LMFDB - Embedded newform 4560.2.d.i.2431.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(2431,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4560.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.d (of order $2$, degree $1$, 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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 36 x^{8} - 128 x^{7} + 232 x^{6} + 104 x^{5} + 324 x^{4} + \cdots + 64 $ x^12 - 4x^11 + 8x^10 + 4x^9 + 36x^8 - 128x^7 + 232x^6 + 104x^5 + 324x^4 - 784x^3 + 800x^2 - 320*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.12
Root			$-1.15765 - 1.15765i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.i.2431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +5.16275i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +5.16275i q^{7} +1.00000 q^{9} -4.48246i q^{11} +1.15254i q^{13} +1.00000 q^{15} +0.995591 q^{17} +(-3.36000 - 2.77676i) q^{19} -5.16275i q^{21} -2.60109i q^{23} +1.00000 q^{25} -1.00000 q^{27} +9.93774i q^{29} -2.14440 q^{31} +4.48246i q^{33} -5.16275i q^{35} -7.02135i q^{37} -1.15254i q^{39} +11.4810i q^{41} -3.55725i q^{43} -1.00000 q^{45} +5.00441i q^{47} -19.6540 q^{49} -0.995591 q^{51} -5.43460i q^{53} +4.48246i q^{55} +(3.36000 + 2.77676i) q^{57} -4.77199 q^{59} -2.92520 q^{61} +5.16275i q^{63} -1.15254i q^{65} -8.46269 q^{67} +2.60109i q^{69} +6.99040 q^{71} +3.86881 q^{73} -1.00000 q^{75} +23.1418 q^{77} -12.8235 q^{79} +1.00000 q^{81} -14.3095i q^{83} -0.995591 q^{85} -9.93774i q^{87} -10.0067i q^{89} -5.95029 q^{91} +2.14440 q^{93} +(3.36000 + 2.77676i) q^{95} -15.9863i q^{97} -4.48246i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 12 q^{5} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 12 * q^5 + 12 * q^9	$ 12 q - 12 q^{3} - 12 q^{5} + 12 q^{9} + 12 q^{15} - 8 q^{17} - 8 q^{19} + 12 q^{25} - 12 q^{27} - 12 q^{45} - 20 q^{49} + 8 q^{51} + 8 q^{57} - 8 q^{59} + 8 q^{67} - 32 q^{71} - 24 q^{73} - 12 q^{75} + 56 q^{77} - 16 q^{79} + 12 q^{81} + 8 q^{85} + 16 q^{91} + 8 q^{95}+O(q^{100}) $ 12 * q - 12 * q^3 - 12 * q^5 + 12 * q^9 + 12 * q^15 - 8 * q^17 - 8 * q^19 + 12 * q^25 - 12 * q^27 - 12 * q^45 - 20 * q^49 + 8 * q^51 + 8 * q^57 - 8 * q^59 + 8 * q^67 - 32 * q^71 - 24 * q^73 - 12 * q^75 + 56 * q^77 - 16 * q^79 + 12 * q^81 + 8 * q^85 + 16 * q^91 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times$.

$n$	$1141$	$1711$	$1921$	$2737$	$3041$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			5.16275i			1.95134i	0.219251	+	0.975668i	$0.429639\pi$
$7$			5.16275i			1.95134i	−0.219251	+	0.975668i	$0.570361\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	4.48246i		−	1.35151i	−0.737126	−	0.675756i	$-0.763817\pi$
$11$		−	4.48246i		−	1.35151i	0.737126	−	0.675756i	$-0.236183\pi$
$12$		0			0
$13$			1.15254i			0.319658i	0.987145	+	0.159829i	$0.0510943\pi$
$13$			1.15254i			0.319658i	−0.987145	+	0.159829i	$0.948906\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	0.995591			0.241466			0.120733	−	0.992685i	$-0.461475\pi$
$17$	0.995591			0.241466			0.120733	+	0.992685i	$0.461475\pi$
