0.432196\pi$
$180$		0			0
$181$			18.9737i			1.41030i	0.709057	+	0.705151i	$0.249121\pi$
$181$			18.9737i			1.41030i	−0.709057	+	0.705151i	$0.750879\pi$
$182$		0			0
$183$	0.707107	+	1.58114i	0.0522708	+	0.116881i
$184$		0			0
$185$	21.2132			1.55963
$186$		0			0
$187$	15.0000			1.09691
$188$		0			0
$189$	−4.94975	−	1.58114i	−0.360041	−	0.115011i
$190$		0			0
$191$			11.1803i			0.808981i	0.914542	+	0.404491i	$0.132551\pi$
$191$			11.1803i			0.808981i	−0.914542	+	0.404491i	$0.867449\pi$
$192$		0			0
$193$		−	6.32456i		−	0.455251i	−0.973749	−	0.227626i	$-0.926904\pi$
$193$		−	6.32456i		−	0.455251i	0.973749	−	0.227626i	$-0.0730963\pi$
$194$		0			0
$195$	−5.00000	−	11.1803i	−0.358057	−	0.800641i
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	3.00000			0.212664			0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000			0.212664			0.106332	+	0.994331i	$0.466089\pi$
$200$		0			0
$201$	10.0000	−	4.47214i	0.705346	−	0.315440i
$202$		0			0
$203$	−5.65685			−0.397033
$204$		0			0
$205$		−	22.1359i		−	1.54604i
$206$		0			0
$207$	10.0000	+	8.94427i	0.695048	+	0.621670i
$208$		0			0
$209$	7.07107	−	6.70820i	0.489116	−	0.464016i
$210$		0			0
$211$			22.1359i			1.52390i	0.647635	+	0.761951i	$0.275758\pi$
$211$			22.1359i			1.52390i	−0.647635	+	0.761951i	$0.724242\pi$
$212$		0			0
$213$	−3.00000	−	6.70820i	−0.205557	−	0.459639i
$214$		0			0
$215$			11.1803i			0.762493i
$216$		0			0
$217$			3.16228i			0.214669i
$218$		0			0
$219$	2.12132	+	4.74342i	0.143346	+	0.320530i
$220$		0			0
$221$	−21.2132			−1.42695
$222$		0			0
$223$			3.16228i			0.211762i	0.994379	+	0.105881i	$0.0337662\pi$
$223$			3.16228i			0.211762i	−0.994379	+	0.105881i	$0.966234\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.41421			−0.0938647			−0.0469323	−	0.998898i	$-0.514945\pi$
$227$	−1.41421			−0.0938647			−0.0469323	+	0.998898i	$0.514945\pi$
$228$		0			0
$229$	−1.00000			−0.0660819			−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000			−0.0660819			−0.0330409	+	0.999454i	$0.510519\pi$
$230$		0			0
$231$	−3.53553	+	1.58114i	−0.232621	+	0.104031i
$232$		0			0
$233$			20.1246i			1.31841i	0.751965	+	0.659204i	$0.229106\pi$
$233$			20.1246i			1.31841i	−0.751965	+	0.659204i	$0.770894\pi$
$234$		0			0
$235$	5.00000			0.326164
$236$		0			0
$237$	−20.0000	+	8.94427i	−1.29914	+	0.580993i
$238$		0			0
$239$			15.6525i			1.01247i	0.862394	+	0.506237i	$0.168964\pi$
$239$			15.6525i			1.01247i	−0.862394	+	0.506237i	$0.831036\pi$
$240$		0			0
$241$			3.16228i			0.203700i	0.994800	+	0.101850i	$0.0324762\pi$
$241$			3.16228i			0.203700i	−0.994800	+	0.101850i	$0.967524\pi$
$242$		0			0
$243$	−13.4350	+	7.90569i	−0.861858	+	0.507151i
$244$		0			0
$245$		−	13.4164i		−	0.857143i
$246$		0			0
$247$	−10.0000	+	9.48683i	−0.636285	+	0.603633i
$248$		0			0
$249$	14.1421	−	6.32456i	0.896221	−	0.400802i
$250$		0			0
$251$		−	6.70820i		−	0.423418i	−0.977333	−	0.211709i	$-0.932097\pi$
$251$		−	6.70820i		−	0.423418i	0.977333	−	0.211709i	$-0.0679029\pi$
$252$		0			0
$253$	10.0000			0.628695
$254$		0			0
$255$	−10.6066	−	23.7171i	−0.664211	−	1.48522i
$256$		0			0
$257$	−22.6274			−1.41146			−0.705730	−	0.708481i	$-0.749381\pi$
$257$	−22.6274			−1.41146			−0.705730	+	0.708481i	$0.749381\pi$
$258$		0			0
$259$			9.48683i			0.589483i
$260$		0			0
$261$	−11.3137	+	12.6491i	−0.700301	+	0.782960i
$262$		0			0
$263$		−	29.0689i		−	1.79246i	−0.443585	−	0.896232i	$-0.646294\pi$
$263$		−	29.0689i		−	1.79246i	0.443585	−	0.896232i	$-0.353706\pi$
$264$		0			0
$265$			9.48683i			0.582772i
$266$		0			0
$267$	9.00000	+	20.1246i	0.550791	+	1.23161i
$268$		0			0
$269$	21.2132			1.29339			0.646696	−	0.762748i	$-0.276150\pi$
$269$	21.2132			1.29339			0.646696	+	0.762748i	$0.276150\pi$
$270$		0			0
$271$	−6.00000			−0.364474			−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000			−0.364474			−0.182237	+	0.983255i	$0.558334\pi$
$272$		0			0
$273$	5.00000	−	2.23607i	0.302614	−	0.135333i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	19.0000			1.14160			0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000			1.14160			0.570800	+	0.821089i	$0.306633\pi$
$278$		0			0
$279$	7.07107	+	6.32456i	0.423334	+	0.378641i
$280$		0			0
$281$	−14.1421			−0.843649			−0.421825	−	0.906677i	$-0.638610\pi$
$281$	−14.1421			−0.843649			−0.421825	+	0.906677i	$0.638610\pi$
$282$		0			0
