LMFDB - Embedded newform 4032.2.v.b.3599.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("4032.1583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.v (of order $4$, degree $2$, 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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3599.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4032.3599
Dual form			4032.2.v.b.1583.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 - 1.41421i) q^{5} -1.00000 q^{7} +O(q^{10})$	$q+(1.41421 - 1.41421i) q^{5} -1.00000 q^{7} +(-2.82843 - 2.82843i) q^{11} +(4.00000 - 4.00000i) q^{13} +5.65685i q^{17} +(-4.00000 - 4.00000i) q^{19} -1.41421i q^{23} +1.00000i q^{25} +(-5.65685 - 5.65685i) q^{29} +(-1.41421 + 1.41421i) q^{35} +(-5.00000 - 5.00000i) q^{37} +8.48528 q^{41} +(-7.00000 + 7.00000i) q^{43} +1.00000 q^{49} +(-7.07107 + 7.07107i) q^{53} -8.00000 q^{55} +(5.65685 + 5.65685i) q^{59} +(4.00000 - 4.00000i) q^{61} -11.3137i q^{65} +(-7.00000 - 7.00000i) q^{67} +7.07107i q^{71} -6.00000i q^{73} +(2.82843 + 2.82843i) q^{77} +(2.82843 - 2.82843i) q^{83} +(8.00000 + 8.00000i) q^{85} -16.9706 q^{89} +(-4.00000 + 4.00000i) q^{91} -11.3137 q^{95} -10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^7	$ 4 q - 4 q^{7} + 16 q^{13} - 16 q^{19} - 20 q^{37} - 28 q^{43} + 4 q^{49} - 32 q^{55} + 16 q^{61} - 28 q^{67} + 32 q^{85} - 16 q^{91} - 40 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 + 16 * q^13 - 16 * q^19 - 20 * q^37 - 28 * q^43 + 4 * q^49 - 32 * q^55 + 16 * q^61 - 28 * q^67 + 32 * q^85 - 16 * q^91 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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			0
$3$		0			0
$4$		0			0
$5$	1.41421	−	1.41421i	0.632456	−	0.632456i	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	1.41421	−	1.41421i	0.632456	−	0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.82843	−	2.82843i	−0.852803	−	0.852803i	0.137675	−	0.990478i	$-0.456037\pi$
$11$	−2.82843	−	2.82843i	−0.852803	−	0.852803i	−0.990478	+	0.137675i	$0.956037\pi$
$12$		0			0
$13$	4.00000	−	4.00000i	1.10940	−	1.10940i	0.116171	−	0.993229i	$-0.462938\pi$
$13$	4.00000	−	4.00000i	1.10940	−	1.10940i	0.993229	−	0.116171i	$-0.0370621\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			5.65685i			1.37199i	0.727607	+	0.685994i	$0.240633\pi$
$17$			5.65685i			1.37199i	−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$	−4.00000	−	4.00000i	−0.917663	−	0.917663i	0.0791961	−	0.996859i	$-0.474765\pi$
$19$	−4.00000	−	4.00000i	−0.917663	−	0.917663i	−0.996859	+	0.0791961i	$0.974765\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.41421i		−	0.294884i	−0.989071	−	0.147442i	$-0.952896\pi$
$23$		−	1.41421i		−	0.294884i	0.989071	−	0.147442i	$-0.0471040\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.65685	−	5.65685i	−1.05045	−	1.05045i	−0.998658	−	0.0517937i	$-0.983506\pi$
$29$	−5.65685	−	5.65685i	−1.05045	−	1.05045i	−0.0517937	−	0.998658i	$-0.516494\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.41421	+	1.41421i	−0.239046	+	0.239046i
$36$		0			0
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.48528			1.32518			0.662589	−	0.748983i	$-0.269458\pi$
$41$	8.48528			1.32518			0.662589	+	0.748983i	$0.269458\pi$
$42$		0			0
$43$	−7.00000	+	7.00000i	−1.06749	+	1.06749i	−0.0699387	+	0.997551i	$0.522280\pi$
$43$	−7.00000	+	7.00000i	−1.06749	+	1.06749i	−0.997551	+	0.0699387i	$0.977720\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.07107	+	7.07107i	−0.971286	+	0.971286i	−0.999599	−	0.0283132i	$-0.990986\pi$
$53$	−7.07107	+	7.07107i	−0.971286	+	0.971286i	0.0283132	+	0.999599i	$0.490986\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.65685	+	5.65685i	0.736460	+	0.736460i	0.971891	−	0.235431i	$-0.0756503\pi$
$59$	5.65685	+	5.65685i	0.736460	+	0.736460i	−0.235431	+	0.971891i	$0.575650\pi$
$60$		0			0
$61$	4.00000	−	4.00000i	0.512148	−	0.512148i	−0.403036	−	0.915184i	$-0.632045\pi$
$61$	4.00000	−	4.00000i	0.512148	−	0.512148i	0.915184	+	0.403036i	$0.132045\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	11.3137i		−	1.40329i
$66$		0			0
$67$	−7.00000	−	7.00000i	−0.855186	−	0.855186i	0.135580	−	0.990766i	$-0.456710\pi$
$67$	−7.00000	−	7.00000i	−0.855186	−	0.855186i	−0.990766	+	0.135580i	$0.956710\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.07107i			0.839181i	0.907713	+	0.419591i	$0.137826\pi$
$71$			7.07107i			0.839181i	−0.907713	+	0.419591i	$0.862174\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.82843	+	2.82843i	0.322329	+	0.322329i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.82843	−	2.82843i	0.310460	−	0.310460i	−0.534628	−	0.845088i	$-0.679548\pi$
$83$	2.82843	−	2.82843i	0.310460	−	0.310460i	0.845088	+	0.534628i	$0.179548\pi$
$84$		0			0
$85$	8.00000	+	8.00000i	0.867722	+	0.867722i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−16.9706			−1.79888			−0.899438	−	0.437048i	$-0.856024\pi$
$89$	−16.9706			−1.79888			−0.899438	+	0.437048i	$0.856024\pi$
$90$		0			0
$91$	−4.00000	+	4.00000i	−0.419314	+	0.419314i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.3137			−1.16076