$18$		0			0
$19$	−3.36000	−	2.77676i	−0.770837	−	0.637032i
$19$	−3.36000	−	2.77676i	−0.770837	−	0.637032i
$20$		0			0
$21$		−	5.16275i		−	1.12660i
$22$		0			0
$23$		−	2.60109i		−	0.542365i	−0.962528	−	0.271182i	$-0.912585\pi$
$23$		−	2.60109i		−	0.542365i	0.962528	−	0.271182i	$-0.0874147\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$			9.93774i			1.84539i	0.385528	+	0.922696i	$0.374019\pi$
$29$			9.93774i			1.84539i	−0.385528	+	0.922696i	$0.625981\pi$
$30$		0			0
$31$	−2.14440			−0.385145			−0.192573	−	0.981283i	$-0.561683\pi$
$31$	−2.14440			−0.385145			−0.192573	+	0.981283i	$0.561683\pi$
$32$		0			0
$33$			4.48246i			0.780295i
$34$		0			0
$35$		−	5.16275i		−	0.872664i
$36$		0			0
$37$		−	7.02135i		−	1.15430i	−0.816637	−	0.577152i	$-0.804164\pi$
$37$		−	7.02135i		−	1.15430i	0.816637	−	0.577152i	$-0.195836\pi$
$38$		0			0
$39$		−	1.15254i		−	0.184555i
$40$		0			0
$41$			11.4810i			1.79304i	0.443005	+	0.896519i	$0.353912\pi$
$41$			11.4810i			1.79304i	−0.443005	+	0.896519i	$0.646088\pi$
$42$		0			0
$43$		−	3.55725i		−	0.542476i	−0.962512	−	0.271238i	$-0.912567\pi$
$43$		−	3.55725i		−	0.542476i	0.962512	−	0.271238i	$-0.0874330\pi$
$44$		0			0
$45$	−1.00000			−0.149071
$46$		0			0
$47$			5.00441i			0.729968i	0.931014	+	0.364984i	$0.118926\pi$
$47$			5.00441i			0.729968i	−0.931014	+	0.364984i	$0.881074\pi$
$48$		0			0
$49$	−19.6540			−2.80772
$50$		0			0
$51$	−0.995591			−0.139411
$52$		0			0
$53$		−	5.43460i		−	0.746500i	−0.927731	−	0.373250i	$-0.878243\pi$
$53$		−	5.43460i		−	0.746500i	0.927731	−	0.373250i	$-0.121757\pi$
$54$		0			0
$55$			4.48246i			0.604414i
$56$		0			0
$57$	3.36000	+	2.77676i	0.445043	+	0.367791i
$58$		0			0
$59$	−4.77199			−0.621260			−0.310630	−	0.950531i	$-0.600540\pi$
$59$	−4.77199			−0.621260			−0.310630	+	0.950531i	$0.600540\pi$
$60$		0			0
$61$	−2.92520			−0.374534			−0.187267	−	0.982309i	$-0.559963\pi$
$61$	−2.92520			−0.374534			−0.187267	+	0.982309i	$0.559963\pi$
$62$		0			0
$63$			5.16275i			0.650446i
$64$		0			0
$65$		−	1.15254i		−	0.142955i
$66$		0			0
$67$	−8.46269			−1.03388			−0.516941	−	0.856021i	$-0.672929\pi$
$67$	−8.46269			−1.03388			−0.516941	+	0.856021i	$0.672929\pi$
$68$		0			0
$69$			2.60109i			0.313134i
$70$		0			0
$71$	6.99040			0.829607			0.414804	−	0.909911i	$-0.363850\pi$
$71$	6.99040			0.829607			0.414804	+	0.909911i	$0.363850\pi$
$72$		0			0
$73$	3.86881			0.452810			0.226405	−	0.974033i	$-0.427303\pi$
$73$	3.86881			0.452810			0.226405	+	0.974033i	$0.427303\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	23.1418			2.63725
$78$		0			0
$79$	−12.8235			−1.44276			−0.721379	−	0.692540i	$-0.756491\pi$
$79$	−12.8235			−1.44276			−0.721379	+	0.692540i	$0.756491\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	14.3095i		−	1.57067i	−0.619072	−	0.785335i	$-0.712491\pi$
$83$		−	14.3095i		−	1.57067i	0.619072	−	0.785335i	$-0.287509\pi$
$84$		0			0
$85$	−0.995591			−0.107987
$86$		0			0
$87$		−	9.93774i		−	1.06544i
$88$		0			0