$283$	−25.0000			−1.48610			−0.743048	−	0.669238i	$-0.766621\pi$
$283$	−25.0000			−1.48610			−0.743048	+	0.669238i	$0.766621\pi$
$284$		0			0
$285$	−15.6066	−	6.43692i	−0.924455	−	0.381290i
$286$		0			0
$287$	9.89949			0.584349
$288$		0			0
$289$	−28.0000			−1.64706
$290$		0			0
$291$	−5.00000	+	2.23607i	−0.293105	+	0.131081i
$292$		0			0
$293$	−19.7990			−1.15667			−0.578335	−	0.815800i	$-0.696297\pi$
$293$	−19.7990			−1.15667			−0.578335	+	0.815800i	$0.696297\pi$
$294$		0			0
$295$		−	3.16228i		−	0.184115i
$296$		0			0
$297$	−3.53553	+	11.0680i	−0.205152	+	0.642229i
$298$		0			0
$299$	−14.1421			−0.817861
$300$		0			0
$301$	−5.00000			−0.288195
$302$		0			0
$303$	14.1421	−	6.32456i	0.812444	−	0.363336i
$304$		0			0
$305$		−	2.23607i		−	0.128037i
$306$		0			0
$307$		−	18.9737i		−	1.08288i	−0.840738	−	0.541442i	$-0.817879\pi$
$307$		−	18.9737i		−	1.08288i	0.840738	−	0.541442i	$-0.182121\pi$
$308$		0			0
$309$	−10.0000	+	4.47214i	−0.568880	+	0.254411i
$310$		0			0
$311$		−	24.5967i		−	1.39475i	−0.716705	−	0.697377i	$-0.754350\pi$
$311$		−	24.5967i		−	1.39475i	0.716705	−	0.697377i	$-0.245650\pi$
$312$		0			0
$313$	20.0000			1.13047			0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000			1.13047			0.565233	+	0.824931i	$0.308786\pi$
$314$		0			0
$315$	5.00000	+	4.47214i	0.281718	+	0.251976i
$316$		0			0
$317$	−1.41421			−0.0794301			−0.0397151	−	0.999211i	$-0.512645\pi$
$317$	−1.41421			−0.0794301			−0.0397151	+	0.999211i	$0.512645\pi$
$318$		0			0
$319$			12.6491i			0.708214i
$320$		0			0
$321$	6.00000	+	13.4164i	0.334887	+	0.748831i
$322$		0			0
$323$	−21.2132	+	20.1246i	−1.18033	+	1.11976i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−10.0000	+	4.47214i	−0.553001	+	0.247310i
$328$		0			0
$329$			2.23607i			0.123278i
$330$		0			0
$331$		−	12.6491i		−	0.695258i	−0.937632	−	0.347629i	$-0.886987\pi$
$331$		−	12.6491i		−	0.695258i	0.937632	−	0.347629i	$-0.113013\pi$
$332$		0			0
$333$	21.2132	+	18.9737i	1.16248	+	1.03975i
$334$		0			0
$335$	−14.1421			−0.772667
$336$		0			0
$337$		−	22.1359i		−	1.20582i	−0.797809	−	0.602911i	$-0.794007\pi$
$337$		−	22.1359i		−	1.20582i	0.797809	−	0.602911i	$-0.205993\pi$
$338$		0			0
$339$	−4.00000	−	8.94427i	−0.217250	−	0.485786i
$340$		0			0
$341$	7.07107			0.382920
$342$		0			0
$343$	13.0000			0.701934
$344$		0			0
$345$	−7.07107	−	15.8114i	−0.380693	−	0.851257i
$346$		0			0
$347$			2.23607i			0.120038i	0.998197	+	0.0600192i	$0.0191162\pi$
$347$			2.23607i			0.120038i	−0.998197	+	0.0600192i	$0.980884\pi$
$348$		0			0
$349$	−7.00000			−0.374701			−0.187351	−	0.982293i	$-0.559990\pi$
$349$	−7.00000			−0.374701			−0.187351	+	0.982293i	$0.559990\pi$
$350$		0			0
$351$	5.00000	−	15.6525i	0.266880	−	0.835467i
$352$		0			0
$353$			8.94427i			0.476056i	0.971258	+	0.238028i	$0.0765009\pi$
$353$			8.94427i			0.476056i	−0.971258	+	0.238028i	$0.923499\pi$
$354$		0			0
$355$			9.48683i			0.503509i
$356$		0			0
$357$	10.6066	−	4.74342i	0.561361	−	0.251048i
$358$		0			0
$359$		−	20.1246i		−	1.06214i	−0.847329	−	0.531068i	$-0.821791\pi$
$359$		−	20.1246i		−	1.06214i	0.847329	−	0.531068i	$-0.178209\pi$
$360$		0			0
$361$	−1.00000	+	18.9737i	−0.0526316	+	0.998614i
$362$		0			0
$363$	−4.24264	−	9.48683i	−0.222681	−	0.497930i
$364$		0			0
$365$		−	6.70820i		−	0.351123i
$366$		0			0
$367$	10.0000			0.521996			0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000			0.521996			0.260998	+	0.965339i	$0.415948\pi$
$368$		0			0
$369$	19.7990	−	22.1359i	1.03069	−	1.15235i
$370$		0			0
$371$	−4.24264			−0.220267
$372$		0			0
$373$			12.6491i			0.654946i	0.944861	+	0.327473i	$0.106197\pi$
$373$			12.6491i			0.654946i	−0.944861	+	0.327473i	$0.893803\pi$
$374$		0			0
$375$	17.6777	−	7.90569i	0.912871	−	0.408248i
$376$		0			0
$377$		−	17.8885i		−	0.921307i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	32.5269			1.66205			0.831024	−	0.556237i	$-0.187755\pi$
$383$	32.5269			1.66205			0.831024	+	0.556237i	$0.187755\pi$
$384$		0			0
$385$	5.00000			0.254824
$386$		0			0
$387$	−10.0000	+	11.1803i	−0.508329	+	0.568329i
$388$		0			0
$389$		−	2.23607i		−	0.113373i	−0.998392	−	0.0566866i	$-0.981946\pi$
$389$		−	2.23607i		−	0.113373i	0.998392	−	0.0566866i	$-0.0180536\pi$
$390$		0			0
$391$	−30.0000			−1.51717
$392$		0			0
$393$	−3.53553	+	1.58114i	−0.178344	+	0.0797579i
$394$		0			0
$395$	28.2843			1.42314
