LMFDB - Embedded newform 1800.3.v.k.793.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,3,Mod(793,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("1800.793");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1800.v (of order $4$, degree $2$, 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:	$49.0464475849$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			793.1
Root			$2.70156i$ of defining polynomial
Character	$\chi$	$=$	1800.793
Dual form			1800.3.v.k.1657.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.70156 + 6.70156i) q^{7} +O(q^{10})$	$q+(-6.70156 + 6.70156i) q^{7} +1.40312 q^{11} +(14.4031 + 14.4031i) q^{13} +(-2.40312 + 2.40312i) q^{17} +22.8062i q^{19} +(-0.104686 - 0.104686i) q^{23} -45.6125i q^{29} -2.59688 q^{31} +(-10.6125 + 10.6125i) q^{37} +44.6281 q^{41} +(-26.7016 - 26.7016i) q^{43} +(-10.4922 + 10.4922i) q^{47} -40.8219i q^{49} +(-3.00000 - 3.00000i) q^{53} +41.1938i q^{59} -57.4031 q^{61} +(-34.7016 + 34.7016i) q^{67} -45.4031 q^{71} +(-11.3875 - 11.3875i) q^{73} +(-9.40312 + 9.40312i) q^{77} +86.4500i q^{79} +(-81.7172 - 81.7172i) q^{83} +91.2250i q^{89} -193.047 q^{91} +(-49.0000 + 49.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{7}+O(q^{10}) $ 4 * q - 14 * q^7	$ 4 q - 14 q^{7} - 20 q^{11} + 32 q^{13} + 16 q^{17} + 38 q^{23} - 36 q^{31} + 60 q^{37} - 52 q^{41} - 94 q^{43} - 106 q^{47} - 12 q^{53} - 204 q^{61} - 126 q^{67} - 156 q^{71} - 148 q^{73} - 12 q^{77} - 186 q^{83} - 388 q^{91} - 196 q^{97}+O(q^{100}) $ 4 * q - 14 * q^7 - 20 * q^11 + 32 * q^13 + 16 * q^17 + 38 * q^23 - 36 * q^31 + 60 * q^37 - 52 * q^41 - 94 * q^43 - 106 * q^47 - 12 * q^53 - 204 * q^61 - 126 * q^67 - 156 * q^71 - 148 * q^73 - 12 * q^77 - 186 * q^83 - 388 * q^91 - 196 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−6.70156	+	6.70156i	−0.957366	+	0.957366i	−0.999128	−	0.0417616i	$-0.986703\pi$
$7$	−6.70156	+	6.70156i	−0.957366	+	0.957366i	0.0417616	+	0.999128i	$0.486703\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.40312			0.127557			0.0637784	−	0.997964i	$-0.479685\pi$
$11$	1.40312			0.127557			0.0637784	+	0.997964i	$0.479685\pi$
$12$		0			0
$13$	14.4031	+	14.4031i	1.10793	+	1.10793i	0.993422	+	0.114511i	$0.0365300\pi$
$13$	14.4031	+	14.4031i	1.10793	+	1.10793i	0.114511	+	0.993422i	$0.463470\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.40312	+	2.40312i	−0.141360	+	0.141360i	−0.774246	−	0.632885i	$-0.781870\pi$
$17$	−2.40312	+	2.40312i	−0.141360	+	0.141360i	0.632885	+	0.774246i	$0.281870\pi$
$18$		0			0
$19$			22.8062i			1.20033i	0.799877	+	0.600164i	$0.204898\pi$
$19$			22.8062i			1.20033i	−0.799877	+	0.600164i	$0.795102\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.104686	−	0.104686i	−0.00455158	−	0.00455158i	0.704827	−	0.709379i	$-0.251024\pi$
$23$	−0.104686	−	0.104686i	−0.00455158	−	0.00455158i	−0.709379	+	0.704827i	$0.751024\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	45.6125i		−	1.57284i	−0.617689	−	0.786422i	$-0.711931\pi$
$29$		−	45.6125i		−	1.57284i	0.617689	−	0.786422i	$-0.288069\pi$
$30$		0			0
$31$	−2.59688			−0.0837702			−0.0418851	−	0.999122i	$-0.513336\pi$
$31$	−2.59688			−0.0837702			−0.0418851	+	0.999122i	$0.513336\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−10.6125	+	10.6125i	−0.286824	+	0.286824i	−0.835823	−	0.548999i	$-0.815009\pi$
$37$	−10.6125	+	10.6125i	−0.286824	+	0.286824i	0.548999	+	0.835823i	$0.315009\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	44.6281			1.08849			0.544245	−	0.838926i	$-0.316816\pi$
$41$	44.6281			1.08849			0.544245	+	0.838926i	$0.316816\pi$
$42$		0			0
$43$	−26.7016	−	26.7016i	−0.620967	−	0.620967i	0.324812	−	0.945779i	$-0.394699\pi$
$43$	−26.7016	−	26.7016i	−0.620967	−	0.620967i	−0.945779	+	0.324812i	$0.894699\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−10.4922	+	10.4922i	−0.223238	+	0.223238i	−0.809861	−	0.586622i	$-0.800457\pi$
$47$	−10.4922	+	10.4922i	−0.223238	+	0.223238i	0.586622	+	0.809861i	$0.300457\pi$
$48$		0			0
$49$		−	40.8219i		−	0.833099i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.00000	−	3.00000i	−0.0566038	−	0.0566038i	0.678238	−	0.734842i	$-0.262744\pi$
$53$	−3.00000	−	3.00000i	−0.0566038	−	0.0566038i	−0.734842	+	0.678238i	$0.762744\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$			41.1938i			0.698199i	0.937086	+	0.349100i	$0.113513\pi$
$59$			41.1938i			0.698199i	−0.937086	+	0.349100i	$0.886487\pi$
$60$		0			0
$61$	−57.4031			−0.941035			−0.470517	−	0.882391i	$-0.655933\pi$
$61$	−57.4031			−0.941035			−0.470517	+	0.882391i	$0.655933\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−34.7016	+	34.7016i	−0.517934	+	0.517934i	−0.916946	−	0.399012i	$-0.869353\pi$
$67$	−34.7016	+	34.7016i	−0.517934	+	0.517934i	0.399012	+	0.916946i	$0.369353\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−45.4031			−0.639481			−0.319740	−	0.947505i	$-0.603596\pi$
$71$	−45.4031			−0.639481			−0.319740	+	0.947505i	$0.603596\pi$
$72$		0			0
$73$	−11.3875	−	11.3875i	−0.155993	−	0.155993i	0.624795	−	0.780789i	$-0.285182\pi$
$73$	−11.3875	−	11.3875i	−0.155993	−	0.155993i	−0.780789	+	0.624795i	$0.785182\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.40312	+	9.40312i	−0.122118	+	0.122118i
$78$		0			0