$89$		−	10.0067i		−	1.06071i	−0.847775	−	0.530356i	$-0.822058\pi$
$89$		−	10.0067i		−	1.06071i	0.847775	−	0.530356i	$-0.177942\pi$
$90$		0			0
$91$	−5.95029			−0.623760
$92$		0			0
$93$	2.14440			0.222364
$94$		0			0
$95$	3.36000	+	2.77676i	0.344729	+	0.284889i
$96$		0			0
$97$		−	15.9863i		−	1.62316i	−0.584242	−	0.811580i	$-0.698608\pi$
$97$		−	15.9863i		−	1.62316i	0.584242	−	0.811580i	$-0.301392\pi$
$98$		0			0
$99$		−	4.48246i		−	0.450504i
$100$		0			0
$101$	13.7451			1.36769			0.683844	−	0.729628i	$-0.260307\pi$
$101$	13.7451			1.36769			0.683844	+	0.729628i	$0.260307\pi$
$102$		0			0
$103$	−0.725070			−0.0714433			−0.0357216	−	0.999362i	$-0.511373\pi$
$103$	−0.725070			−0.0714433			−0.0357216	+	0.999362i	$0.511373\pi$
$104$		0			0
$105$			5.16275i			0.503833i
$106$		0			0
$107$	−3.03391			−0.293299			−0.146650	−	0.989189i	$-0.546849\pi$
$107$	−3.03391			−0.293299			−0.146650	+	0.989189i	$0.546849\pi$
$108$		0			0
$109$			16.9561i			1.62410i	0.583588	+	0.812050i	$0.301648\pi$
$109$			16.9561i			1.62410i	−0.583588	+	0.812050i	$0.698352\pi$
$110$		0			0
$111$			7.02135i			0.666437i
$112$		0			0
$113$			9.73087i			0.915403i	0.889106	+	0.457702i	$0.151327\pi$
$113$			9.73087i			0.915403i	−0.889106	+	0.457702i	$0.848673\pi$
$114$		0			0
$115$			2.60109i			0.242553i
$116$		0			0
$117$			1.15254i			0.106553i
$118$		0			0
$119$			5.13999i			0.471182i
$120$		0			0
$121$	−9.09241			−0.826583
$122$		0			0
$123$		−	11.4810i		−	1.03521i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	3.70373			0.328653			0.164327	−	0.986406i	$-0.447455\pi$
$127$	3.70373			0.328653			0.164327	+	0.986406i	$0.447455\pi$
$128$		0			0
$129$			3.55725i			0.313199i
$130$		0			0
$131$			2.37756i			0.207728i	0.994592	+	0.103864i	$0.0331207\pi$
$131$			2.37756i			0.207728i	−0.994592	+	0.103864i	$0.966879\pi$
$132$		0			0
$133$	14.3357	−	17.3469i	1.24306	−	1.50416i
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	9.36587			0.800180			0.400090	−	0.916476i	$-0.368979\pi$
$137$	9.36587			0.800180			0.400090	+	0.916476i	$0.368979\pi$
$138$		0			0
$139$		−	15.7808i		−	1.33851i	−0.743033	−	0.669254i	$-0.766614\pi$
$139$		−	15.7808i		−	1.33851i	0.743033	−	0.669254i	$-0.233386\pi$
$140$		0			0
$141$		−	5.00441i		−	0.421447i
$142$		0			0
$143$	5.16622			0.432021
$144$		0			0
$145$		−	9.93774i		−	0.825284i
$146$		0			0
$147$	19.6540			1.62104
$148$		0			0
$149$	−5.43760			−0.445466			−0.222733	−	0.974880i	$-0.571498\pi$
$149$	−5.43760			−0.445466			−0.222733	+	0.974880i	$0.571498\pi$
$150$		0			0
$151$	−5.41557			−0.440713			−0.220357	−	0.975419i	$-0.570722\pi$
$151$	−5.41557			−0.440713			−0.220357	+	0.975419i	$0.570722\pi$
$152$		0			0
$153$	0.995591			0.0804887
$154$		0			0
$155$	2.14440			0.172242
$156$		0			0
$157$	20.4933			1.63554			0.817772	−	0.575542i	$-0.195209\pi$
$157$	20.4933			1.63554			0.817772	+	0.575542i	$0.195209\pi$
$158$		0			0
$159$			5.43460i			0.430992i
$160$		0			0
$161$	13.4288			1.05834
$162$		0			0
$163$		−	1.99765i		−	0.156468i	−0.996935	−	0.0782342i	$-0.975072\pi$