$396$		0			0
$397$	−15.0000			−0.752828			−0.376414	−	0.926451i	$-0.622843\pi$
$397$	−15.0000			−0.752828			−0.376414	+	0.926451i	$0.622843\pi$
$398$		0			0
$399$	2.87868	−	6.97948i	0.144114	−	0.349411i
$400$		0			0
$401$	28.2843			1.41245			0.706225	−	0.707988i	$-0.250397\pi$
$401$	28.2843			1.41245			0.706225	+	0.707988i	$0.250397\pi$
$402$		0			0
$403$	−10.0000			−0.498135
$404$		0			0
$405$	20.0000	−	2.23607i	0.993808	−	0.111111i
$406$		0			0
$407$	21.2132			1.05150
$408$		0			0
$409$			3.16228i			0.156365i	0.996939	+	0.0781823i	$0.0249116\pi$
$409$			3.16228i			0.156365i	−0.996939	+	0.0781823i	$0.975088\pi$
$410$		0			0
$411$	−3.53553	+	1.58114i	−0.174395	+	0.0779918i
$412$		0			0
$413$	1.41421			0.0695889
$414$		0			0
$415$	−20.0000			−0.981761
$416$		0			0
$417$	−4.94975	−	11.0680i	−0.242390	−	0.542001i
$418$		0			0
$419$		−	22.3607i		−	1.09239i	−0.837658	−	0.546195i	$-0.816076\pi$
$419$		−	22.3607i		−	1.09239i	0.837658	−	0.546195i	$-0.183924\pi$
$420$		0			0
$421$		−	22.1359i		−	1.07884i	−0.842037	−	0.539420i	$-0.818644\pi$
$421$		−	22.1359i		−	1.07884i	0.842037	−	0.539420i	$-0.181356\pi$
$422$		0			0
$423$	5.00000	+	4.47214i	0.243108	+	0.217443i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.00000			0.0483934
$428$		0			0
$429$	−5.00000	−	11.1803i	−0.241402	−	0.539792i
$430$		0			0
$431$	−4.24264			−0.204361			−0.102180	−	0.994766i	$-0.532582\pi$
$431$	−4.24264			−0.204361			−0.102180	+	0.994766i	$0.532582\pi$
$432$		0			0
$433$			18.9737i			0.911816i	0.890027	+	0.455908i	$0.150685\pi$
$433$			18.9737i			0.911816i	−0.890027	+	0.455908i	$0.849315\pi$
$434$		0			0
$435$	20.0000	−	8.94427i	0.958927	−	0.428845i
$436$		0			0
$437$	−14.1421	+	13.4164i	−0.676510	+	0.641794i
$438$		0			0
$439$		−	6.32456i		−	0.301855i	−0.988545	−	0.150927i	$-0.951774\pi$
$439$		−	6.32456i		−	0.301855i	0.988545	−	0.150927i	$-0.0482259\pi$
$440$		0			0
$441$	12.0000	−	13.4164i	0.571429	−	0.638877i
$442$		0			0
$443$			11.1803i			0.531194i	0.964084	+	0.265597i	$0.0855691\pi$
$443$			11.1803i			0.531194i	−0.964084	+	0.265597i	$0.914431\pi$
$444$		0			0
$445$		−	28.4605i		−	1.34916i
$446$		0			0
$447$	−17.6777	+	7.90569i	−0.836125	+	0.373927i
$448$		0			0
$449$	5.65685			0.266963			0.133482	−	0.991051i	$-0.457384\pi$
$449$	5.65685			0.266963			0.133482	+	0.991051i	$0.457384\pi$
$450$		0			0
$451$		−	22.1359i		−	1.04234i
$452$		0			0
$453$	−25.0000	+	11.1803i	−1.17460	+	0.525298i
$454$		0			0
$455$	−7.07107			−0.331497
$456$		0			0
$457$	−11.0000			−0.514558			−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000			−0.514558			−0.257279	+	0.966337i	$0.582826\pi$
$458$		0			0
$459$	10.6066	−	33.2039i	0.495074	−	1.54983i
$460$		0			0
$461$		−	24.5967i		−	1.14558i	−0.819700	−	0.572792i	$-0.805860\pi$
$461$		−	24.5967i		−	1.14558i	0.819700	−	0.572792i	$-0.194140\pi$
$462$		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
$465$	−5.00000	−	11.1803i	−0.231869	−	0.518476i
$466$		0			0
$467$		−	2.23607i		−	0.103473i	−0.998661	−	0.0517364i	$-0.983524\pi$
$467$		−	2.23607i		−	0.103473i	0.998661	−	0.0517364i	$-0.0164756\pi$
$468$		0			0
$469$		−	6.32456i		−	0.292041i
$470$		0			0
$471$	−2.82843	−	6.32456i	−0.130327	−	0.291420i
$472$		0			0
$473$			11.1803i			0.514073i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−8.48528	+	9.48683i	−0.388514	+	0.434372i
$478$		0			0
$479$			22.3607i			1.02169i	0.859674	+	0.510843i	$0.170667\pi$
$479$			22.3607i			1.02169i	−0.859674	+	0.510843i	$0.829333\pi$
$480$		0			0
$481$	−30.0000			−1.36788
$482$		0			0
$483$	7.07107	−	3.16228i	0.321745	−	0.143889i
$484$		0			0
$485$	7.07107			0.321081
$486$		0			0
$487$			28.4605i			1.28967i	0.764323	+	0.644834i	$0.223074\pi$
$487$			28.4605i			1.28967i	−0.764323	+	0.644834i	$0.776926\pi$
$488$		0			0
$489$	−12.7279	−	28.4605i	−0.575577	−	1.28703i
$490$		0			0
$491$			31.3050i			1.41277i	0.707826	+	0.706386i	$0.249676\pi$
$491$			31.3050i			1.41277i	−0.707826	+	0.706386i	$0.750324\pi$
$492$		0			0
$493$		−	37.9473i		−	1.70906i
$494$		0			0
$495$	10.0000	−	11.1803i	0.449467	−	0.502519i
$496$		0			0
$497$	−4.24264			−0.190308
$498$		0			0
$499$	−17.0000			−0.761025			−0.380512	−	0.924776i	$-0.624252\pi$
$499$	−17.0000			−0.761025			−0.380512	+	0.924776i	$0.624252\pi$
$500$		0			0
$501$	14.0000	+	31.3050i	0.625474	+	1.39860i
$502$		0			0