$79$			86.4500i			1.09430i	0.837033	+	0.547152i	$0.184288\pi$
$79$			86.4500i			1.09430i	−0.837033	+	0.547152i	$0.815712\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−81.7172	−	81.7172i	−0.984544	−	0.984544i	0.0153380	−	0.999882i	$-0.495118\pi$
$83$	−81.7172	−	81.7172i	−0.984544	−	0.984544i	−0.999882	+	0.0153380i	$0.995118\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			91.2250i			1.02500i	0.858687	+	0.512500i	$0.171281\pi$
$89$			91.2250i			1.02500i	−0.858687	+	0.512500i	$0.828719\pi$
$90$		0			0
$91$	−193.047			−2.12139
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−49.0000	+	49.0000i	−0.505155	+	0.505155i	−0.913035	−	0.407881i	$-0.866268\pi$
$97$	−49.0000	+	49.0000i	−0.505155	+	0.505155i	0.407881	+	0.913035i	$0.366268\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−60.0312			−0.594369			−0.297184	−	0.954820i	$-0.596048\pi$
$101$	−60.0312			−0.594369			−0.297184	+	0.954820i	$0.596048\pi$
$102$		0			0
$103$	−61.5078	−	61.5078i	−0.597163	−	0.597163i	0.342393	−	0.939557i	$-0.388762\pi$
$103$	−61.5078	−	61.5078i	−0.597163	−	0.597163i	−0.939557	+	0.342393i	$0.888762\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	67.5391	−	67.5391i	0.631206	−	0.631206i	−0.317164	−	0.948371i	$-0.602731\pi$
$107$	67.5391	−	67.5391i	0.631206	−	0.631206i	0.948371	+	0.317164i	$0.102731\pi$
$108$		0			0
$109$		−	19.0469i		−	0.174742i	−0.996176	−	0.0873709i	$-0.972153\pi$
$109$		−	19.0469i		−	0.174742i	0.996176	−	0.0873709i	$-0.0278465\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.82187	+	8.82187i	0.0780696	+	0.0780696i	0.745063	−	0.666994i	$-0.232419\pi$
$113$	8.82187	+	8.82187i	0.0780696	+	0.0780696i	−0.666994	+	0.745063i	$0.732419\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	32.2094i		−	0.270667i
$120$		0			0
$121$	−119.031			−0.983729
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−14.7016	+	14.7016i	−0.115760	+	0.115760i	−0.762614	−	0.646854i	$-0.776084\pi$
$127$	−14.7016	+	14.7016i	−0.115760	+	0.115760i	0.646854	+	0.762614i	$0.276084\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−237.884			−1.81591			−0.907956	−	0.419066i	$-0.862357\pi$
$131$	−237.884			−1.81591			−0.907956	+	0.419066i	$0.862357\pi$
$132$		0			0
$133$	−152.837	−	152.837i	−1.14915	−	1.14915i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	86.6125	−	86.6125i	0.632208	−	0.632208i	−0.316413	−	0.948621i	$-0.602479\pi$
$137$	86.6125	−	86.6125i	0.632208	−	0.632208i	0.948621	+	0.316413i	$0.102479\pi$
$138$		0			0
$139$			246.806i			1.77558i	0.460244	+	0.887792i	$0.347762\pi$
$139$			246.806i			1.77558i	−0.460244	+	0.887792i	$0.652238\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	20.2094	+	20.2094i	0.141324	+	0.141324i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$			121.853i			0.817806i	0.912578	+	0.408903i	$0.134089\pi$
$149$			121.853i			0.817806i	−0.912578	+	0.408903i	$0.865911\pi$
$150$		0			0
$151$	245.528			1.62601			0.813007	−	0.582254i	$-0.197829\pi$
$151$	245.528			1.62601			0.813007	+	0.582254i	$0.197829\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	146.466	−	146.466i	0.932902	−	0.932902i	−0.0649843	−	0.997886i	$-0.520700\pi$
$157$	146.466	−	146.466i	0.932902	−	0.932902i	0.997886	+	0.0649843i	$0.0206997\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.40312			0.00871506
$162$		0			0
$163$	174.973	+	174.973i	1.07346	+	1.07346i	0.997079	+	0.0763776i	$0.0243354\pi$
$163$	174.973	+	174.973i	1.07346	+	1.07346i	0.0763776	+	0.997079i	$0.475665\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−203.330	+	203.330i	−1.21754	+	1.21754i	−0.249053	+	0.968490i	$0.580119\pi$
$167$	−203.330	+	203.330i	−1.21754	+	1.21754i	−0.968490	+	0.249053i	$0.919881\pi$
$168$		0			0
$169$			245.900i			1.45503i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−168.612	−	168.612i	−0.974639	−	0.974639i	0.0250476	−	0.999686i	$-0.492026\pi$
$173$	−168.612	−	168.612i	−0.974639	−	0.974639i	−0.999686	+	0.0250476i	$0.992026\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			27.5813i			0.154085i	0.997028	+	0.0770426i	$0.0245477\pi$
$179$			27.5813i			0.154085i	−0.997028	+	0.0770426i	$0.975452\pi$
$180$		0			0
$181$	−88.3875			−0.488329			−0.244164	−	0.969734i	$-0.578514\pi$
$181$	−88.3875			−0.488329			−0.244164	+	0.969734i	$0.578514\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−3.37188	+	3.37188i	−0.0180315	+	0.0180315i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	79.4969			0.416214			0.208107	−	0.978106i	$-0.433270\pi$
$191$	79.4969			0.416214			0.208107	+	0.978106i	$0.433270\pi$
$192$		0			0
$193$	−104.822	−	104.822i	−0.543118	−	0.543118i	0.381323	−	0.924442i	$-0.375468\pi$
$193$	−104.822	−	104.822i	−0.543118	−	0.543118i	−0.924442	+	0.381323i	$0.875468\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	68.1625	−	68.1625i	0.346003	−	0.346003i	−0.512616	−	0.858618i	$-0.671323\pi$
$197$	68.1625	−	68.1625i	0.346003	−	0.346003i	0.858618	+	0.512616i	$0.171323\pi$
$198$		0			0
$199$		−	250.512i		−	1.25886i	−0.777059	−	0.629428i	$-0.783289\pi$
$199$		−	250.512i		−	1.25886i	0.777059	−	0.629428i	$-0.216711\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	305.675	+	305.675i	1.50579	+	1.50579i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			32.0000i			0.153110i