$163$		−	1.99765i		−	0.156468i	0.996935	−	0.0782342i	$-0.0249282\pi$
$164$		0			0
$165$		−	4.48246i		−	0.348959i
$166$		0			0
$167$	−6.57140			−0.508510			−0.254255	−	0.967137i	$-0.581830\pi$
$167$	−6.57140			−0.508510			−0.254255	+	0.967137i	$0.581830\pi$
$168$		0			0
$169$	11.6716			0.897819
$170$		0			0
$171$	−3.36000	−	2.77676i	−0.256946	−	0.212344i
$172$		0			0
$173$		−	20.5228i		−	1.56032i	−0.625579	−	0.780160i	$-0.715137\pi$
$173$		−	20.5228i		−	1.56032i	0.625579	−	0.780160i	$-0.284863\pi$
$174$		0			0
$175$			5.16275i			0.390267i
$176$		0			0
$177$	4.77199			0.358685
$178$		0			0
$179$	−12.7469			−0.952746			−0.476373	−	0.879243i	$-0.658049\pi$
$179$	−12.7469			−0.952746			−0.476373	+	0.879243i	$0.658049\pi$
$180$		0			0
$181$		−	20.4121i		−	1.51722i	−0.651545	−	0.758610i	$-0.725879\pi$
$181$		−	20.4121i		−	1.51722i	0.651545	−	0.758610i	$-0.274121\pi$
$182$		0			0
$183$	2.92520			0.216237
$184$		0			0
$185$			7.02135i			0.516220i
$186$		0			0
$187$		−	4.46269i		−	0.326344i
$188$		0			0
$189$		−	5.16275i		−	0.375535i
$190$		0			0
$191$		−	6.85573i		−	0.496063i	−0.968752	−	0.248032i	$-0.920216\pi$
$191$		−	6.85573i		−	0.496063i	0.968752	−	0.248032i	$-0.0797837\pi$
$192$		0			0
$193$		−	20.6986i		−	1.48992i	−0.667109	−	0.744960i	$-0.732469\pi$
$193$		−	20.6986i		−	1.48992i	0.667109	−	0.744960i	$-0.267531\pi$
$194$		0			0
$195$			1.15254i			0.0825353i
$196$		0			0
$197$	−5.77774			−0.411647			−0.205824	−	0.978589i	$-0.565987\pi$
$197$	−5.77774			−0.411647			−0.205824	+	0.978589i	$0.565987\pi$
$198$		0			0
$199$		−	6.45943i		−	0.457897i	−0.973439	−	0.228948i	$-0.926471\pi$
$199$		−	6.45943i		−	0.457897i	0.973439	−	0.228948i	$-0.0735287\pi$
$200$		0			0
$201$	8.46269			0.596912
$202$		0			0
$203$	−51.3061			−3.60098
$204$		0			0
$205$		−	11.4810i		−	0.801871i
$206$		0			0
$207$		−	2.60109i		−	0.180788i
$208$		0			0
$209$	−12.4467	+	15.0611i	−0.860956	+	1.04180i
$210$		0			0
$211$	−18.4870			−1.27270			−0.636349	−	0.771402i	$-0.719556\pi$
$211$	−18.4870			−1.27270			−0.636349	+	0.771402i	$0.719556\pi$
$212$		0			0
$213$	−6.99040			−0.478974
$214$		0			0
$215$			3.55725i			0.242603i
$216$		0			0
$217$		−	11.0710i		−	0.751548i
$218$		0			0
$219$	−3.86881			−0.261430
$220$		0			0
$221$			1.14746i			0.0771866i
$222$		0			0
$223$	12.8114			0.857917			0.428958	−	0.903324i	$-0.358881\pi$
$223$	12.8114			0.857917			0.428958	+	0.903324i	$0.358881\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	26.8133			1.77966			0.889829	−	0.456294i	$-0.150823\pi$
$227$	26.8133			1.77966			0.889829	+	0.456294i	$0.150823\pi$
$228$		0			0
$229$	−6.85061			−0.452701			−0.226350	−	0.974046i	$-0.572679\pi$
$229$	−6.85061			−0.452701			−0.226350	+	0.974046i	$0.572679\pi$
$230$		0			0
$231$	−23.1418			−1.52262
$232$		0			0
$233$	−6.33196			−0.414820			−0.207410	−	0.978254i	$-0.566503\pi$
$233$	−6.33196			−0.414820			−0.207410	+	0.978254i	$0.566503\pi$
$234$		0			0
$235$		−	5.00441i		−	0.326452i
$236$		0			0
$237$	12.8235			0.832977
$238$		0			0