$503$			40.2492i			1.79462i	0.441397	+	0.897312i	$0.354483\pi$
$503$			40.2492i			1.79462i	−0.441397	+	0.897312i	$0.645517\pi$
$504$		0			0
$505$	−20.0000			−0.889988
$506$		0			0
$507$	−2.12132	−	4.74342i	−0.0942111	−	0.210663i
$508$		0			0
$509$	26.8701			1.19099			0.595497	−	0.803357i	$-0.296955\pi$
$509$	26.8701			1.19099			0.595497	+	0.803357i	$0.296955\pi$
$510$		0			0
$511$	3.00000			0.132712
$512$		0			0
$513$	−9.84924	−	20.3959i	−0.434855	−	0.900501i
$514$		0			0
$515$	14.1421			0.623177
$516$		0			0
$517$	5.00000			0.219900
$518$		0			0
$519$	2.00000	+	4.47214i	0.0877903	+	0.196305i
$520$		0			0
$521$	31.1127			1.36307			0.681536	−	0.731785i	$-0.261312\pi$
$521$	31.1127			1.36307			0.681536	+	0.731785i	$0.261312\pi$
$522$		0			0
$523$			37.9473i			1.65932i	0.558268	+	0.829660i	$0.311466\pi$
$523$			37.9473i			1.65932i	−0.558268	+	0.829660i	$0.688534\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.2132			−0.924062
$528$		0			0
$529$	3.00000			0.130435
$530$		0			0
$531$	2.82843	−	3.16228i	0.122743	−	0.137231i
$532$		0			0
$533$			31.3050i			1.35597i
$534$		0			0
$535$		−	18.9737i		−	0.820303i
$536$		0			0
$537$	−4.00000	−	8.94427i	−0.172613	−	0.385974i
$538$		0			0
$539$		−	13.4164i		−	0.577886i
$540$		0			0
$541$	3.00000			0.128980			0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000			0.128980			0.0644900	+	0.997918i	$0.479458\pi$
$542$		0			0
$543$	30.0000	−	13.4164i	1.28742	−	0.575753i
$544$		0			0
$545$	14.1421			0.605783
$546$		0			0
$547$		−	22.1359i		−	0.946465i	−0.880938	−	0.473232i	$-0.843087\pi$
$547$		−	22.1359i		−	0.946465i	0.880938	−	0.473232i	$-0.156913\pi$
$548$		0			0
$549$	2.00000	−	2.23607i	0.0853579	−	0.0954331i
$550$		0			0
$551$	−16.9706	−	17.8885i	−0.722970	−	0.762078i
$552$		0			0
$553$			12.6491i			0.537895i
$554$		0			0
$555$	−15.0000	−	33.5410i	−0.636715	−	1.42374i
$556$		0			0
$557$		−	2.23607i		−	0.0947452i	−0.998877	−	0.0473726i	$-0.984915\pi$
$557$		−	2.23607i		−	0.0947452i	0.998877	−	0.0473726i	$-0.0150848\pi$
$558$		0			0
$559$		−	15.8114i		−	0.668750i
$560$		0			0
$561$	−10.6066	−	23.7171i	−0.447811	−	1.00134i
$562$		0			0
$563$	18.3848			0.774826			0.387413	−	0.921906i	$-0.373369\pi$
$563$	18.3848			0.774826			0.387413	+	0.921906i	$0.373369\pi$
$564$		0			0
$565$			12.6491i			0.532152i
$566$		0			0
$567$	1.00000	+	8.94427i	0.0419961	+	0.375624i
$568$		0			0
$569$	−1.41421			−0.0592869			−0.0296435	−	0.999561i	$-0.509437\pi$
$569$	−1.41421			−0.0592869			−0.0296435	+	0.999561i	$0.509437\pi$
$570$		0			0
$571$	−26.0000			−1.08807			−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000			−1.08807			−0.544033	+	0.839064i	$0.683103\pi$
$572$		0			0
$573$	17.6777	−	7.90569i	0.738495	−	0.330265i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−45.0000			−1.87337			−0.936687	−	0.350167i	$-0.886125\pi$
$577$	−45.0000			−1.87337			−0.936687	+	0.350167i	$0.886125\pi$
$578$		0			0
$579$	−10.0000	+	4.47214i	−0.415586	+	0.185856i
$580$		0			0
$581$		−	8.94427i		−	0.371071i
$582$		0			0
$583$			9.48683i			0.392904i
$584$		0			0
$585$	−14.1421	+	15.8114i	−0.584705	+	0.653720i
$586$		0			0
$587$		−	38.0132i		−	1.56897i	−0.620147	−	0.784485i	$-0.712927\pi$
$587$		−	38.0132i		−	1.56897i	0.620147	−	0.784485i	$-0.287073\pi$
$588$		0			0
$589$	−10.0000	+	9.48683i	−0.412043	+	0.390898i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	13.4164i		−	0.550946i	−0.961309	−	0.275473i	$-0.911166\pi$
$593$		−	13.4164i		−	0.550946i	0.961309	−	0.275473i	$-0.0888344\pi$
$594$		0			0
$595$	−15.0000			−0.614940
$596$		0			0
$597$	−2.12132	−	4.74342i	−0.0868199	−	0.194135i
$598$		0			0
$599$	−14.1421			−0.577832			−0.288916	−	0.957354i	$-0.593295\pi$
$599$	−14.1421			−0.577832			−0.288916	+	0.957354i	$0.593295\pi$
$600$		0			0
$601$			41.1096i			1.67690i	0.544982	+	0.838448i	$0.316537\pi$
$601$			41.1096i			1.67690i	−0.544982	+	0.838448i	$0.683463\pi$
$602$		0			0
$603$	−14.1421	−	12.6491i	−0.575912	−	0.515112i
$604$		0			0
$605$			13.4164i			0.545455i
$606$		0			0
$607$			15.8114i			0.641764i	0.947119	+	0.320882i	$0.103979\pi$
$607$			15.8114i			0.641764i	−0.947119	+	0.320882i	$0.896021\pi$
$608$		0			0
$609$	4.00000	+	8.94427i	0.162088	+	0.362440i
$610$		0			0
$611$	−7.07107			−0.286065
$612$		0			0
$613$	33.0000			1.33286			0.666429	−	0.745569i	$-0.267822\pi$