$210$		0			0
$211$	−332.628			−1.57644			−0.788218	−	0.615396i	$-0.788996\pi$
$211$	−332.628			−1.57644			−0.788218	+	0.615396i	$0.788996\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	17.4031	−	17.4031i	0.0801987	−	0.0801987i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−69.2250			−0.313235
$222$		0			0
$223$	−187.602	−	187.602i	−0.841263	−	0.841263i	0.147761	−	0.989023i	$-0.452793\pi$
$223$	−187.602	−	187.602i	−0.841263	−	0.841263i	−0.989023	+	0.147761i	$0.952793\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−152.167	+	152.167i	−0.670340	+	0.670340i	−0.957794	−	0.287454i	$-0.907191\pi$
$227$	−152.167	+	152.167i	−0.670340	+	0.670340i	0.287454	+	0.957794i	$0.407191\pi$
$228$		0			0
$229$		−	196.062i		−	0.856168i	−0.903739	−	0.428084i	$-0.859189\pi$
$229$		−	196.062i		−	0.856168i	0.903739	−	0.428084i	$-0.140811\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−60.7906	−	60.7906i	−0.260904	−	0.260904i	0.564517	−	0.825421i	$-0.309062\pi$
$233$	−60.7906	−	60.7906i	−0.260904	−	0.260904i	−0.825421	+	0.564517i	$0.809062\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			178.388i			0.746391i	0.927753	+	0.373196i	$0.121738\pi$
$239$			178.388i			0.746391i	−0.927753	+	0.373196i	$0.878262\pi$
$240$		0			0
$241$	89.8219			0.372705			0.186352	−	0.982483i	$-0.440333\pi$
$241$	89.8219			0.372705			0.186352	+	0.982483i	$0.440333\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−328.481	+	328.481i	−1.32988	+	1.32988i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.46561			0.0217753			0.0108877	−	0.999941i	$-0.496534\pi$
$251$	5.46561			0.0217753			0.0108877	+	0.999941i	$0.496534\pi$
$252$		0			0
$253$	−0.146888	−	0.146888i	−0.000580585	−	0.000580585i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−340.078	+	340.078i	−1.32326	+	1.32326i	−0.412141	+	0.911120i	$0.635219\pi$
$257$	−340.078	+	340.078i	−1.32326	+	1.32326i	−0.911120	+	0.412141i	$0.864781\pi$
$258$		0			0
$259$		−	142.241i		−	0.549192i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−29.3609	−	29.3609i	−0.111638	−	0.111638i	0.649081	−	0.760719i	$-0.275154\pi$
$263$	−29.3609	−	29.3609i	−0.111638	−	0.111638i	−0.760719	+	0.649081i	$0.775154\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	403.047i		−	1.49832i	−0.662392	−	0.749158i	$-0.730458\pi$
$269$		−	403.047i		−	1.49832i	0.662392	−	0.749158i	$-0.269542\pi$
$270$		0			0
$271$	−308.984			−1.14016			−0.570082	−	0.821588i	$-0.693089\pi$
$271$	−308.984			−1.14016			−0.570082	+	0.821588i	$0.693089\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.59688	+	9.59688i	−0.0346458	+	0.0346458i	−0.724217	−	0.689572i	$-0.757799\pi$
$277$	−9.59688	+	9.59688i	−0.0346458	+	0.0346458i	0.689572	+	0.724217i	$0.257799\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	268.628			0.955972			0.477986	−	0.878367i	$-0.341367\pi$
$281$	268.628			0.955972			0.477986	+	0.878367i	$0.341367\pi$
$282$		0			0
$283$	116.942	+	116.942i	0.413223	+	0.413223i	0.882860	−	0.469637i	$-0.155615\pi$
$283$	116.942	+	116.942i	0.413223	+	0.413223i	−0.469637	+	0.882860i	$0.655615\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−299.078	+	299.078i	−1.04208	+	1.04208i
$288$		0			0
$289$			277.450i			0.960035i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−133.691	−	133.691i	−0.456282	−	0.456282i	0.441151	−	0.897433i	$-0.354570\pi$
$293$	−133.691	−	133.691i	−0.456282	−	0.456282i	−0.897433	+	0.441151i	$0.854570\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	3.01562i		−	0.0100857i
$300$		0			0
$301$	357.884			1.18898
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	109.298	−	109.298i	0.356021	−	0.356021i	−0.506323	−	0.862344i	$-0.668996\pi$
$307$	109.298	−	109.298i	0.356021	−	0.356021i	0.862344	+	0.506323i	$0.168996\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−428.691			−1.37843			−0.689213	−	0.724559i	$-0.742044\pi$
$311$	−428.691			−1.37843			−0.689213	+	0.724559i	$0.742044\pi$
$312$		0			0
$313$	−299.334	−	299.334i	−0.956340	−	0.956340i	0.0427462	−	0.999086i	$-0.486389\pi$
$313$	−299.334	−	299.334i	−0.956340	−	0.956340i	−0.999086	+	0.0427462i	$0.986389\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	412.109	−	412.109i	1.30003	−	1.30003i	0.371661	−	0.928369i	$-0.378788\pi$
$317$	412.109	−	412.109i	1.30003	−	1.30003i	0.928369	−	0.371661i	$-0.121212\pi$
$318$		0			0
$319$		−	64.0000i		−	0.200627i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−54.8062	−	54.8062i	−0.169679	−	0.169679i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	140.628i		−	0.427441i
$330$		0			0
$331$	12.9219			0.0390390			0.0195195	−	0.999809i	$-0.493786\pi$
$331$	12.9219			0.0390390			0.0195195	+	0.999809i	$0.493786\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	61.3250	−	61.3250i	0.181973	−	0.181973i	−0.610242	−	0.792215i	$-0.708928\pi$
$337$	61.3250	−	61.3250i	0.181973	−	0.181973i	0.792215	+	0.610242i	$0.208928\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.64374			−0.0106855
$342$		0			0
$343$	−54.8062	−	54.8062i	−0.159785	−	0.159785i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	179.183	−	179.183i	0.516377	−	0.516377i	−0.400096	−	0.916473i	$-0.631023\pi$
$347$	179.183	−	179.183i	0.516377	−	0.516377i	0.916473	+	0.400096i	$0.131023\pi$