$239$		−	13.8565i		−	0.896304i	−0.893957	−	0.448152i	$-0.852082\pi$
$239$		−	13.8565i		−	0.896304i	0.893957	−	0.448152i	$-0.147918\pi$
$240$		0			0
$241$		−	5.30041i		−	0.341430i	−0.985320	−	0.170715i	$-0.945392\pi$
$241$		−	5.30041i		−	0.341430i	0.985320	−	0.170715i	$-0.0546077\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	19.6540			1.25565
$246$		0			0
$247$	3.20033	−	3.87255i	0.203632	−	0.246404i
$248$		0			0
$249$			14.3095i			0.906826i
$250$		0			0
$251$		−	14.4249i		−	0.910493i	−0.890366	−	0.455246i	$-0.849551\pi$
$251$		−	14.4249i		−	0.910493i	0.890366	−	0.455246i	$-0.150449\pi$
$252$		0			0
$253$	−11.6593			−0.733012
$254$		0			0
$255$	0.995591			0.0623463
$256$		0			0
$257$		−	18.7940i		−	1.17234i	−0.810189	−	0.586169i	$-0.800635\pi$
$257$		−	18.7940i		−	1.17234i	0.810189	−	0.586169i	$-0.199365\pi$
$258$		0			0
$259$	36.2495			2.25243
$260$		0			0
$261$			9.93774i			0.615131i
$262$		0			0
$263$			7.48658i			0.461643i	0.972996	+	0.230821i	$0.0741413\pi$
$263$			7.48658i			0.461643i	−0.972996	+	0.230821i	$0.925859\pi$
$264$		0			0
$265$			5.43460i			0.333845i
$266$		0			0
$267$			10.0067i			0.612402i
$268$		0			0
$269$		−	7.70524i		−	0.469797i	−0.972020	−	0.234898i	$-0.924524\pi$
$269$		−	7.70524i		−	0.469797i	0.972020	−	0.234898i	$-0.0754757\pi$
$270$		0			0
$271$		−	16.1502i		−	0.981053i	−0.871426	−	0.490527i	$-0.836804\pi$
$271$		−	16.1502i		−	0.981053i	0.871426	−	0.490527i	$-0.163196\pi$
$272$		0			0
$273$	5.95029			0.360128
$274$		0			0
$275$		−	4.48246i		−	0.270302i
$276$		0			0
$277$	−17.0926			−1.02699			−0.513496	−	0.858092i	$-0.671650\pi$
$277$	−17.0926			−1.02699			−0.513496	+	0.858092i	$0.671650\pi$
$278$		0			0
$279$	−2.14440			−0.128382
$280$		0			0
$281$			14.3099i			0.853660i	0.904332	+	0.426830i	$0.140370\pi$
$281$			14.3099i			0.853660i	−0.904332	+	0.426830i	$0.859630\pi$
$282$		0			0
$283$		−	18.0771i		−	1.07457i	−0.843401	−	0.537285i	$-0.819450\pi$
$283$		−	18.0771i		−	1.07457i	0.843401	−	0.537285i	$-0.180550\pi$
$284$		0			0
$285$	−3.36000	−	2.77676i	−0.199029	−	0.164481i
$286$		0			0
$287$	−59.2738			−3.49882
$288$		0			0
$289$	−16.0088			−0.941694
$290$		0			0
$291$			15.9863i			0.937132i
$292$		0			0
$293$			5.33637i			0.311754i	0.987776	+	0.155877i	$0.0498203\pi$
$293$			5.33637i			0.311754i	−0.987776	+	0.155877i	$0.950180\pi$
$294$		0			0
$295$	4.77199			0.277836
$296$		0			0
$297$			4.48246i			0.260098i
$298$		0			0
$299$	2.99787			0.173371
$300$		0			0
$301$	18.3652			1.05855
$302$		0			0
$303$	−13.7451			−0.789635
$304$		0			0
$305$	2.92520			0.167497
$306$		0			0
$307$	−4.91660			−0.280605			−0.140303	−	0.990109i	$-0.544808\pi$
$307$	−4.91660			−0.280605			−0.140303	+	0.990109i	$0.544808\pi$
$308$		0			0
$309$	0.725070			0.0412478
$310$		0			0
$311$		−	15.3870i		−	0.872518i	−0.899821	−	0.436259i	$-0.856303\pi$
$311$		−	15.3870i		−	0.872518i	0.899821	−	0.436259i	$-0.143697\pi$
$312$		0			0
$313$	−32.3545			−1.82878			−0.914391	−	0.404832i	$-0.867330\pi$
$313$	−32.3545			−1.82878			−0.914391	+	0.404832i	$0.867330\pi$
$314$		0			0