$613$	33.0000			1.33286			0.666429	+	0.745569i	$0.267822\pi$
$614$		0			0
$615$	−35.0000	+	15.6525i	−1.41134	+	0.631169i
$616$		0			0
$617$			15.6525i			0.630145i	0.949068	+	0.315072i	$0.102029\pi$
$617$			15.6525i			0.630145i	−0.949068	+	0.315072i	$0.897971\pi$
$618$		0			0
$619$	−32.0000			−1.28619			−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000			−1.28619			−0.643094	+	0.765787i	$0.722350\pi$
$620$		0			0
$621$	7.07107	−	22.1359i	0.283752	−	0.888285i
$622$		0			0
$623$	12.7279			0.509933
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$	−15.6066	−	6.43692i	−0.623268	−	0.257066i
$628$		0			0
$629$	−63.6396			−2.53748
$630$		0			0
$631$	33.0000			1.31371			0.656855	−	0.754017i	$-0.271887\pi$
$631$	33.0000			1.31371			0.656855	+	0.754017i	$0.271887\pi$
$632$		0			0
$633$	35.0000	−	15.6525i	1.39113	−	0.622130i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			18.9737i			0.751764i
$638$		0			0
$639$	−8.48528	+	9.48683i	−0.335673	+	0.375293i
$640$		0			0
$641$	−35.3553			−1.39645			−0.698226	−	0.715877i	$-0.746027\pi$
$641$	−35.3553			−1.39645			−0.698226	+	0.715877i	$0.746027\pi$
$642$		0			0
$643$	7.00000			0.276053			0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000			0.276053			0.138027	+	0.990429i	$0.455924\pi$
$644$		0			0
$645$	17.6777	−	7.90569i	0.696058	−	0.311286i
$646$		0			0
$647$			6.70820i			0.263727i	0.991268	+	0.131863i	$0.0420960\pi$
$647$			6.70820i			0.263727i	−0.991268	+	0.131863i	$0.957904\pi$
$648$		0			0
$649$		−	3.16228i		−	0.124130i
$650$		0			0
$651$	5.00000	−	2.23607i	0.195965	−	0.0876384i
$652$		0			0
$653$		−	42.4853i		−	1.66258i	−0.555840	−	0.831289i	$-0.687603\pi$
$653$		−	42.4853i		−	1.66258i	0.555840	−	0.831289i	$-0.312397\pi$
$654$		0			0
$655$	5.00000			0.195366
$656$		0			0
$657$	6.00000	−	6.70820i	0.234082	−	0.261712i
$658$		0			0
$659$	14.1421			0.550899			0.275450	−	0.961315i	$-0.411173\pi$
$659$	14.1421			0.550899			0.275450	+	0.961315i	$0.411173\pi$
$660$		0			0
$661$			25.2982i			0.983987i	0.870599	+	0.491993i	$0.163732\pi$
$661$			25.2982i			0.983987i	−0.870599	+	0.491993i	$0.836268\pi$
$662$		0			0
$663$	15.0000	+	33.5410i	0.582552	+	1.30263i
$664$		0			0
$665$	−7.07107	+	6.70820i	−0.274204	+	0.260133i
$666$		0			0
$667$		−	25.2982i		−	0.979551i
$668$		0			0
$669$	5.00000	−	2.23607i	0.193311	−	0.0864514i
$670$		0			0
$671$		−	2.23607i		−	0.0863224i
$672$		0			0
$673$		−	22.1359i		−	0.853278i	−0.904422	−	0.426639i	$-0.859698\pi$
$673$		−	22.1359i		−	0.853278i	0.904422	−	0.426639i	$-0.140302\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	36.7696			1.41317			0.706584	−	0.707629i	$-0.250235\pi$
$677$	36.7696			1.41317			0.706584	+	0.707629i	$0.250235\pi$
$678$		0			0
$679$			3.16228i			0.121357i
$680$		0			0
$681$	1.00000	+	2.23607i	0.0383201	+	0.0856863i
$682$		0			0
$683$	−4.24264			−0.162340			−0.0811701	−	0.996700i	$-0.525866\pi$
$683$	−4.24264			−0.162340			−0.0811701	+	0.996700i	$0.525866\pi$
$684$		0			0
$685$	5.00000			0.191040
$686$		0			0
$687$	0.707107	+	1.58114i	0.0269778	+	0.0603242i
$688$		0			0
$689$		−	13.4164i		−	0.511124i
$690$		0			0
$691$	43.0000			1.63580			0.817899	−	0.575362i	$-0.195139\pi$
$691$	43.0000			1.63580			0.817899	+	0.575362i	$0.195139\pi$
$692$		0			0
$693$	5.00000	+	4.47214i	0.189934	+	0.169882i
$694$		0			0
$695$			15.6525i			0.593732i
$696$		0			0
$697$			66.4078i			2.51538i
$698$		0			0
$699$	31.8198	−	14.2302i	1.20354	−	0.538237i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−30.0000	+	28.4605i	−1.13147	+	1.07341i
$704$		0			0
$705$	−3.53553	−	7.90569i	−0.133156	−	0.297746i
$706$		0			0
$707$		−	8.94427i		−	0.336384i
$708$		0			0
$709$	−32.0000			−1.20179			−0.600893	−	0.799330i	$-0.705188\pi$
$709$	−32.0000			−1.20179			−0.600893	+	0.799330i	$0.705188\pi$
$710$		0			0
$711$	28.2843	+	25.2982i	1.06074	+	0.948757i
$712$		0			0
$713$	−14.1421			−0.529627
$714$		0			0
$715$			15.8114i			0.591312i
$716$		0			0
$717$	24.7487	−	11.0680i	0.924259	−	0.413341i
$718$		0			0
$719$		−	33.5410i		−	1.25087i	−0.780277	−	0.625434i	$-0.784922\pi$
$719$		−	33.5410i		−	1.25087i	0.780277	−	0.625434i	$-0.215078\pi$
$720$		0			0
$721$			6.32456i			0.235539i
$722$		0			0
$723$	5.00000	−	2.23607i	0.185952	−	0.0831603i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	49.0000			1.81731			0.908655	−	0.417548i	$-0.137111\pi$