$348$		0			0
$349$			240.962i			0.690437i	0.938522	+	0.345218i	$0.112195\pi$
$349$			240.962i			0.690437i	−0.938522	+	0.345218i	$0.887805\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	141.900	+	141.900i	0.401983	+	0.401983i	0.878931	−	0.476948i	$-0.158257\pi$
$353$	141.900	+	141.900i	0.401983	+	0.401983i	−0.476948	+	0.878931i	$0.658257\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	238.325i		−	0.663858i	−0.943304	−	0.331929i	$-0.892301\pi$
$359$		−	238.325i		−	0.663858i	0.943304	−	0.331929i	$-0.107699\pi$
$360$		0			0
$361$	−159.125			−0.440789
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−259.477	+	259.477i	−0.707021	+	0.707021i	−0.965908	−	0.258887i	$-0.916644\pi$
$367$	−259.477	+	259.477i	−0.707021	+	0.707021i	0.258887	+	0.965908i	$0.416644\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	40.2094			0.108381
$372$		0			0
$373$	310.350	+	310.350i	0.832037	+	0.832037i	0.987795	−	0.155758i	$-0.0497819\pi$
$373$	310.350	+	310.350i	0.832037	+	0.832037i	−0.155758	+	0.987795i	$0.549782\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	656.962	−	656.962i	1.74261	−	1.74261i
$378$		0			0
$379$		−	90.1562i		−	0.237879i	−0.992901	−	0.118940i	$-0.962051\pi$
$379$		−	90.1562i		−	0.237879i	0.992901	−	0.118940i	$-0.0379495\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−209.298	−	209.298i	−0.546471	−	0.546471i	0.378947	−	0.925418i	$-0.376286\pi$
$383$	−209.298	−	209.298i	−0.546471	−	0.546471i	−0.925418	+	0.378947i	$0.876286\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			193.372i			0.497100i	0.968619	+	0.248550i	$0.0799540\pi$
$389$			193.372i			0.497100i	−0.968619	+	0.248550i	$0.920046\pi$
$390$		0			0
$391$	0.503149			0.00128683
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−80.2250	+	80.2250i	−0.202078	+	0.202078i	−0.800890	−	0.598812i	$-0.795640\pi$
$397$	−80.2250	+	80.2250i	−0.202078	+	0.202078i	0.598812	+	0.800890i	$0.295640\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	727.800			1.81496			0.907481	−	0.420093i	$-0.138002\pi$
$401$	727.800			1.81496			0.907481	+	0.420093i	$0.138002\pi$
$402$		0			0
$403$	−37.4031	−	37.4031i	−0.0928117	−	0.0928117i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−14.8907	+	14.8907i	−0.0365864	+	0.0365864i
$408$		0			0
$409$		−	355.403i		−	0.868956i	−0.900682	−	0.434478i	$-0.856933\pi$
$409$		−	355.403i		−	0.868956i	0.900682	−	0.434478i	$-0.143067\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−276.062	−	276.062i	−0.668432	−	0.668432i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			767.644i			1.83209i	0.401081	+	0.916043i	$0.368635\pi$
$419$			767.644i			1.83209i	−0.401081	+	0.916043i	$0.631365\pi$
$420$		0			0
$421$	446.722			1.06110			0.530549	−	0.847655i	$-0.321986\pi$
$421$	446.722			1.06110			0.530549	+	0.847655i	$0.321986\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	384.691	−	384.691i	0.900915	−	0.900915i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−303.791			−0.704851			−0.352425	−	0.935840i	$-0.614643\pi$
$431$	−303.791			−0.704851			−0.352425	+	0.935840i	$0.614643\pi$
$432$		0			0
$433$	610.350	+	610.350i	1.40958	+	1.40958i	0.761959	+	0.647625i	$0.224238\pi$
$433$	610.350	+	610.350i	1.40958	+	1.40958i	0.647625	+	0.761959i	$0.275762\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.38750	−	2.38750i	0.00546339	−	0.00546339i
$438$		0			0
$439$			737.925i			1.68092i	0.541872	+	0.840461i	$0.317716\pi$
$439$			737.925i			1.68092i	−0.541872	+	0.840461i	$0.682284\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	481.508	+	481.508i	1.08693	+	1.08693i	0.995843	+	0.0910817i	$0.0290324\pi$
$443$	481.508	+	481.508i	1.08693	+	1.08693i	0.0910817	+	0.995843i	$0.470968\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	568.428i		−	1.26599i	−0.774157	−	0.632993i	$-0.781826\pi$
$449$		−	568.428i		−	1.26599i	0.774157	−	0.632993i	$-0.218174\pi$
$450$		0			0
$451$	62.6188			0.138844
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	108.791	−	108.791i	0.238054	−	0.238054i	−0.577990	−	0.816044i	$-0.696163\pi$
$457$	108.791	−	108.791i	0.238054	−	0.238054i	0.816044	+	0.577990i	$0.196163\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−348.281			−0.755491			−0.377745	−	0.925910i	$-0.623301\pi$
$461$	−348.281			−0.755491			−0.377745	+	0.925910i	$0.623301\pi$
$462$		0			0
$463$	307.330	+	307.330i	0.663779	+	0.663779i	0.956269	−	0.292490i	$-0.0944837\pi$
$463$	307.330	+	307.330i	0.663779	+	0.663779i	−0.292490	+	0.956269i	$0.594484\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	50.3453	−	50.3453i	0.107806	−	0.107806i	−0.651146	−	0.758952i	$-0.725712\pi$
$467$	50.3453	−	50.3453i	0.107806	−	0.107806i	0.758952	+	0.651146i	$0.225712\pi$
$468$		0			0
$469$		−	465.109i		−	0.991704i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−37.4656	−	37.4656i	−0.0792085	−	0.0792085i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$			445.862i			0.930819i	0.885095	+	0.465410i	$0.154093\pi$
$479$			445.862i			0.930819i	−0.885095	+	0.465410i	$0.845907\pi$
$480$		0			0
$481$	−305.706			−0.635564
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	322.136	−	322.136i	0.661470	−	0.661470i	−0.294256	−	0.955727i	$-0.595072\pi$