$315$		−	5.16275i		−	0.290888i
$316$		0			0
$317$		−	3.44342i		−	0.193402i	−0.995313	−	0.0967009i	$-0.969171\pi$
$317$		−	3.44342i		−	0.193402i	0.995313	−	0.0967009i	$-0.0308290\pi$
$318$		0			0
$319$	44.5455			2.49407
$320$		0			0
$321$	3.03391			0.169336
$322$		0			0
$323$	−3.34519	−	2.76451i	−0.186131	−	0.153822i
$324$		0			0
$325$			1.15254i			0.0639316i
$326$		0			0
$327$		−	16.9561i		−	0.937674i
$328$		0			0
$329$	−25.8365			−1.42441
$330$		0			0
$331$	−22.6438			−1.24462			−0.622309	−	0.782772i	$-0.713805\pi$
$331$	−22.6438			−1.24462			−0.622309	+	0.782772i	$0.713805\pi$
$332$		0			0
$333$		−	7.02135i		−	0.384768i
$334$		0			0
$335$	8.46269			0.462366
$336$		0			0
$337$			3.77848i			0.205827i	0.994690	+	0.102913i	$0.0328165\pi$
$337$			3.77848i			0.205827i	−0.994690	+	0.102913i	$0.967184\pi$
$338$		0			0
$339$		−	9.73087i		−	0.528508i
$340$		0			0
$341$			9.61217i			0.520528i
$342$		0			0
$343$		−	65.3295i		−	3.52746i
$344$		0			0
$345$		−	2.60109i		−	0.140038i
$346$		0			0
$347$			18.3008i			0.982440i	0.871036	+	0.491220i	$0.163449\pi$
$347$			18.3008i			0.982440i	−0.871036	+	0.491220i	$0.836551\pi$
$348$		0			0
$349$	30.3080			1.62235			0.811176	−	0.584802i	$-0.198828\pi$
$349$	30.3080			1.62235			0.811176	+	0.584802i	$0.198828\pi$
$350$		0			0
$351$		−	1.15254i		−	0.0615182i
$352$		0			0
$353$	12.5930			0.670259			0.335130	−	0.942172i	$-0.391220\pi$
$353$	12.5930			0.670259			0.335130	+	0.942172i	$0.391220\pi$
$354$		0			0
$355$	−6.99040			−0.371012
$356$		0			0
$357$		−	5.13999i		−	0.272037i
$358$		0			0
$359$			19.1724i			1.01188i	0.862568	+	0.505941i	$0.168855\pi$
$359$			19.1724i			1.01188i	−0.862568	+	0.505941i	$0.831145\pi$
$360$		0			0
$361$	3.57923	+	18.6598i	0.188381	+	0.982096i
$362$		0			0
$363$	9.09241			0.477228
$364$		0			0
$365$	−3.86881			−0.202503
$366$		0			0
$367$		−	27.6933i		−	1.44558i	−0.691070	−	0.722788i	$-0.742860\pi$
$367$		−	27.6933i		−	1.44558i	0.691070	−	0.722788i	$-0.257140\pi$
$368$		0			0
$369$			11.4810i			0.597679i
$370$		0			0
$371$	28.0575			1.45667
$372$		0			0
$373$			29.3563i			1.52001i	0.649915	+	0.760007i	$0.274805\pi$
$373$			29.3563i			1.52001i	−0.649915	+	0.760007i	$0.725195\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	−11.4537			−0.589894
$378$		0			0
$379$	17.5516			0.901567			0.450783	−	0.892633i	$-0.351145\pi$
$379$	17.5516			0.901567			0.450783	+	0.892633i	$0.351145\pi$
$380$		0			0
$381$	−3.70373			−0.189748
$382$		0			0
$383$	−21.5033			−1.09877			−0.549383	−	0.835571i	$-0.685137\pi$
$383$	−21.5033			−1.09877			−0.549383	+	0.835571i	$0.685137\pi$
$384$		0			0
$385$	−23.1418			−1.17942
$386$		0			0
$387$		−	3.55725i		−	0.180825i
$388$		0			0
$389$	13.5202			0.685502			0.342751	−	0.939426i	$-0.388641\pi$
$389$	13.5202			0.685502			0.342751	+	0.939426i	$0.388641\pi$
$390$		0			0
$391$		−	2.58962i		−	0.130963i
$392$		0			0
$393$		−	2.37756i		−	0.119932i
$394$		0			0
$395$	12.8235			0.645221
$396$		0			0
$397$	−19.7409			−0.990767			−0.495383	−	0.868674i	$-0.664972\pi$