$727$	49.0000			1.81731			0.908655	+	0.417548i	$0.137111\pi$
$728$		0			0
$729$	22.0000	+	15.6525i	0.814815	+	0.579721i
$730$		0			0
$731$		−	33.5410i		−	1.24056i
$732$		0			0
$733$	−8.00000			−0.295487			−0.147743	−	0.989026i	$-0.547201\pi$
$733$	−8.00000			−0.295487			−0.147743	+	0.989026i	$0.547201\pi$
$734$		0			0
$735$	−21.2132	+	9.48683i	−0.782461	+	0.349927i
$736$		0			0
$737$	−14.1421			−0.520932
$738$		0			0
$739$	−47.0000			−1.72892			−0.864461	−	0.502699i	$-0.832340\pi$
$739$	−47.0000			−1.72892			−0.864461	+	0.502699i	$0.832340\pi$
$740$		0			0
$741$	22.0711	+	9.10318i	0.810801	+	0.334414i
$742$		0			0
$743$	−45.2548			−1.66024			−0.830119	−	0.557586i	$-0.811728\pi$
$743$	−45.2548			−1.66024			−0.830119	+	0.557586i	$0.811728\pi$
$744$		0			0
$745$	25.0000			0.915929
$746$		0			0
$747$	−20.0000	−	17.8885i	−0.731762	−	0.654508i
$748$		0			0
$749$	8.48528			0.310045
$750$		0			0
$751$			31.6228i			1.15393i	0.816768	+	0.576966i	$0.195763\pi$
$751$			31.6228i			1.15393i	−0.816768	+	0.576966i	$0.804237\pi$
$752$		0			0
$753$	−10.6066	+	4.74342i	−0.386526	+	0.172860i
$754$		0			0
$755$	35.3553			1.28671
$756$		0			0
$757$	51.0000			1.85363			0.926813	−	0.375523i	$-0.122537\pi$
$757$	51.0000			1.85363			0.926813	+	0.375523i	$0.122537\pi$
$758$		0			0
$759$	−7.07107	−	15.8114i	−0.256664	−	0.573917i
$760$		0			0
$761$		−	29.0689i		−	1.05375i	−0.849944	−	0.526873i	$-0.823364\pi$
$761$		−	29.0689i		−	1.05375i	0.849944	−	0.526873i	$-0.176636\pi$
$762$		0			0
$763$			6.32456i			0.228964i
$764$		0			0
$765$	−30.0000	+	33.5410i	−1.08465	+	1.21268i
$766$		0			0
$767$			4.47214i			0.161479i
$768$		0			0
$769$	7.00000			0.252426			0.126213	−	0.992003i	$-0.459718\pi$
$769$	7.00000			0.252426			0.126213	+	0.992003i	$0.459718\pi$
$770$		0			0
$771$	16.0000	+	35.7771i	0.576226	+	1.28848i
$772$		0			0
$773$	−18.3848			−0.661254			−0.330627	−	0.943761i	$-0.607260\pi$
$773$	−18.3848			−0.661254			−0.330627	+	0.943761i	$0.607260\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	15.0000	−	6.70820i	0.538122	−	0.240655i
$778$		0			0
$779$	29.6985	+	31.3050i	1.06406	+	1.12162i
$780$		0			0
$781$			9.48683i			0.339466i
$782$		0			0
$783$	28.0000	+	8.94427i	1.00064	+	0.319642i
$784$		0			0
$785$			8.94427i			0.319235i
$786$		0			0
$787$		−	25.2982i		−	0.901784i	−0.892578	−	0.450892i	$-0.851106\pi$
$787$		−	25.2982i		−	0.901784i	0.892578	−	0.450892i	$-0.148894\pi$
$788$		0			0
$789$	−45.9619	+	20.5548i	−1.63629	+	0.731770i
$790$		0			0
$791$	−5.65685			−0.201135
$792$		0			0
$793$			3.16228i			0.112296i
$794$		0			0
$795$	15.0000	−	6.70820i	0.531995	−	0.237915i
$796$		0			0
$797$	19.7990			0.701316			0.350658	−	0.936504i	$-0.385958\pi$
$797$	19.7990			0.701316			0.350658	+	0.936504i	$0.385958\pi$
$798$		0			0
$799$	−15.0000			−0.530662
$800$		0			0
$801$	25.4558	−	28.4605i	0.899438	−	1.00560i
$802$		0			0
$803$		−	6.70820i		−	0.236727i
$804$		0			0
$805$	−10.0000			−0.352454
$806$		0			0
$807$	−15.0000	−	33.5410i	−0.528025	−	1.18070i
$808$		0			0
$809$		−	2.23607i		−	0.0786160i	−0.999227	−	0.0393080i	$-0.987485\pi$
$809$		−	2.23607i		−	0.0786160i	0.999227	−	0.0393080i	$-0.0125153\pi$
$810$		0			0
$811$			47.4342i			1.66564i	0.553545	+	0.832819i	$0.313275\pi$
$811$			47.4342i			1.66564i	−0.553545	+	0.832819i	$0.686725\pi$
$812$		0			0
$813$	4.24264	+	9.48683i	0.148796	+	0.332718i
$814$		0			0
$815$			40.2492i			1.40987i
$816$		0			0
$817$	−15.0000	−	15.8114i	−0.524784	−	0.553170i
$818$		0			0
$819$	−7.07107	−	6.32456i	−0.247083	−	0.220998i
$820$		0			0
$821$			51.4296i			1.79490i	0.441112	+	0.897452i	$0.354584\pi$
$821$			51.4296i			1.79490i	−0.441112	+	0.897452i	$0.645416\pi$
$822$		0			0
$823$	23.0000			0.801730			0.400865	−	0.916137i	$-0.368710\pi$
$823$	23.0000			0.801730			0.400865	+	0.916137i	$0.368710\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−25.4558			−0.885186			−0.442593	−	0.896723i	$-0.645941\pi$
$827$	−25.4558			−0.885186			−0.442593	+	0.896723i	$0.645941\pi$
$828$		0			0
$829$		−	18.9737i		−	0.658983i	−0.944159	−	0.329491i	$-0.893123\pi$
$829$		−	18.9737i		−	0.658983i	0.944159	−	0.329491i	$-0.106877\pi$
$830$		0			0
$831$	−13.4350	−	30.0416i	−0.466056	−	1.04213i
$832$		0			0
$833$			40.2492i			1.39455i
$834$		0			0
$835$		−	44.2719i		−	1.53209i
$836$		0			0
$837$	5.00000	−	15.6525i	0.172825	−	0.541029i