$487$	322.136	−	322.136i	0.661470	−	0.661470i	0.955727	+	0.294256i	$0.0950720\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	362.953			0.739212			0.369606	−	0.929189i	$-0.379493\pi$
$491$	362.953			0.739212			0.369606	+	0.929189i	$0.379493\pi$
$492$		0			0
$493$	109.612	+	109.612i	0.222338	+	0.222338i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	304.272	−	304.272i	0.612217	−	0.612217i
$498$		0			0
$499$			555.831i			1.11389i	0.830549	+	0.556945i	$0.188027\pi$
$499$			555.831i			1.11389i	−0.830549	+	0.556945i	$0.811973\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−173.717	−	173.717i	−0.345362	−	0.345362i	0.513017	−	0.858379i	$-0.328528\pi$
$503$	−173.717	−	173.717i	−0.345362	−	0.345362i	−0.858379	+	0.513017i	$0.828528\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$			213.737i			0.419916i	0.977710	+	0.209958i	$0.0673328\pi$
$509$			213.737i			0.419916i	−0.977710	+	0.209958i	$0.932667\pi$
$510$		0			0
$511$	152.628			0.298685
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−14.7218	+	14.7218i	−0.0284755	+	0.0284755i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−312.094			−0.599028			−0.299514	−	0.954092i	$-0.596825\pi$
$521$	−312.094			−0.599028			−0.299514	+	0.954092i	$0.596825\pi$
$522$		0			0
$523$	197.655	+	197.655i	0.377925	+	0.377925i	0.870353	−	0.492428i	$-0.163891\pi$
$523$	197.655	+	197.655i	0.377925	+	0.377925i	−0.492428	+	0.870353i	$0.663891\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.24062	−	6.24062i	0.0118418	−	0.0118418i
$528$		0			0
$529$		−	528.978i		−	0.999959i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	642.784	+	642.784i	1.20597	+	1.20597i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	57.2782i		−	0.106267i
$540$		0			0
$541$	172.031			0.317988			0.158994	−	0.987280i	$-0.449175\pi$
$541$	172.031			0.317988			0.158994	+	0.987280i	$0.449175\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	60.9422	−	60.9422i	0.111412	−	0.111412i	−0.649203	−	0.760615i	$-0.724898\pi$
$547$	60.9422	−	60.9422i	0.111412	−	0.111412i	0.760615	+	0.649203i	$0.224898\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1040.25			1.88793
$552$		0			0
$553$	−579.350	−	579.350i	−1.04765	−	1.04765i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−441.575	+	441.575i	−0.792774	+	0.792774i	−0.981944	−	0.189171i	$-0.939420\pi$
$557$	−441.575	+	441.575i	−0.792774	+	0.792774i	0.189171	+	0.981944i	$0.439420\pi$
$558$		0			0
$559$		−	769.172i		−	1.37598i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	502.764	+	502.764i	0.893009	+	0.893009i	0.994805	−	0.101796i	$-0.0324590\pi$
$563$	502.764	+	502.764i	0.893009	+	0.893009i	−0.101796	+	0.994805i	$0.532459\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	518.753i		−	0.911693i	−0.890059	−	0.455846i	$-0.849337\pi$
$569$		−	518.753i		−	0.911693i	0.890059	−	0.455846i	$-0.150663\pi$
$570$		0			0
$571$	1034.07			1.81098			0.905492	−	0.424363i	$-0.139502\pi$
$571$	1034.07			1.81098			0.905492	+	0.424363i	$0.139502\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−220.172	+	220.172i	−0.381580	+	0.381580i	−0.871671	−	0.490091i	$-0.836964\pi$
$577$	−220.172	+	220.172i	−0.381580	+	0.381580i	0.490091	+	0.871671i	$0.336964\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1095.27			1.88514
$582$		0			0
$583$	−4.20937	−	4.20937i	−0.00722019	−	0.00722019i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−547.517	+	547.517i	−0.932738	+	0.932738i	−0.997876	−	0.0651384i	$-0.979251\pi$
$587$	−547.517	+	547.517i	−0.932738	+	0.932738i	0.0651384	+	0.997876i	$0.479251\pi$
$588$		0			0
$589$		−	59.2250i		−	0.100552i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−165.450	−	165.450i	−0.279005	−	0.279005i	0.553707	−	0.832712i	$-0.313213\pi$
$593$	−165.450	−	165.450i	−0.279005	−	0.279005i	−0.832712	+	0.553707i	$0.813213\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	997.987i		−	1.66609i	−0.553206	−	0.833045i	$-0.686596\pi$
$599$		−	997.987i		−	1.66609i	0.553206	−	0.833045i	$-0.313404\pi$
$600$		0			0
$601$	691.372			1.15037			0.575185	−	0.818024i	$-0.304930\pi$
$601$	691.372			1.15037			0.575185	+	0.818024i	$0.304930\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−115.833	+	115.833i	−0.190828	+	0.190828i	−0.796054	−	0.605226i	$-0.793083\pi$
$607$	−115.833	+	115.833i	−0.190828	+	0.190828i	0.605226	+	0.796054i	$0.293083\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−302.241			−0.494665
$612$		0			0
$613$	72.7906	+	72.7906i	0.118745	+	0.118745i	0.763982	−	0.645237i	$-0.223242\pi$
$613$	72.7906	+	72.7906i	0.118745	+	0.118745i	−0.645237	+	0.763982i	$0.723242\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17.5344	−	17.5344i	0.0284188	−	0.0284188i	−0.692755	−	0.721173i	$-0.743603\pi$
$617$	17.5344	−	17.5344i	0.0284188	−	0.0284188i	0.721173	+	0.692755i	$0.243603\pi$
$618$		0			0
$619$		−	1026.99i		−	1.65912i	−0.558419	−	0.829559i	$-0.688592\pi$
$619$		−	1026.99i		−	1.65912i	0.558419	−	0.829559i	$-0.311408\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−611.350	−	611.350i	−0.981300	−	0.981300i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	51.0063i		−	0.0810911i
$630$		0			0
$631$	246.241			0.390239			0.195119	−	0.980780i	$-0.437491\pi$