$397$	−19.7409			−0.990767			−0.495383	+	0.868674i	$0.664972\pi$
$398$		0			0
$399$	−14.3357	+	17.3469i	−0.717683	+	0.868429i
$400$		0			0
$401$		−	13.0148i		−	0.649927i	−0.945727	−	0.324963i	$-0.894648\pi$
$401$		−	13.0148i		−	0.649927i	0.945727	−	0.324963i	$-0.105352\pi$
$402$		0			0
$403$		−	2.47151i		−	0.123115i
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	−31.4729			−1.56005
$408$		0			0
$409$			7.21274i			0.356647i	0.983972	+	0.178323i	$0.0570673\pi$
$409$			7.21274i			0.356647i	−0.983972	+	0.178323i	$0.942933\pi$
$410$		0			0
$411$	−9.36587			−0.461984
$412$		0			0
$413$		−	24.6366i		−	1.21229i
$414$		0			0
$415$			14.3095i			0.702425i
$416$		0			0
$417$			15.7808i			0.772788i
$418$		0			0
$419$			22.8697i			1.11726i	0.829417	+	0.558630i	$0.188673\pi$
$419$			22.8697i			1.11726i	−0.829417	+	0.558630i	$0.811327\pi$
$420$		0			0
$421$			0.690374i			0.0336468i	0.999858	+	0.0168234i	$0.00535530\pi$
$421$			0.690374i			0.0336468i	−0.999858	+	0.0168234i	$0.994645\pi$
$422$		0			0
$423$			5.00441i			0.243323i
$424$		0			0
$425$	0.995591			0.0482932
$426$		0			0
$427$		−	15.1021i		−	0.730842i
$428$		0			0
$429$	−5.16622			−0.249428
$430$		0			0
$431$	−17.5189			−0.843854			−0.421927	−	0.906630i	$-0.638646\pi$
$431$	−17.5189			−0.843854			−0.421927	+	0.906630i	$0.638646\pi$
$432$		0			0
$433$			6.74698i			0.324239i	0.986771	+	0.162120i	$0.0518330\pi$
$433$			6.74698i			0.324239i	−0.986771	+	0.162120i	$0.948167\pi$
$434$		0			0
$435$			9.93774i			0.476478i
$436$		0			0
$437$	−7.22260	+	8.73967i	−0.345504	+	0.418075i
$438$		0			0
$439$	−27.0047			−1.28887			−0.644433	−	0.764661i	$-0.722906\pi$
$439$	−27.0047			−1.28887			−0.644433	+	0.764661i	$0.722906\pi$
$440$		0			0
$441$	−19.6540			−0.935905
$442$		0			0
$443$		−	25.3626i		−	1.20501i	−0.798113	−	0.602507i	$-0.794168\pi$
$443$		−	25.3626i		−	1.20501i	0.798113	−	0.602507i	$-0.205832\pi$
$444$		0			0
$445$			10.0067i			0.474365i
$446$		0			0
$447$	5.43760			0.257190
$448$		0			0
$449$		−	16.1395i		−	0.761671i	−0.924643	−	0.380836i	$-0.875636\pi$
$449$		−	16.1395i		−	0.761671i	0.924643	−	0.380836i	$-0.124364\pi$
$450$		0			0
$451$	51.4633			2.42331
$452$		0			0
$453$	5.41557			0.254446
$454$		0			0
$455$	5.95029			0.278954
$456$		0			0
$457$	−23.8033			−1.11347			−0.556735	−	0.830690i	$-0.687946\pi$
$457$	−23.8033			−1.11347			−0.556735	+	0.830690i	$0.687946\pi$
$458$		0			0
$459$	−0.995591			−0.0464702
$460$		0			0
$461$	−23.2704			−1.08381			−0.541905	−	0.840440i	$-0.682297\pi$
$461$	−23.2704			−1.08381			−0.541905	+	0.840440i	$0.682297\pi$
$462$		0			0
$463$			9.25723i			0.430220i	0.976590	+	0.215110i	$0.0690110\pi$
$463$			9.25723i			0.430220i	−0.976590	+	0.215110i	$0.930989\pi$
$464$		0			0
$465$	−2.14440			−0.0994440
$466$		0			0
$467$			39.5695i			1.83106i	0.402253	+	0.915529i	$0.368227\pi$
$467$			39.5695i			1.83106i	−0.402253	+	0.915529i	$0.631773\pi$
$468$		0			0
$469$		−	43.6908i		−	2.01745i
$470$		0			0
$471$	−20.4933			−0.944282
$472$		0			0
$473$	−15.9452			−0.733162