$96$		0			0
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.07107	+	7.07107i	−0.703598	+	0.703598i	−0.965181	−	0.261583i	$-0.915755\pi$
$101$	−7.07107	+	7.07107i	−0.703598	+	0.703598i	0.261583	+	0.965181i	$0.415755\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.07107	−	7.07107i	−0.683586	−	0.683586i	0.277220	−	0.960806i	$-0.410587\pi$
$107$	−7.07107	−	7.07107i	−0.683586	−	0.683586i	−0.960806	+	0.277220i	$0.910587\pi$
$108$		0			0
$109$	13.0000	−	13.0000i	1.24517	−	1.24517i	0.287348	−	0.957826i	$-0.407226\pi$
$109$	13.0000	−	13.0000i	1.24517	−	1.24517i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.41421i			0.133038i	0.997785	+	0.0665190i	$0.0211893\pi$
$113$			1.41421i			0.133038i	−0.997785	+	0.0665190i	$0.978811\pi$
$114$		0			0
$115$	−2.00000	−	2.00000i	−0.186501	−	0.186501i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	5.65685i		−	0.518563i
$120$		0			0
$121$			5.00000i			0.454545i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.48528	+	8.48528i	0.758947	+	0.758947i
$126$		0			0
$127$			18.0000i			1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$			18.0000i			1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−14.1421	+	14.1421i	−1.23560	+	1.23560i	−0.273824	+	0.961780i	$0.588289\pi$
$131$	−14.1421	+	14.1421i	−1.23560	+	1.23560i	−0.961780	+	0.273824i	$0.911711\pi$
$132$		0			0
$133$	4.00000	+	4.00000i	0.346844	+	0.346844i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.24264			−0.362473			−0.181237	−	0.983440i	$-0.558010\pi$
$137$	−4.24264			−0.362473			−0.181237	+	0.983440i	$0.558010\pi$
$138$		0			0
$139$	14.0000	−	14.0000i	1.18746	−	1.18746i	0.209698	−	0.977766i	$-0.432752\pi$
$139$	14.0000	−	14.0000i	1.18746	−	1.18746i	0.977766	−	0.209698i	$-0.0672482\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−22.6274			−1.89220
$144$		0			0
$145$	−16.0000			−1.32873
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.3137	+	11.3137i	−0.926855	+	0.926855i	−0.997501	−	0.0706463i	$-0.977494\pi$
$149$	−11.3137	+	11.3137i	−0.926855	+	0.926855i	0.0706463	+	0.997501i	$0.477494\pi$
$150$		0			0
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.00000	+	2.00000i	−0.159617	+	0.159617i	−0.782397	−	0.622780i	$-0.786003\pi$
$157$	−2.00000	+	2.00000i	−0.159617	+	0.159617i	0.622780	+	0.782397i	$0.286003\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1.41421i			0.111456i
$162$		0			0
$163$	−13.0000	−	13.0000i	−1.01824	−	1.01824i	−0.999831	−	0.0184080i	$-0.994140\pi$
$163$	−13.0000	−	13.0000i	−1.01824	−	1.01824i	−0.0184080	−	0.999831i	$-0.505860\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	14.1421i		−	1.09435i	−0.837018	−	0.547176i	$-0.815703\pi$
$167$		−	14.1421i		−	1.09435i	0.837018	−	0.547176i	$-0.184297\pi$
$168$		0			0
$169$		−	19.0000i		−	1.46154i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.41421	−	1.41421i	−0.107521	−	0.107521i	0.651300	−	0.758820i	$-0.274224\pi$
$173$	−1.41421	−	1.41421i	−0.107521	−	0.107521i	−0.758820	+	0.651300i	$0.774224\pi$
$174$		0			0
$175$		−	1.00000i		−	0.0755929i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7.07107	−	7.07107i	0.528516	−	0.528516i	−0.391613	−	0.920130i	$-0.628083\pi$
$179$	7.07107	−	7.07107i	0.528516	−	0.528516i	0.920130	+	0.391613i	$0.128083\pi$
$180$		0			0
$181$	−8.00000	−	8.00000i	−0.594635	−	0.594635i	0.344245	−	0.938880i	$-0.388135\pi$
$181$	−8.00000	−	8.00000i	−0.594635	−	0.594635i	−0.938880	+	0.344245i	$0.888135\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−14.1421			−1.03975
$186$		0			0
$187$	16.0000	−	16.0000i	1.17004	−	1.17004i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.7279			−0.920960			−0.460480	−	0.887670i	$-0.652323\pi$
$191$	−12.7279			−0.920960			−0.460480	+	0.887670i	$0.652323\pi$
$192$		0			0
$193$	14.0000			1.00774			0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.82843	+	2.82843i	−0.201517	+	0.201517i	−0.800650	−	0.599133i	$-0.795512\pi$
$197$	−2.82843	+	2.82843i	−0.201517	+	0.201517i	0.599133	+	0.800650i	$0.295512\pi$
$198$		0			0
$199$	16.0000			1.13421			0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000			1.13421			0.567105	+	0.823646i	$0.308063\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	5.65685	+	5.65685i	0.397033	+	0.397033i
$204$		0			0
$205$	12.0000	−	12.0000i	0.838116	−	0.838116i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			22.6274i			1.56517i
$210$		0			0
$211$	−19.0000	−	19.0000i	−1.30801	−	1.30801i	−0.922847	−	0.385167i	$-0.874144\pi$
$211$	−19.0000	−	19.0000i	−1.30801	−	1.30801i	−0.385167	−	0.922847i	$-0.625856\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			19.7990i			1.35028i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	22.6274	+	22.6274i	1.52208	+	1.52208i
$222$		0			0
$223$		−	12.0000i		−	0.803579i	−0.915732	−	0.401790i	$-0.868388\pi$
$223$		−	12.0000i		−	0.803579i	0.915732	−	0.401790i	$-0.131612\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.65685	+	5.65685i	−0.375459	+	0.375459i	−0.869461	−	0.494002i	$-0.835534\pi$
$227$	−5.65685	+	5.65685i	−0.375459	+	0.375459i	0.494002	+	0.869461i	$0.335534\pi$