$631$	246.241			0.390239			0.195119	+	0.980780i	$0.437491\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	587.962	−	587.962i	0.923018	−	0.923018i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−825.822			−1.28833			−0.644167	−	0.764885i	$-0.722796\pi$
$641$	−825.822			−1.28833			−0.644167	+	0.764885i	$0.722796\pi$
$642$		0			0
$643$	−15.2453	−	15.2453i	−0.0237096	−	0.0237096i	0.695153	−	0.718862i	$-0.255337\pi$
$643$	−15.2453	−	15.2453i	−0.0237096	−	0.0237096i	−0.718862	+	0.695153i	$0.755337\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	359.414	−	359.414i	0.555509	−	0.555509i	−0.372517	−	0.928025i	$-0.621505\pi$
$647$	359.414	−	359.414i	0.555509	−	0.555509i	0.928025	+	0.372517i	$0.121505\pi$
$648$		0			0
$649$			57.8000i			0.0890600i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−247.187	−	247.187i	−0.378541	−	0.378541i	0.492034	−	0.870576i	$-0.336253\pi$
$653$	−247.187	−	247.187i	−0.378541	−	0.378541i	−0.870576	+	0.492034i	$0.836253\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	400.606i		−	0.607900i	−0.952688	−	0.303950i	$-0.901694\pi$
$659$		−	400.606i		−	0.607900i	0.952688	−	0.303950i	$-0.0983056\pi$
$660$		0			0
$661$	121.947			0.184488			0.0922442	−	0.995736i	$-0.470596\pi$
$661$	121.947			0.184488			0.0922442	+	0.995736i	$0.470596\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−4.77501	+	4.77501i	−0.00715893	+	0.00715893i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−80.5437			−0.120035
$672$		0			0
$673$	645.450	+	645.450i	0.959064	+	0.959064i	0.999194	−	0.0401306i	$-0.0127774\pi$
$673$	645.450	+	645.450i	0.959064	+	0.959064i	−0.0401306	+	0.999194i	$0.512777\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−684.087	+	684.087i	−1.01047	+	1.01047i	−0.0105243	+	0.999945i	$0.503350\pi$
$677$	−684.087	+	684.087i	−1.01047	+	1.01047i	−0.999945	+	0.0105243i	$0.996650\pi$
$678$		0			0
$679$		−	656.753i		−	0.967236i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	164.252	+	164.252i	0.240485	+	0.240485i	0.817051	−	0.576565i	$-0.195607\pi$
$683$	164.252	+	164.252i	0.240485	+	0.240485i	−0.576565	+	0.817051i	$0.695607\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	86.4187i		−	0.125426i
$690$		0			0
$691$	−241.403			−0.349353			−0.174677	−	0.984626i	$-0.555888\pi$
$691$	−241.403			−0.349353			−0.174677	+	0.984626i	$0.555888\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−107.247	+	107.247i	−0.153869	+	0.153869i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−353.822			−0.504739			−0.252369	−	0.967631i	$-0.581210\pi$
$701$	−353.822			−0.504739			−0.252369	+	0.967631i	$0.581210\pi$
$702$		0			0
$703$	−242.031	−	242.031i	−0.344283	−	0.344283i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	402.303	−	402.303i	0.569028	−	0.569028i
$708$		0			0
$709$		−	362.762i		−	0.511654i	−0.966723	−	0.255827i	$-0.917652\pi$
$709$		−	362.762i		−	0.511654i	0.966723	−	0.255827i	$-0.0823477\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.271857	+	0.271857i	0.000381287	+	0.000381287i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1191.41i		−	1.65704i	−0.559959	−	0.828520i	$-0.689183\pi$
$719$		−	1191.41i		−	1.65704i	0.559959	−	0.828520i	$-0.310817\pi$
$720$		0			0
$721$	824.397			1.14341
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−304.502	+	304.502i	−0.418847	+	0.418847i	−0.884806	−	0.465959i	$-0.845709\pi$
$727$	−304.502	+	304.502i	−0.418847	+	0.418847i	0.465959	+	0.884806i	$0.345709\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	128.334			0.175560
$732$		0			0
$733$	117.263	+	117.263i	0.159976	+	0.159976i	0.782556	−	0.622580i	$-0.213915\pi$
$733$	117.263	+	117.263i	0.159976	+	0.159976i	−0.622580	+	0.782556i	$0.713915\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−48.6906	+	48.6906i	−0.0660659	+	0.0660659i
$738$		0			0
$739$			692.669i			0.937305i	0.883383	+	0.468653i	$0.155260\pi$
$739$			692.669i			0.937305i	−0.883383	+	0.468653i	$0.844740\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	826.301	+	826.301i	1.11212	+	1.11212i	0.992864	+	0.119251i	$0.0380493\pi$
$743$	826.301	+	826.301i	1.11212	+	1.11212i	0.119251	+	0.992864i	$0.461951\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			905.234i			1.20859i
$750$		0			0
$751$	61.4031			0.0817618			0.0408809	−	0.999164i	$-0.486984\pi$
$751$	61.4031			0.0817618			0.0408809	+	0.999164i	$0.486984\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−366.372	+	366.372i	−0.483979	+	0.483979i	−0.906400	−	0.422421i	$-0.861180\pi$
$757$	−366.372	+	366.372i	−0.483979	+	0.483979i	0.422421	+	0.906400i	$0.361180\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	895.675			1.17697			0.588486	−	0.808508i	$-0.299724\pi$
$761$	895.675			1.17697			0.588486	+	0.808508i	$0.299724\pi$
$762$		0			0
$763$	127.644	+	127.644i	0.167292	+	0.167292i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−593.319	+	593.319i	−0.773558	+	0.773558i
$768$		0			0
$769$			241.675i			0.314272i	0.987577	+	0.157136i	$0.0502261\pi$
$769$			241.675i			0.314272i	−0.987577	+	0.157136i	$0.949774\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	359.459	+	359.459i	0.465019	+	0.465019i	0.900296	−	0.435278i	$-0.143350\pi$