$838$		0			0
$839$	−19.7990			−0.683537			−0.341769	−	0.939784i	$-0.611026\pi$
$839$	−19.7990			−0.683537			−0.341769	+	0.939784i	$0.611026\pi$
$840$		0			0
$841$	3.00000			0.103448
$842$		0			0
$843$	10.0000	+	22.3607i	0.344418	+	0.770143i
$844$		0			0
$845$			6.70820i			0.230769i
$846$		0			0
$847$	−6.00000			−0.206162
$848$		0			0
$849$	17.6777	+	39.5285i	0.606696	+	1.35661i
$850$		0			0
$851$	−42.4264			−1.45436
$852$		0			0
$853$	28.0000			0.958702			0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000			0.958702			0.479351	+	0.877623i	$0.340872\pi$
$854$		0			0
$855$	0.857864	+	29.2278i	0.0293383	+	0.999570i
$856$		0			0
$857$	1.41421			0.0483086			0.0241543	−	0.999708i	$-0.492311\pi$
$857$	1.41421			0.0483086			0.0241543	+	0.999708i	$0.492311\pi$
$858$		0			0
$859$	−13.0000			−0.443554			−0.221777	−	0.975097i	$-0.571186\pi$
$859$	−13.0000			−0.443554			−0.221777	+	0.975097i	$0.571186\pi$
$860$		0			0
$861$	−7.00000	−	15.6525i	−0.238559	−	0.533435i
$862$		0			0
$863$	−38.1838			−1.29979			−0.649895	−	0.760024i	$-0.725187\pi$
$863$	−38.1838			−1.29979			−0.649895	+	0.760024i	$0.725187\pi$
$864$		0			0
$865$		−	6.32456i		−	0.215041i
$866$		0			0
$867$	19.7990	+	44.2719i	0.672409	+	1.50355i
$868$		0			0
$869$	28.2843			0.959478
$870$		0			0
$871$	20.0000			0.677674
$872$		0			0
$873$	7.07107	+	6.32456i	0.239319	+	0.214054i
$874$		0			0
$875$		−	11.1803i		−	0.377964i
$876$		0			0
$877$		−	31.6228i		−	1.06783i	−0.845540	−	0.533913i	$-0.820721\pi$
$877$		−	31.6228i		−	1.06783i	0.845540	−	0.533913i	$-0.179279\pi$
$878$		0			0
$879$	14.0000	+	31.3050i	0.472208	+	1.05589i
$880$		0			0
$881$			6.70820i			0.226005i	0.993595	+	0.113003i	$0.0360468\pi$
$881$			6.70820i			0.226005i	−0.993595	+	0.113003i	$0.963953\pi$
$882$		0			0
$883$	33.0000			1.11054			0.555269	−	0.831671i	$-0.312615\pi$
$883$	33.0000			1.11054			0.555269	+	0.831671i	$0.312615\pi$
$884$		0			0
$885$	−5.00000	+	2.23607i	−0.168073	+	0.0751646i
$886$		0			0
$887$	32.5269			1.09215			0.546073	−	0.837737i	$-0.316122\pi$
$887$	32.5269			1.09215			0.546073	+	0.837737i	$0.316122\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	20.0000	−	2.23607i	0.670025	−	0.0749111i
$892$		0			0
$893$	−7.07107	+	6.70820i	−0.236624	+	0.224481i
$894$		0			0
$895$			12.6491i			0.422813i
$896$		0			0
$897$	10.0000	+	22.3607i	0.333890	+	0.746601i
$898$		0			0
$899$		−	17.8885i		−	0.596616i
$900$		0			0
$901$		−	28.4605i		−	0.948157i
$902$		0			0
$903$	3.53553	+	7.90569i	0.117655	+	0.263085i
$904$		0			0
$905$	−42.4264			−1.41030
$906$		0			0
$907$		−	15.8114i		−	0.525009i	−0.964931	−	0.262504i	$-0.915452\pi$
$907$		−	15.8114i		−	0.525009i	0.964931	−	0.262504i	$-0.0845484\pi$
$908$		0			0
$909$	−20.0000	−	17.8885i	−0.663358	−	0.593326i
$910$		0			0
$911$	53.7401			1.78049			0.890245	−	0.455483i	$-0.150533\pi$
$911$	53.7401			1.78049			0.890245	+	0.455483i	$0.150533\pi$
$912$		0			0
$913$	−20.0000			−0.661903
$914$		0			0
$915$	−3.53553	+	1.58114i	−0.116881	+	0.0522708i
$916$		0			0
$917$			2.23607i			0.0738415i
$918$		0			0
$919$	18.0000			0.593765			0.296883	−	0.954914i	$-0.404053\pi$
$919$	18.0000			0.593765			0.296883	+	0.954914i	$0.404053\pi$
$920$		0			0
$921$	−30.0000	+	13.4164i	−0.988534	+	0.442086i
$922$		0			0
$923$		−	13.4164i		−	0.441606i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	14.1421	+	12.6491i	0.464489	+	0.415451i
$928$		0			0
$929$		−	44.7214i		−	1.46726i	−0.679549	−	0.733630i	$-0.737825\pi$
$929$		−	44.7214i		−	1.46726i	0.679549	−	0.733630i	$-0.262175\pi$
$930$		0			0
$931$	18.0000	+	18.9737i	0.589926	+	0.621837i
$932$		0			0
$933$	−38.8909	+	17.3925i	−1.27323	+	0.569406i
$934$		0			0
$935$			33.5410i			1.09691i
$936$		0			0
$937$	21.0000			0.686040			0.343020	−	0.939328i	$-0.388550\pi$
$937$	21.0000			0.686040			0.343020	+	0.939328i	$0.388550\pi$
$938$		0			0
$939$	−14.1421	−	31.6228i	−0.461511	−	1.03197i
$940$		0			0
$941$	−28.2843			−0.922041			−0.461020	−	0.887390i	$-0.652517\pi$
$941$	−28.2843			−0.922041			−0.461020	+	0.887390i	$0.652517\pi$
$942$		0			0
$943$			44.2719i			1.44169i
$944$		0			0
$945$	3.53553	−	11.0680i	0.115011	−	0.360041i
$946$		0			0
$947$			58.1378i			1.88922i	0.328190	+	0.944612i	$0.393561\pi$
$947$			58.1378i			1.88922i	−0.328190	+	0.944612i	$0.606439\pi$
$948$		0			0
$949$			9.48683i			0.307956i