$228$		0			0
$229$	4.00000	+	4.00000i	0.264327	+	0.264327i	0.826809	−	0.562482i	$-0.190153\pi$
$229$	4.00000	+	4.00000i	0.264327	+	0.264327i	−0.562482	+	0.826809i	$0.690153\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.24264			−0.277945			−0.138972	−	0.990296i	$-0.544380\pi$
$233$	−4.24264			−0.277945			−0.138972	+	0.990296i	$0.544380\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.24264			0.274434			0.137217	−	0.990541i	$-0.456184\pi$
$239$	4.24264			0.274434			0.137217	+	0.990541i	$0.456184\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.41421	−	1.41421i	0.0903508	−	0.0903508i
$246$		0			0
$247$	−32.0000			−2.03611
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.82843	−	2.82843i	−0.178529	−	0.178529i	0.612185	−	0.790714i	$-0.290291\pi$
$251$	−2.82843	−	2.82843i	−0.178529	−	0.178529i	−0.790714	+	0.612185i	$0.790291\pi$
$252$		0			0
$253$	−4.00000	+	4.00000i	−0.251478	+	0.251478i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	11.3137i		−	0.705730i	−0.935674	−	0.352865i	$-0.885208\pi$
$257$		−	11.3137i		−	0.705730i	0.935674	−	0.352865i	$-0.114792\pi$
$258$		0			0
$259$	5.00000	+	5.00000i	0.310685	+	0.310685i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	9.89949i		−	0.610429i	−0.952284	−	0.305215i	$-0.901272\pi$
$263$		−	9.89949i		−	0.610429i	0.952284	−	0.305215i	$-0.0987282\pi$
$264$		0			0
$265$			20.0000i			1.22859i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.41421	−	1.41421i	−0.0862261	−	0.0862261i	0.662678	−	0.748904i	$-0.269420\pi$
$269$	−1.41421	−	1.41421i	−0.0862261	−	0.0862261i	−0.748904	+	0.662678i	$0.769420\pi$
$270$		0			0
$271$			12.0000i			0.728948i	0.931214	+	0.364474i	$0.118751\pi$
$271$			12.0000i			0.728948i	−0.931214	+	0.364474i	$0.881249\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.82843	−	2.82843i	0.170561	−	0.170561i
$276$		0			0
$277$	−11.0000	−	11.0000i	−0.660926	−	0.660926i	0.294672	−	0.955598i	$-0.404789\pi$
$277$	−11.0000	−	11.0000i	−0.660926	−	0.660926i	−0.955598	+	0.294672i	$0.904789\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4.24264			0.253095			0.126547	−	0.991961i	$-0.459610\pi$
$281$	4.24264			0.253095			0.126547	+	0.991961i	$0.459610\pi$
$282$		0			0
$283$	2.00000	−	2.00000i	0.118888	−	0.118888i	−0.645160	−	0.764048i	$-0.723209\pi$
$283$	2.00000	−	2.00000i	0.118888	−	0.118888i	0.764048	+	0.645160i	$0.223209\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.48528			−0.500870
$288$		0			0
$289$	−15.0000			−0.882353
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.89949	−	9.89949i	0.578335	−	0.578335i	−0.356110	−	0.934444i	$-0.615897\pi$
$293$	9.89949	−	9.89949i	0.578335	−	0.578335i	0.934444	+	0.356110i	$0.115897\pi$
$294$		0			0
$295$	16.0000			0.931556
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5.65685	−	5.65685i	−0.327144	−	0.327144i
$300$		0			0
$301$	7.00000	−	7.00000i	0.403473	−	0.403473i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	11.3137i		−	0.647821i
$306$		0			0
$307$	8.00000	+	8.00000i	0.456584	+	0.456584i	0.897532	−	0.440948i	$-0.145358\pi$
$307$	8.00000	+	8.00000i	0.456584	+	0.456584i	−0.440948	+	0.897532i	$0.645358\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			11.3137i			0.641542i	0.947157	+	0.320771i	$0.103942\pi$
$311$			11.3137i			0.641542i	−0.947157	+	0.320771i	$0.896058\pi$
$312$		0			0
$313$			18.0000i			1.01742i	0.860938	+	0.508710i	$0.169877\pi$
$313$			18.0000i			1.01742i	−0.860938	+	0.508710i	$0.830123\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.41421	−	1.41421i	−0.0794301	−	0.0794301i	0.666276	−	0.745706i	$-0.267887\pi$
$317$	−1.41421	−	1.41421i	−0.0794301	−	0.0794301i	−0.745706	+	0.666276i	$0.767887\pi$
$318$		0			0
$319$			32.0000i			1.79166i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	22.6274	−	22.6274i	1.25902	−	1.25902i
$324$		0			0
$325$	4.00000	+	4.00000i	0.221880	+	0.221880i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−13.0000	+	13.0000i	−0.714545	+	0.714545i	−0.967483	−	0.252938i	$-0.918603\pi$
$331$	−13.0000	+	13.0000i	−0.714545	+	0.714545i	0.252938	+	0.967483i	$0.418603\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−19.7990			−1.08173
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.07107	−	7.07107i	−0.379595	−	0.379595i	0.491361	−	0.870956i	$-0.336500\pi$
$347$	−7.07107	−	7.07107i	−0.379595	−	0.379595i	−0.870956	+	0.491361i	$0.836500\pi$
$348$		0			0
$349$	−2.00000	+	2.00000i	−0.107058	+	0.107058i	−0.758607	−	0.651549i	$-0.774119\pi$
$349$	−2.00000	+	2.00000i	−0.107058	+	0.107058i	0.651549	+	0.758607i	$0.274119\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2.82843i		−	0.150542i	−0.997163	−	0.0752710i	$-0.976018\pi$
$353$		−	2.82843i		−	0.150542i	0.997163	−	0.0752710i	$-0.0239822\pi$
$354$		0			0
$355$	10.0000	+	10.0000i	0.530745	+	0.530745i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			32.5269i			1.71670i	0.513061	+	0.858352i	$0.328512\pi$
$359$			32.5269i			1.71670i	−0.513061	+	0.858352i	$0.671488\pi$
$360$		0			0
$361$			13.0000i			0.684211i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.48528	−	8.48528i	−0.444140	−	0.444140i