$773$	359.459	+	359.459i	0.465019	+	0.465019i	−0.435278	+	0.900296i	$0.643350\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1017.80i			1.30655i
$780$		0			0
$781$	−63.7062			−0.0815701
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−171.789	+	171.789i	−0.218283	+	0.218283i	−0.807775	−	0.589491i	$-0.799328\pi$
$787$	−171.789	+	171.789i	−0.218283	+	0.218283i	0.589491	+	0.807775i	$0.299328\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−118.241			−0.149482
$792$		0			0
$793$	−826.784	−	826.784i	−1.04260	−	1.04260i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−547.891	+	547.891i	−0.687441	+	0.687441i	−0.961666	−	0.274224i	$-0.911579\pi$
$797$	−547.891	+	547.891i	−0.687441	+	0.687441i	0.274224	+	0.961666i	$0.411579\pi$
$798$		0			0
$799$		−	50.4281i		−	0.0631140i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−15.9781	−	15.9781i	−0.0198980	−	0.0198980i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			225.925i			0.279264i	0.990203	+	0.139632i	$0.0445920\pi$
$809$			225.925i			0.279264i	−0.990203	+	0.139632i	$0.955408\pi$
$810$		0			0
$811$	1083.03			1.33543			0.667715	−	0.744417i	$-0.267272\pi$
$811$	1083.03			1.33543			0.667715	+	0.744417i	$0.267272\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	608.962	−	608.962i	0.745364	−	0.745364i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−125.297			−0.152615			−0.0763074	−	0.997084i	$-0.524313\pi$
$821$	−125.297			−0.152615			−0.0763074	+	0.997084i	$0.524313\pi$
$822$		0			0
$823$	408.480	+	408.480i	0.496330	+	0.496330i	0.910293	−	0.413963i	$-0.135856\pi$
$823$	408.480	+	408.480i	0.496330	+	0.496330i	−0.413963	+	0.910293i	$0.635856\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	705.508	−	705.508i	0.853093	−	0.853093i	−0.137420	−	0.990513i	$-0.543881\pi$
$827$	705.508	−	705.508i	0.853093	−	0.853093i	0.990513	+	0.137420i	$0.0438810\pi$
$828$		0			0
$829$			1040.20i			1.25476i	0.778713	+	0.627380i	$0.215873\pi$
$829$			1040.20i			1.25476i	−0.778713	+	0.627380i	$0.784127\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	98.1000	+	98.1000i	0.117767	+	0.117767i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$			154.512i			0.184163i	0.995751	+	0.0920813i	$0.0293520\pi$
$839$			154.512i			0.184163i	−0.995751	+	0.0920813i	$0.970648\pi$
$840$		0			0
$841$	−1239.50			−1.47384
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	797.695	−	797.695i	0.941789	−	0.941789i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.22197			0.00261101
$852$		0			0
$853$	572.141	+	572.141i	0.670739	+	0.670739i	0.957886	−	0.287147i	$-0.0927069\pi$
$853$	572.141	+	572.141i	0.670739	+	0.670739i	−0.287147	+	0.957886i	$0.592707\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−81.2625	+	81.2625i	−0.0948221	+	0.0948221i	−0.752927	−	0.658105i	$-0.771358\pi$
$857$	−81.2625	+	81.2625i	−0.0948221	+	0.0948221i	0.658105	+	0.752927i	$0.271358\pi$
$858$		0			0
$859$		−	214.094i		−	0.249236i	−0.992205	−	0.124618i	$-0.960229\pi$
$859$		−	214.094i		−	0.249236i	0.992205	−	0.124618i	$-0.0397705\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−72.8172	−	72.8172i	−0.0843768	−	0.0843768i	0.663659	−	0.748036i	$-0.269003\pi$
$863$	−72.8172	−	72.8172i	−0.0843768	−	0.0843768i	−0.748036	+	0.663659i	$0.769003\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			121.300i			0.139586i
$870$		0			0
$871$	−999.622			−1.14767
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	922.841	−	922.841i	1.05227	−	1.05227i	0.0537133	−	0.998556i	$-0.482894\pi$
$877$	922.841	−	922.841i	1.05227	−	1.05227i	0.998556	−	0.0537133i	$-0.0171057\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−364.922			−0.414213			−0.207107	−	0.978318i	$-0.566405\pi$
$881$	−364.922			−0.414213			−0.207107	+	0.978318i	$0.566405\pi$
$882$		0			0
$883$	274.786	+	274.786i	0.311196	+	0.311196i	0.845373	−	0.534177i	$-0.179378\pi$
$883$	274.786	+	274.786i	0.311196	+	0.311196i	−0.534177	+	0.845373i	$0.679378\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	609.464	−	609.464i	0.687107	−	0.687107i	−0.274484	−	0.961592i	$-0.588507\pi$
$887$	609.464	−	609.464i	0.687107	−	0.687107i	0.961592	+	0.274484i	$0.0885072\pi$
$888$		0			0
$889$		−	197.047i		−	0.221650i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−239.287	−	239.287i	−0.267959	−	0.267959i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			118.450i			0.131757i
$900$		0			0
$901$	14.4187			0.0160030
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−112.439	+	112.439i	−0.123968	+	0.123968i	−0.766369	−	0.642401i	$-0.777938\pi$
$907$	−112.439	+	112.439i	−0.123968	+	0.123968i	0.642401	+	0.766369i	$0.277938\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1469.90			−1.61350			−0.806752	−	0.590890i	$-0.798777\pi$
$911$	−1469.90			−1.61350			−0.806752	+	0.590890i	$0.798777\pi$
$912$		0			0
$913$	−114.659	−	114.659i	−0.125585	−	0.125585i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1594.20	−	1594.20i	1.73849	−	1.73849i
$918$		0			0
$919$			503.662i			0.548055i	0.961722	+	0.274027i	$0.0883559\pi$
$919$			503.662i			0.548055i	−0.961722	+	0.274027i	$0.911644\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−653.947	−	653.947i	−0.708501	−	0.708501i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1016.43i			1.09411i	0.837097	+	0.547055i	$0.184251\pi$