$950$		0			0
$951$	1.00000	+	2.23607i	0.0324272	+	0.0725095i
$952$		0			0
$953$	25.4558			0.824596			0.412298	−	0.911049i	$-0.364726\pi$
$953$	25.4558			0.824596			0.412298	+	0.911049i	$0.364726\pi$
$954$		0			0
$955$	−25.0000			−0.808981
$956$		0			0
$957$	20.0000	−	8.94427i	0.646508	−	0.289127i
$958$		0			0
$959$			2.23607i			0.0722064i
$960$		0			0
$961$	21.0000			0.677419
$962$		0			0
$963$	16.9706	−	18.9737i	0.546869	−	0.611418i
$964$		0			0
$965$	14.1421			0.455251
$966$		0			0
$967$	26.0000			0.836104			0.418052	−	0.908423i	$-0.362713\pi$
$967$	26.0000			0.836104			0.418052	+	0.908423i	$0.362713\pi$
$968$		0			0
$969$	46.8198	+	19.3108i	1.50407	+	0.620351i
$970$		0			0
$971$	−16.9706			−0.544611			−0.272306	−	0.962211i	$-0.587786\pi$
$971$	−16.9706			−0.544611			−0.272306	+	0.962211i	$0.587786\pi$
$972$		0			0
$973$	−7.00000			−0.224410
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.0416			0.769160			0.384580	−	0.923092i	$-0.374346\pi$
$977$	24.0416			0.769160			0.384580	+	0.923092i	$0.374346\pi$
$978$		0			0
$979$		−	28.4605i		−	0.909601i
$980$		0			0
$981$	14.1421	+	12.6491i	0.451524	+	0.403855i
$982$		0			0
$983$	48.0833			1.53362			0.766809	−	0.641875i	$-0.221843\pi$
$983$	48.0833			1.53362			0.766809	+	0.641875i	$0.221843\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	3.53553	−	1.58114i	0.112537	−	0.0503282i
$988$		0			0
$989$		−	22.3607i		−	0.711028i
$990$		0			0
$991$			28.4605i			0.904078i	0.891998	+	0.452039i	$0.149303\pi$
$991$			28.4605i			0.904078i	−0.891998	+	0.452039i	$0.850697\pi$
$992$		0			0
$993$	−20.0000	+	8.94427i	−0.634681	+	0.283838i
$994$		0			0
$995$			6.70820i			0.212664i
$996$		0			0
$997$	25.0000			0.791758			0.395879	−	0.918303i	$-0.370440\pi$
$997$	25.0000			0.791758			0.395879	+	0.918303i	$0.370440\pi$
$998$		0			0
$999$	15.0000	−	46.9574i	0.474579	−	1.48567i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.f.f.113.1		4
3.2	odd	2	inner	912.2.f.f.113.3		4
4.3	odd	2		57.2.d.a.56.2	yes	4
12.11	even	2		57.2.d.a.56.4	yes	4
19.18	odd	2	inner	912.2.f.f.113.4		4
57.56	even	2	inner	912.2.f.f.113.2		4
76.75	even	2		57.2.d.a.56.3	yes	4
228.227	odd	2		57.2.d.a.56.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.d.a.56.1	✓	4	228.227	odd	2
57.2.d.a.56.2	yes	4	4.3	odd	2
57.2.d.a.56.3	yes	4	76.75	even	2
57.2.d.a.56.4	yes	4	12.11	even	2
912.2.f.f.113.1		4	1.1	even	1	trivial
912.2.f.f.113.2		4	57.56	even	2	inner
912.2.f.f.113.3		4	3.2	odd	2	inner
912.2.f.f.113.4		4	19.18	odd	2	inner

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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.58114i) q^{3} +2.23607i q^{5} -1.00000 q^{7} +(-2.00000 + 2.23607i) q^{9} +O(q^{10})\)	\(q+(-0.707107 - 1.58114i) q^{3} +2.23607i q^{5} -1.00000 q^{7} +(-2.00000 + 2.23607i) q^{9} +2.23607i q^{11} -3.16228i q^{13} +(3.53553 - 1.58114i) q^{15} -6.70820i q^{17} +(-3.00000 - 3.16228i) q^{19} +(0.707107 + 1.58114i) q^{21} -4.47214i q^{23} +(4.94975 + 1.58114i) q^{27} +5.65685 q^{29} -3.16228i q^{31} +(3.53553 - 1.58114i) q^{33} -2.23607i q^{35} -9.48683i q^{37} +(-5.00000 + 2.23607i) q^{39} -9.89949 q^{41} +5.00000 q^{43} +(-5.00000 - 4.47214i) q^{45} -2.23607i q^{47} -6.00000 q^{49} +(-10.6066 + 4.74342i) q^{51} +4.24264 q^{53} -5.00000 q^{55} +(-2.87868 + 6.97948i) q^{57} -1.41421 q^{59} -1.00000 q^{61} +(2.00000 - 2.23607i) q^{63} +7.07107 q^{65} +6.32456i q^{67} +(-7.07107 + 3.16228i) q^{69} +4.24264 q^{71} -3.00000 q^{73} -2.23607i q^{77} -12.6491i q^{79} +(-1.00000 - 8.94427i) q^{81} +8.94427i q^{83} +15.0000 q^{85} +(-4.00000 - 8.94427i) q^{87} -12.7279 q^{89} +3.16228i q^{91} +(-5.00000 + 2.23607i) q^{93} +(7.07107 - 6.70820i) q^{95} -3.16228i q^{97} +(-5.00000 - 4.47214i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 8 q^{9}+O(q^{10}) \) 4 * q - 4 * q^7 - 8 * q^9	\( 4 q - 4 q^{7} - 8 q^{9} - 12 q^{19} - 20 q^{39} + 20 q^{43} - 20 q^{45} - 24 q^{49} - 20 q^{55} - 20 q^{57} - 4 q^{61} + 8 q^{63} - 12 q^{73} - 4 q^{81} + 60 q^{85} - 16 q^{87} - 20 q^{93} - 20 q^{99}+O(q^{100}) \) 4 * q - 4 * q^7 - 8 * q^9 - 12 * q^19 - 20 * q^39 + 20 * q^43 - 20 * q^45 - 24 * q^49 - 20 * q^55 - 20 * q^57 - 4 * q^61 + 8 * q^63 - 12 * q^73 - 4 * q^81 + 60 * q^85 - 16 * q^87 - 20 * q^93 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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