$366$		0			0
$367$		−	24.0000i		−	1.25279i	−0.779506	−	0.626395i	$-0.784530\pi$
$367$		−	24.0000i		−	1.25279i	0.779506	−	0.626395i	$-0.215470\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	7.07107	−	7.07107i	0.367112	−	0.367112i
$372$		0			0
$373$	−5.00000	−	5.00000i	−0.258890	−	0.258890i	0.565712	−	0.824603i	$-0.308601\pi$
$373$	−5.00000	−	5.00000i	−0.258890	−	0.258890i	−0.824603	+	0.565712i	$0.808601\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−45.2548			−2.33074
$378$		0			0
$379$	17.0000	−	17.0000i	0.873231	−	0.873231i	−0.119592	−	0.992823i	$-0.538159\pi$
$379$	17.0000	−	17.0000i	0.873231	−	0.873231i	0.992823	+	0.119592i	$0.0381586\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.48528			0.433578			0.216789	−	0.976219i	$-0.430442\pi$
$383$	8.48528			0.433578			0.216789	+	0.976219i	$0.430442\pi$
$384$		0			0
$385$	8.00000			0.407718
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.07107	+	7.07107i	−0.358517	+	0.358517i	−0.863266	−	0.504749i	$-0.831585\pi$
$389$	−7.07107	+	7.07107i	−0.358517	+	0.358517i	0.504749	+	0.863266i	$0.331585\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−2.00000	+	2.00000i	−0.100377	+	0.100377i	−0.755512	−	0.655135i	$-0.772612\pi$
$397$	−2.00000	+	2.00000i	−0.100377	+	0.100377i	0.655135	+	0.755512i	$0.272612\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	24.0416i		−	1.20058i	−0.799782	−	0.600291i	$-0.795051\pi$
$401$		−	24.0416i		−	1.20058i	0.799782	−	0.600291i	$-0.204949\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			28.2843i			1.40200i
$408$		0			0
$409$		−	18.0000i		−	0.890043i	−0.895520	−	0.445021i	$-0.853196\pi$
$409$		−	18.0000i		−	0.890043i	0.895520	−	0.445021i	$-0.146804\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.65685	−	5.65685i	−0.278356	−	0.278356i
$414$		0			0
$415$		−	8.00000i		−	0.392705i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.82843	−	2.82843i	0.138178	−	0.138178i	−0.634635	−	0.772812i	$-0.718849\pi$
$419$	2.82843	−	2.82843i	0.138178	−	0.138178i	0.772812	+	0.634635i	$0.218849\pi$
$420$		0			0
$421$	1.00000	+	1.00000i	0.0487370	+	0.0487370i	0.731055	−	0.682318i	$-0.239028\pi$
$421$	1.00000	+	1.00000i	0.0487370	+	0.0487370i	−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.65685			−0.274398
$426$		0			0
$427$	−4.00000	+	4.00000i	−0.193574	+	0.193574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	21.2132			1.02180			0.510902	−	0.859639i	$-0.329311\pi$
$431$	21.2132			1.02180			0.510902	+	0.859639i	$0.329311\pi$
$432$		0			0
$433$	−34.0000			−1.63394			−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000			−1.63394			−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.65685	+	5.65685i	−0.270604	+	0.270604i
$438$		0			0
$439$	28.0000			1.33637			0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000			1.33637			0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	14.1421	+	14.1421i	0.671913	+	0.671913i	0.958157	−	0.286244i	$-0.0924067\pi$
$443$	14.1421	+	14.1421i	0.671913	+	0.671913i	−0.286244	+	0.958157i	$0.592407\pi$
$444$		0			0
$445$	−24.0000	+	24.0000i	−1.13771	+	1.13771i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	32.5269i		−	1.53504i	−0.641025	−	0.767520i	$-0.721491\pi$
$449$		−	32.5269i		−	1.53504i	0.641025	−	0.767520i	$-0.278509\pi$
$450$		0			0
$451$	−24.0000	−	24.0000i	−1.13012	−	1.13012i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			11.3137i			0.530395i
$456$		0			0
$457$		−	36.0000i		−	1.68401i	−0.539471	−	0.842004i	$-0.681376\pi$
$457$		−	36.0000i		−	1.68401i	0.539471	−	0.842004i	$-0.318624\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.07107	+	7.07107i	0.329332	+	0.329332i	0.852333	−	0.523000i	$-0.175187\pi$
$461$	7.07107	+	7.07107i	0.329332	+	0.329332i	−0.523000	+	0.852333i	$0.675187\pi$
$462$		0			0
$463$		−	24.0000i		−	1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$		−	24.0000i		−	1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.3137	−	11.3137i	0.523536	−	0.523536i	−0.395101	−	0.918637i	$-0.629291\pi$
$467$	11.3137	−	11.3137i	0.523536	−	0.523536i	0.918637	+	0.395101i	$0.129291\pi$
$468$		0			0
$469$	7.00000	+	7.00000i	0.323230	+	0.323230i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	39.5980			1.82072
$474$		0			0
$475$	4.00000	−	4.00000i	0.183533	−	0.183533i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−33.9411			−1.55081			−0.775405	−	0.631464i	$-0.782454\pi$
$479$	−33.9411			−1.55081			−0.775405	+	0.631464i	$0.782454\pi$
$480$		0			0
$481$	−40.0000			−1.82384
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−14.1421	+	14.1421i	−0.642161	+	0.642161i
$486$		0			0
$487$	−38.0000			−1.72194			−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000			−1.72194			−0.860972	+	0.508652i	$0.830144\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	14.1421	+	14.1421i	0.638226	+	0.638226i	0.950118	−	0.311892i	$-0.100963\pi$
$491$	14.1421	+	14.1421i	0.638226	+	0.638226i	−0.311892	+	0.950118i	$0.600963\pi$
$492$		0			0
$493$	32.0000	−	32.0000i	1.44121	−	1.44121i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	7.07107i		−	0.317181i