$929$			1016.43i			1.09411i	−0.837097	+	0.547055i	$0.815749\pi$
$930$		0			0
$931$	930.994			0.999993
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	678.841	−	678.841i	0.724483	−	0.724483i	−0.245032	−	0.969515i	$-0.578798\pi$
$937$	678.841	−	678.841i	0.724483	−	0.724483i	0.969515	+	0.245032i	$0.0787985\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	603.987			0.641857			0.320928	−	0.947103i	$-0.396005\pi$
$941$	603.987			0.641857			0.320928	+	0.947103i	$0.396005\pi$
$942$		0			0
$943$	−4.67196	−	4.67196i	−0.00495435	−	0.00495435i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.02658	−	1.02658i	0.00108403	−	0.00108403i	−0.706565	−	0.707649i	$-0.749756\pi$
$947$	1.02658	−	1.02658i	0.00108403	−	0.00108403i	0.707649	+	0.706565i	$0.249756\pi$
$948$		0			0
$949$		−	328.031i		−	0.345660i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	531.459	+	531.459i	0.557670	+	0.557670i	0.928643	−	0.370974i	$-0.120976\pi$
$953$	531.459	+	531.459i	0.557670	+	0.557670i	−0.370974	+	0.928643i	$0.620976\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1160.88i			1.21051i
$960$		0			0
$961$	−954.256			−0.992983
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	303.392	−	303.392i	0.313746	−	0.313746i	−0.532613	−	0.846359i	$-0.678790\pi$
$967$	303.392	−	303.392i	0.313746	−	0.313746i	0.846359	+	0.532613i	$0.178790\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1156.04			1.19057			0.595284	−	0.803516i	$-0.297040\pi$
$971$	1156.04			1.19057			0.595284	+	0.803516i	$0.297040\pi$
$972$		0			0
$973$	−1653.99	−	1653.99i	−1.69988	−	1.69988i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−287.000	+	287.000i	−0.293756	+	0.293756i	−0.838562	−	0.544806i	$-0.816603\pi$
$977$	−287.000	+	287.000i	−0.293756	+	0.293756i	0.544806	+	0.838562i	$0.316603\pi$
$978$		0			0
$979$			128.000i			0.130746i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−868.523	−	868.523i	−0.883544	−	0.883544i	0.110349	−	0.993893i	$-0.464803\pi$
$983$	−868.523	−	868.523i	−0.883544	−	0.883544i	−0.993893	+	0.110349i	$0.964803\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			5.59058i			0.00565276i
$990$		0			0
$991$	−1263.28			−1.27476			−0.637379	−	0.770551i	$-0.719981\pi$
$991$	−1263.28			−1.27476			−0.637379	+	0.770551i	$0.719981\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−992.547	+	992.547i	−0.995533	+	0.995533i	−0.999990	−	0.00445673i	$-0.998581\pi$
$997$	−992.547	+	992.547i	−0.995533	+	0.995533i	0.00445673	+	0.999990i	$0.498581\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.3.v.k.793.1		4
3.2	odd	2		200.3.l.e.193.1		4
5.2	odd	4	inner	1800.3.v.k.1657.1		4
5.3	odd	4		360.3.v.c.217.2		4
5.4	even	2		360.3.v.c.73.2		4
12.11	even	2		400.3.p.i.193.2		4
15.2	even	4		200.3.l.e.57.1		4
15.8	even	4		40.3.l.b.17.2	✓	4
15.14	odd	2		40.3.l.b.33.2	yes	4
20.3	even	4		720.3.bh.l.577.2		4
20.19	odd	2		720.3.bh.l.433.2		4
60.23	odd	4		80.3.p.d.17.1		4
60.47	odd	4		400.3.p.i.257.2		4
60.59	even	2		80.3.p.d.33.1		4
120.29	odd	2		320.3.p.l.193.1		4
120.53	even	4		320.3.p.l.257.1		4
120.59	even	2		320.3.p.i.193.2		4
120.83	odd	4		320.3.p.i.257.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.3.l.b.17.2	✓	4	15.8	even	4
40.3.l.b.33.2	yes	4	15.14	odd	2
80.3.p.d.17.1		4	60.23	odd	4
80.3.p.d.33.1		4	60.59	even	2
200.3.l.e.57.1		4	15.2	even	4
200.3.l.e.193.1		4	3.2	odd	2
320.3.p.i.193.2		4	120.59	even	2
320.3.p.i.257.2		4	120.83	odd	4
320.3.p.l.193.1		4	120.29	odd	2
320.3.p.l.257.1		4	120.53	even	4
360.3.v.c.73.2		4	5.4	even	2
360.3.v.c.217.2		4	5.3	odd	4
400.3.p.i.193.2		4	12.11	even	2
400.3.p.i.257.2		4	60.47	odd	4
720.3.bh.l.433.2		4	20.19	odd	2
720.3.bh.l.577.2		4	20.3	even	4
1800.3.v.k.793.1		4	1.1	even	1	trivial
1800.3.v.k.1657.1		4	5.2	odd	4	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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1800.v (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-6.70156 + 6.70156i) q^{7} +O(q^{10})\)	\(q+(-6.70156 + 6.70156i) q^{7} +1.40312 q^{11} +(14.4031 + 14.4031i) q^{13} +(-2.40312 + 2.40312i) q^{17} +22.8062i q^{19} +(-0.104686 - 0.104686i) q^{23} -45.6125i q^{29} -2.59688 q^{31} +(-10.6125 + 10.6125i) q^{37} +44.6281 q^{41} +(-26.7016 - 26.7016i) q^{43} +(-10.4922 + 10.4922i) q^{47} -40.8219i q^{49} +(-3.00000 - 3.00000i) q^{53} +41.1938i q^{59} -57.4031 q^{61} +(-34.7016 + 34.7016i) q^{67} -45.4031 q^{71} +(-11.3875 - 11.3875i) q^{73} +(-9.40312 + 9.40312i) q^{77} +86.4500i q^{79} +(-81.7172 - 81.7172i) q^{83} +91.2250i q^{89} -193.047 q^{91} +(-49.0000 + 49.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{7}+O(q^{10}) \) 4 * q - 14 * q^7	\( 4 q - 14 q^{7} - 20 q^{11} + 32 q^{13} + 16 q^{17} + 38 q^{23} - 36 q^{31} + 60 q^{37} - 52 q^{41} - 94 q^{43} - 106 q^{47} - 12 q^{53} - 204 q^{61} - 126 q^{67} - 156 q^{71} - 148 q^{73} - 12 q^{77} - 186 q^{83} - 388 q^{91} - 196 q^{97}+O(q^{100}) \) 4 * q - 14 * q^7 - 20 * q^11 + 32 * q^13 + 16 * q^17 + 38 * q^23 - 36 * q^31 + 60 * q^37 - 52 * q^41 - 94 * q^43 - 106 * q^47 - 12 * q^53 - 204 * q^61 - 126 * q^67 - 156 * q^71 - 148 * q^73 - 12 * q^77 - 186 * q^83 - 388 * q^91 - 196 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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