$498$		0			0
$499$	−13.0000	−	13.0000i	−0.581960	−	0.581960i	0.353482	−	0.935441i	$-0.384998\pi$
$499$	−13.0000	−	13.0000i	−0.581960	−	0.581960i	−0.935441	+	0.353482i	$0.884998\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	31.1127i		−	1.38725i	−0.720338	−	0.693623i	$-0.756013\pi$
$503$		−	31.1127i		−	1.38725i	0.720338	−	0.693623i	$-0.243987\pi$
$504$		0			0
$505$			20.0000i			0.889988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.89949	−	9.89949i	−0.438787	−	0.438787i	0.452816	−	0.891604i	$-0.350419\pi$
$509$	−9.89949	−	9.89949i	−0.438787	−	0.438787i	−0.891604	+	0.452816i	$0.850419\pi$
$510$		0			0
$511$			6.00000i			0.265424i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.65685	−	5.65685i	0.249271	−	0.249271i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−33.9411			−1.48699			−0.743494	−	0.668743i	$-0.766833\pi$
$521$	−33.9411			−1.48699			−0.743494	+	0.668743i	$0.766833\pi$
$522$		0			0
$523$	20.0000	−	20.0000i	0.874539	−	0.874539i	−0.118424	−	0.992963i	$-0.537784\pi$
$523$	20.0000	−	20.0000i	0.874539	−	0.874539i	0.992963	+	0.118424i	$0.0377843\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$		0			0
$532$		0			0
$533$	33.9411	−	33.9411i	1.47015	−	1.47015i
$534$		0			0
$535$	−20.0000			−0.864675
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.82843	−	2.82843i	−0.121829	−	0.121829i
$540$		0			0
$541$	7.00000	−	7.00000i	0.300954	−	0.300954i	−0.540433	−	0.841387i	$-0.681740\pi$
$541$	7.00000	−	7.00000i	0.300954	−	0.300954i	0.841387	+	0.540433i	$0.181740\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	36.7696i		−	1.57503i
$546$		0			0
$547$	17.0000	+	17.0000i	0.726868	+	0.726868i	0.969994	−	0.243127i	$-0.0781732\pi$
$547$	17.0000	+	17.0000i	0.726868	+	0.726868i	−0.243127	+	0.969994i	$0.578173\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			45.2548i			1.92792i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.3137	+	11.3137i	0.479377	+	0.479377i	0.904932	−	0.425555i	$-0.139921\pi$
$557$	11.3137	+	11.3137i	0.479377	+	0.479377i	−0.425555	+	0.904932i	$0.639921\pi$
$558$		0			0
$559$			56.0000i			2.36855i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.65685	+	5.65685i	−0.238408	+	0.238408i	−0.816191	−	0.577783i	$-0.803918\pi$
$563$	−5.65685	+	5.65685i	−0.238408	+	0.238408i	0.577783	+	0.816191i	$0.303918\pi$
$564$		0			0
$565$	2.00000	+	2.00000i	0.0841406	+	0.0841406i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.7279			0.533582			0.266791	−	0.963754i	$-0.414037\pi$
$569$	12.7279			0.533582			0.266791	+	0.963754i	$0.414037\pi$
$570$		0			0
$571$	11.0000	−	11.0000i	0.460336	−	0.460336i	−0.438430	−	0.898765i	$-0.644465\pi$
$571$	11.0000	−	11.0000i	0.460336	−	0.460336i	0.898765	+	0.438430i	$0.144465\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.41421			0.0589768
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.82843	+	2.82843i	−0.117343	+	0.117343i
$582$		0			0
$583$	40.0000			1.65663
$584$		0			0
$585$		0			0
$586$		0			0
$587$	31.1127	+	31.1127i	1.28416	+	1.28416i	0.938279	+	0.345880i	$0.112419\pi$
$587$	31.1127	+	31.1127i	1.28416	+	1.28416i	0.345880	+	0.938279i	$0.387581\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	19.7990i		−	0.813047i	−0.913640	−	0.406524i	$-0.866741\pi$
$593$		−	19.7990i		−	0.813047i	0.913640	−	0.406524i	$-0.133259\pi$
$594$		0			0
$595$	−8.00000	−	8.00000i	−0.327968	−	0.327968i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1.41421i		−	0.0577832i	−0.999583	−	0.0288916i	$-0.990802\pi$
$599$		−	1.41421i		−	0.0577832i	0.999583	−	0.0288916i	$-0.00919776\pi$
$600$		0			0
$601$			30.0000i			1.22373i	0.790964	+	0.611863i	$0.209580\pi$
$601$			30.0000i			1.22373i	−0.790964	+	0.611863i	$0.790420\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.07107	+	7.07107i	0.287480	+	0.287480i
$606$		0			0
$607$		−	12.0000i		−	0.487065i	−0.969893	−	0.243532i	$-0.921694\pi$
$607$		−	12.0000i		−	0.487065i	0.969893	−	0.243532i	$-0.0783062\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	25.0000	+	25.0000i	1.00974	+	1.00974i	0.999952	+	0.00978840i	$0.00311579\pi$
$613$	25.0000	+	25.0000i	1.00974	+	1.00974i	0.00978840	+	0.999952i	$0.496884\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	21.2132			0.854011			0.427006	−	0.904249i	$-0.359568\pi$
$617$	21.2132			0.854011			0.427006	+	0.904249i	$0.359568\pi$
$618$		0			0
$619$	8.00000	−	8.00000i	0.321547	−	0.321547i	−0.527813	−	0.849360i	$-0.676988\pi$
$619$	8.00000	−	8.00000i	0.321547	−	0.321547i	0.849360	+	0.527813i	$0.176988\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	16.9706			0.679911
$624$		0			0
$625$	19.0000			0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	28.2843	−	28.2843i	1.12777	−	1.12777i
$630$		0			0
$631$	34.0000			1.35352			0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000			1.35352			0.676759	+	0.736204i	$0.263384\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	25.4558	+	25.4558i	1.01018	+	1.01018i
$636$		0			0
$637$	4.00000	−	4.00000i	0.158486	−	0.158486i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	15.5563i		−	0.614439i	−0.951639	−	0.307219i	$-0.900601\pi$
$641$		−	15.5563i		−	0.614439i	0.951639	−	0.307219i	$-0.0993986\pi$
$642$		0			0
$643$	−10.0000	−	10.0000i	−0.394362	−	0.394362i	0.481877	−	0.876239i	$-0.339955\pi$
$643$	−10.0000	−	10.0000i	−0.394362	−	0.394362i	−0.876239	+	0.481877i	$0.839955\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	31.1127i		−	1.22317i	−0.791180	−	0.611583i	$-0.790533\pi$
$647$		−	31.1127i		−	1.22317i	0.791180	−	0.611583i	$-0.209467\pi$
$648$		0			0
$649$		−	32.0000i		−	1.25611i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.82843	+	2.82843i	0.110685	+	0.110685i	0.760280	−	0.649595i	$-0.225062\pi$
$653$	2.82843	+	2.82843i	0.110685	+	0.110685i	−0.649595	+	0.760280i	$0.725062\pi$
$654$		0			0
$655$			40.0000i			1.56293i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.89949	+	9.89949i	−0.385630	+	0.385630i	−0.873125	−	0.487496i	$-0.837910\pi$
$659$	−9.89949	+	9.89949i	−0.385630	+	0.385630i	0.487496	+	0.873125i	$0.337910\pi$
$660$		0			0
$661$	−20.0000	−	20.0000i	−0.777910	−	0.777910i	0.201565	−	0.979475i	$-0.435397\pi$
$661$	−20.0000	−	20.0000i	−0.777910	−	0.777910i	−0.979475	+	0.201565i	$0.935397\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	11.3137			0.438727
$666$		0			0
$667$	−8.00000	+	8.00000i	−0.309761	+	0.309761i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−22.6274			−0.873522
$672$		0			0
$673$	−28.0000			−1.07932			−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000			−1.07932			−0.539660	+	0.841883i	$0.681447\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−24.0416	+	24.0416i	−0.923995	+	0.923995i	−0.997309	−	0.0733140i	$-0.976642\pi$
$677$	−24.0416	+	24.0416i	−0.923995	+	0.923995i	0.0733140	+	0.997309i	$0.476642\pi$
$678$		0			0
$679$	10.0000			0.383765
$680$		0			0
$681$		0			0
$682$		0			0
$683$	14.1421	+	14.1421i	0.541134	+	0.541134i	0.923861	−	0.382727i	$-0.125015\pi$
$683$	14.1421	+	14.1421i	0.541134	+	0.541134i	−0.382727	+	0.923861i	$0.625015\pi$
$684$		0			0
$685$	−6.00000	+	6.00000i	−0.229248	+	0.229248i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			56.5685i			2.15509i
$690$		0			0
$691$	−22.0000	−	22.0000i	−0.836919	−	0.836919i	0.151533	−	0.988452i	$-0.451579\pi$
$691$	−22.0000	−	22.0000i	−0.836919	−	0.836919i	−0.988452	+	0.151533i	$0.951579\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	39.5980i		−	1.50204i
$696$		0			0
$697$			48.0000i			1.81813i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	15.5563	+	15.5563i	0.587555	+	0.587555i	0.936969	−	0.349413i	$-0.113619\pi$
$701$	15.5563	+	15.5563i	0.587555	+	0.587555i	−0.349413	+	0.936969i	$0.613619\pi$
$702$		0			0
$703$			40.0000i			1.50863i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7.07107	−	7.07107i	0.265935	−	0.265935i
$708$		0			0
$709$	31.0000	+	31.0000i	1.16423	+	1.16423i	0.983540	+	0.180689i	$0.0578328\pi$
$709$	31.0000	+	31.0000i	1.16423	+	1.16423i	0.180689	+	0.983540i	$0.442167\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−32.0000	+	32.0000i	−1.19673	+	1.19673i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	42.4264			1.58224			0.791119	−	0.611662i	$-0.209499\pi$
$719$	42.4264			1.58224			0.791119	+	0.611662i	$0.209499\pi$
$720$		0			0
$721$	−4.00000			−0.148968
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.65685	−	5.65685i	0.210090	−	0.210090i
$726$		0			0
$727$	28.0000			1.03846			0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000			1.03846			0.519231	+	0.854634i	$0.326218\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−39.5980	−	39.5980i	−1.46458	−	1.46458i
$732$		0			0
$733$	−26.0000	+	26.0000i	−0.960332	+	0.960332i	−0.999243	−	0.0389108i	$-0.987611\pi$
$733$	−26.0000	+	26.0000i	−0.960332	+	0.960332i	0.0389108	+	0.999243i	$0.487611\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			39.5980i			1.45861i
$738$		0			0
$739$	29.0000	+	29.0000i	1.06678	+	1.06678i	0.997604	+	0.0691779i	$0.0220376\pi$
$739$	29.0000	+	29.0000i	1.06678	+	1.06678i	0.0691779	+	0.997604i	$0.477962\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	26.8701i		−	0.985767i	−0.870095	−	0.492883i	$-0.835943\pi$
$743$		−	26.8701i		−	0.985767i	0.870095	−	0.492883i	$-0.164057\pi$
$744$		0			0
$745$			32.0000i			1.17239i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7.07107	+	7.07107i	0.258371	+	0.258371i
$750$		0			0
$751$			24.0000i			0.875772i	0.899030	+	0.437886i	$0.144273\pi$
$751$			24.0000i			0.875772i	−0.899030	+	0.437886i	$0.855727\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14.1421	−	14.1421i	0.514685	−	0.514685i
$756$		0			0
$757$	37.0000	+	37.0000i	1.34479	+	1.34479i	0.891224	+	0.453564i	$0.149848\pi$
$757$	37.0000	+	37.0000i	1.34479	+	1.34479i	0.453564	+	0.891224i	$0.350152\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	42.4264			1.53796			0.768978	−	0.639275i	$-0.220766\pi$
$761$	42.4264			1.53796			0.768978	+	0.639275i	$0.220766\pi$
$762$		0			0
$763$	−13.0000	+	13.0000i	−0.470632	+	0.470632i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	45.2548			1.63406
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−32.5269	+	32.5269i	−1.16991	+	1.16991i	−0.187682	+	0.982230i	$0.560097\pi$
$773$	−32.5269	+	32.5269i	−1.16991	+	1.16991i	−0.982230	+	0.187682i	$0.939903\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−33.9411	−	33.9411i	−1.21607	−	1.21607i
$780$		0			0
$781$	20.0000	−	20.0000i	0.715656	−	0.715656i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			5.65685i			0.201902i
$786$		0			0
$787$	2.00000	+	2.00000i	0.0712923	+	0.0712923i	0.741854	−	0.670562i	$-0.233947\pi$
$787$	2.00000	+	2.00000i	0.0712923	+	0.0712923i	−0.670562	+	0.741854i	$0.733947\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1.41421i		−	0.0502836i
$792$		0			0
$793$		−	32.0000i		−	1.13635i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.41421	−	1.41421i	−0.0500940	−	0.0500940i	0.681616	−	0.731710i	$-0.261277\pi$
$797$	−1.41421	−	1.41421i	−0.0500940	−	0.0500940i	−0.731710	+	0.681616i	$0.761277\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−16.9706	+	16.9706i	−0.598878	+	0.598878i
$804$		0			0
$805$	2.00000	+	2.00000i	0.0704907	+	0.0704907i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	4.24264			0.149163			0.0745817	−	0.997215i	$-0.476238\pi$
$809$	4.24264			0.149163			0.0745817	+	0.997215i	$0.476238\pi$
$810$		0			0
$811$	8.00000	−	8.00000i	0.280918	−	0.280918i	−0.552557	−	0.833475i	$-0.686348\pi$
$811$	8.00000	−	8.00000i	0.280918	−	0.280918i	0.833475	+	0.552557i	$0.186348\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−36.7696			−1.28798
$816$		0			0
$817$	56.0000			1.95919
$818$		0			0
$819$		0			0
$820$		0			0
$821$	26.8701	−	26.8701i	0.937771	−	0.937771i	−0.0604026	−	0.998174i	$-0.519238\pi$
$821$	26.8701	−	26.8701i	0.937771	−	0.937771i	0.998174	+	0.0604026i	$0.0192385\pi$
$822$		0			0
$823$	−14.0000			−0.488009			−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000			−0.488009			−0.244005	+	0.969774i	$0.578461\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.89949	+	9.89949i	0.344239	+	0.344239i	0.857958	−	0.513719i	$-0.171733\pi$
$827$	9.89949	+	9.89949i	0.344239	+	0.344239i	−0.513719	+	0.857958i	$0.671733\pi$
$828$		0			0
$829$	22.0000	−	22.0000i	0.764092	−	0.764092i	−0.212968	−	0.977059i	$-0.568313\pi$
$829$	22.0000	−	22.0000i	0.764092	−	0.764092i	0.977059	+	0.212968i	$0.0683129\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			5.65685i			0.195998i
$834$		0			0
$835$	−20.0000	−	20.0000i	−0.692129	−	0.692129i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			2.82843i			0.0976481i	0.998807	+	0.0488241i	$0.0155474\pi$
$839$			2.82843i			0.0976481i	−0.998807	+	0.0488241i	$0.984453\pi$
$840$		0			0
$841$			35.0000i			1.20690i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−26.8701	−	26.8701i	−0.924358	−	0.924358i
$846$		0			0
$847$		−	5.00000i		−	0.171802i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.07107	+	7.07107i	−0.242393	+	0.242393i
$852$		0			0
$853$	10.0000	+	10.0000i	0.342393	+	0.342393i	0.857266	−	0.514873i	$-0.172161\pi$
$853$	10.0000	+	10.0000i	0.342393	+	0.342393i	−0.514873	+	0.857266i	$0.672161\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	8.48528			0.289852			0.144926	−	0.989443i	$-0.453706\pi$
$857$	8.48528			0.289852			0.144926	+	0.989443i	$0.453706\pi$
$858$		0			0
$859$	−28.0000	+	28.0000i	−0.955348	+	0.955348i	−0.999045	−	0.0436972i	$-0.986086\pi$
$859$	−28.0000	+	28.0000i	−0.955348	+	0.955348i	0.0436972	+	0.999045i	$0.486086\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	29.6985			1.01095			0.505474	−	0.862842i	$-0.331318\pi$
$863$	29.6985			1.01095			0.505474	+	0.862842i	$0.331318\pi$
$864$		0			0
$865$	−4.00000			−0.136004
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−56.0000			−1.89749
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8.48528	−	8.48528i	−0.286855	−	0.286855i
$876$		0			0
$877$	1.00000	−	1.00000i	0.0337676	−	0.0337676i	−0.690021	−	0.723789i	$-0.742399\pi$
$877$	1.00000	−	1.00000i	0.0337676	−	0.0337676i	0.723789	+	0.690021i	$0.242399\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			31.1127i			1.04821i	0.851653	+	0.524107i	$0.175601\pi$
$881$			31.1127i			1.04821i	−0.851653	+	0.524107i	$0.824399\pi$
$882$		0			0
$883$	35.0000	+	35.0000i	1.17784	+	1.17784i	0.980293	+	0.197551i	$0.0632989\pi$
$883$	35.0000	+	35.0000i	1.17784	+	1.17784i	0.197551	+	0.980293i	$0.436701\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	14.1421i		−	0.474846i	−0.971406	−	0.237423i	$-0.923697\pi$
$887$		−	14.1421i		−	0.474846i	0.971406	−	0.237423i	$-0.0763028\pi$
$888$		0			0
$889$		−	18.0000i		−	0.603701i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	20.0000i		−	0.668526i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−40.0000	−	40.0000i	−1.33259	−	1.33259i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−22.6274			−0.752161
$906$		0			0
$907$	17.0000	−	17.0000i	0.564476	−	0.564476i	−0.366100	−	0.930576i	$-0.619307\pi$
$907$	17.0000	−	17.0000i	0.564476	−	0.564476i	0.930576	+	0.366100i	$0.119307\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	21.2132			0.702825			0.351412	−	0.936221i	$-0.385702\pi$
$911$	21.2132			0.702825			0.351412	+	0.936221i	$0.385702\pi$
$912$		0			0
$913$	−16.0000			−0.529523
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14.1421	−	14.1421i	0.467014	−	0.467014i
$918$		0			0
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	28.2843	+	28.2843i	0.930988	+	0.930988i
$924$		0			0
$925$	5.00000	−	5.00000i	0.164399	−	0.164399i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	53.7401i		−	1.76316i	−0.472038	−	0.881578i	$-0.656482\pi$
$929$		−	53.7401i		−	1.76316i	0.472038	−	0.881578i	$-0.343518\pi$
$930$		0			0
$931$	−4.00000	−	4.00000i	−0.131095	−	0.131095i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	45.2548i		−	1.47999i
$936$		0			0
$937$			18.0000i			0.588034i	0.955800	+	0.294017i	$0.0949923\pi$
$937$			18.0000i			0.588034i	−0.955800	+	0.294017i	$0.905008\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	7.07107	+	7.07107i	0.230510	+	0.230510i	0.812906	−	0.582395i	$-0.197884\pi$
$941$	7.07107	+	7.07107i	0.230510	+	0.230510i	−0.582395	+	0.812906i	$0.697884\pi$
$942$		0			0
$943$		−	12.0000i		−	0.390774i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.41421	+	1.41421i	−0.0459558	+	0.0459558i	−0.729711	−	0.683755i	$-0.760346\pi$
$947$	−1.41421	+	1.41421i	−0.0459558	+	0.0459558i	0.683755	+	0.729711i	$0.260346\pi$
$948$		0			0
$949$	−24.0000	−	24.0000i	−0.779073	−	0.779073i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	12.7279			0.412298			0.206149	−	0.978521i	$-0.433907\pi$
$953$	12.7279			0.412298			0.206149	+	0.978521i	$0.433907\pi$
$954$		0			0
$955$	−18.0000	+	18.0000i	−0.582466	+	0.582466i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.24264			0.137002
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	19.7990	−	19.7990i	0.637352	−	0.637352i
$966$		0			0
$967$	40.0000			1.28631			0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000			1.28631			0.643157	+	0.765735i	$0.277624\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−19.7990	−	19.7990i	−0.635380	−	0.635380i	0.314032	−	0.949412i	$-0.398320\pi$
$971$	−19.7990	−	19.7990i	−0.635380	−	0.635380i	−0.949412	+	0.314032i	$0.898320\pi$
$972$		0			0
$973$	−14.0000	+	14.0000i	−0.448819	+	0.448819i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1.41421i			0.0452447i	0.999744	+	0.0226224i	$0.00720153\pi$
$977$			1.41421i			0.0452447i	−0.999744	+	0.0226224i	$0.992798\pi$
$978$		0			0
$979$	48.0000	+	48.0000i	1.53409	+	1.53409i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			28.2843i			0.902128i	0.892492	+	0.451064i	$0.148955\pi$
$983$			28.2843i			0.902128i	−0.892492	+	0.451064i	$0.851045\pi$
$984$		0			0
$985$			8.00000i			0.254901i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9.89949	+	9.89949i	0.314786	+	0.314786i
$990$		0			0
$991$		−	30.0000i		−	0.952981i	−0.879180	−	0.476491i	$-0.841909\pi$
$991$		−	30.0000i		−	0.952981i	0.879180	−	0.476491i	$-0.158091\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	22.6274	−	22.6274i	0.717337	−	0.717337i
$996$		0			0
$997$	−20.0000	−	20.0000i	−0.633406	−	0.633406i	0.315514	−	0.948921i	$-0.397823\pi$
$997$	−20.0000	−	20.0000i	−0.633406	−	0.633406i	−0.948921	+	0.315514i	$0.897823\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.b.3599.2		4
3.2	odd	2	inner	4032.2.v.b.3599.1		4
4.3	odd	2		1008.2.v.a.827.1	yes	4
12.11	even	2		1008.2.v.a.827.2	yes	4
16.3	odd	4	inner	4032.2.v.b.1583.1		4
16.13	even	4		1008.2.v.a.323.1	✓	4
48.29	odd	4		1008.2.v.a.323.2	yes	4
48.35	even	4	inner	4032.2.v.b.1583.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.a.323.1	✓	4	16.13	even	4
1008.2.v.a.323.2	yes	4	48.29	odd	4
1008.2.v.a.827.1	yes	4	4.3	odd	2
1008.2.v.a.827.2	yes	4	12.11	even	2
4032.2.v.b.1583.1		4	16.3	odd	4	inner
4032.2.v.b.1583.2		4	48.35	even	4	inner
4032.2.v.b.3599.1		4	3.2	odd	2	inner
4032.2.v.b.3599.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.41421 - 1.41421i) q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+(1.41421 - 1.41421i) q^{5} -1.00000 q^{7} +(-2.82843 - 2.82843i) q^{11} +(4.00000 - 4.00000i) q^{13} +5.65685i q^{17} +(-4.00000 - 4.00000i) q^{19} -1.41421i q^{23} +1.00000i q^{25} +(-5.65685 - 5.65685i) q^{29} +(-1.41421 + 1.41421i) q^{35} +(-5.00000 - 5.00000i) q^{37} +8.48528 q^{41} +(-7.00000 + 7.00000i) q^{43} +1.00000 q^{49} +(-7.07107 + 7.07107i) q^{53} -8.00000 q^{55} +(5.65685 + 5.65685i) q^{59} +(4.00000 - 4.00000i) q^{61} -11.3137i q^{65} +(-7.00000 - 7.00000i) q^{67} +7.07107i q^{71} -6.00000i q^{73} +(2.82843 + 2.82843i) q^{77} +(2.82843 - 2.82843i) q^{83} +(8.00000 + 8.00000i) q^{85} -16.9706 q^{89} +(-4.00000 + 4.00000i) q^{91} -11.3137 q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^7	\( 4 q - 4 q^{7} + 16 q^{13} - 16 q^{19} - 20 q^{37} - 28 q^{43} + 4 q^{49} - 32 q^{55} + 16 q^{61} - 28 q^{67} + 32 q^{85} - 16 q^{91} - 40 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 + 16 * q^13 - 16 * q^19 - 20 * q^37 - 28 * q^43 + 4 * q^49 - 32 * q^55 + 16 * q^61 - 28 * q^67 + 32 * q^85 - 16 * q^91 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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