$474$		0			0
$475$	−3.36000	−	2.77676i	−0.154167	−	0.127406i
$476$		0			0
$477$		−	5.43460i		−	0.248833i
$478$		0			0
$479$		−	7.06286i		−	0.322711i	−0.986896	−	0.161355i	$-0.948414\pi$
$479$		−	7.06286i		−	0.322711i	0.986896	−	0.161355i	$-0.0515865\pi$
$480$		0			0
$481$	8.09241			0.368982
$482$		0			0
$483$	−13.4288			−0.611031
$484$		0			0
$485$			15.9863i			0.725899i
$486$		0			0
$487$	10.2623			0.465031			0.232515	−	0.972593i	$-0.425304\pi$
$487$	10.2623			0.465031			0.232515	+	0.972593i	$0.425304\pi$
$488$		0			0
$489$			1.99765i			0.0903370i
$490$		0			0
$491$		−	17.4753i		−	0.788647i	−0.918972	−	0.394324i	$-0.870979\pi$
$491$		−	17.4753i		−	0.788647i	0.918972	−	0.394324i	$-0.129021\pi$
$492$		0			0
$493$			9.89392i			0.445600i
$494$		0			0
$495$			4.48246i			0.201471i
$496$		0			0
$497$			36.0897i			1.61884i
$498$		0			0
$499$			14.4274i			0.645858i	0.946423	+	0.322929i	$0.104667\pi$
$499$			14.4274i			0.645858i	−0.946423	+	0.322929i	$0.895333\pi$
$500$		0			0
$501$	6.57140			0.293588
$502$		0			0
$503$		−	6.05709i		−	0.270073i	−0.990841	−	0.135036i	$-0.956885\pi$
$503$		−	6.05709i		−	0.270073i	0.990841	−	0.135036i	$-0.0431151\pi$
$504$		0			0
$505$	−13.7451			−0.611649
$506$		0			0
$507$	−11.6716			−0.518356
$508$		0			0
$509$		−	39.3218i		−	1.74291i	−0.490476	−	0.871455i	$-0.663177\pi$
$509$		−	39.3218i		−	1.74291i	0.490476	−	0.871455i	$-0.336823\pi$
$510$		0			0
$511$			19.9737i			0.883585i
$512$		0			0
$513$	3.36000	+	2.77676i	0.148348	+	0.122597i
$514$		0			0
$515$	0.725070			0.0319504
$516$		0			0
$517$	22.4320			0.986560
$518$		0			0
$519$			20.5228i			0.900852i
$520$		0			0
$521$			7.48802i			0.328056i	0.986456	+	0.164028i	$0.0524487\pi$
$521$			7.48802i			0.328056i	−0.986456	+	0.164028i	$0.947551\pi$
$522$		0			0
$523$	35.7456			1.56305			0.781524	−	0.623876i	$-0.214443\pi$
$523$	35.7456			1.56305			0.781524	+	0.623876i	$0.214443\pi$
$524$		0			0
$525$		−	5.16275i		−	0.225321i
$526$		0			0
$527$	−2.13494			−0.0929995
$528$		0			0
$529$	16.2343			0.705840
$530$		0			0
$531$	−4.77199			−0.207087
$532$		0			0
$533$	−13.2324			−0.573159
$534$		0			0
$535$	3.03391			0.131167
$536$		0			0
$537$	12.7469			0.550068
$538$		0			0
$539$			88.0982i			3.79466i
$540$		0			0
$541$	−10.5821			−0.454962			−0.227481	−	0.973783i	$-0.573049\pi$
$541$	−10.5821			−0.454962			−0.227481	+	0.973783i	$0.573049\pi$
$542$		0			0
$543$			20.4121i			0.875967i
$544$		0			0
$545$		−	16.9561i		−	0.726319i
$546$		0			0
$547$	31.1571			1.33218			0.666091	−	0.745870i	$-0.267966\pi$
$547$	31.1571			1.33218			0.666091	+	0.745870i	$0.267966\pi$
$548$		0			0
$549$	−2.92520			−0.124845
$550$		0			0
$551$	27.5947	−	33.3908i	1.17557	−	1.42250i
$552$		0			0
$553$		−	66.2046i		−	2.81531i
$554$		0			0
$555$		−	7.02135i		−	0.298040i
$556$		0			0
$557$	−17.3752			−0.736210			−0.368105	−	0.929784i	$-0.619993\pi$
$557$	−17.3752			−0.736210			−0.368105	+	0.929784i	$0.619993\pi$
$558$		0			0
$559$	4.09989			0.173407
$560$		0			0
$561$			4.46269i			0.188415i
$562$		0			0
$563$	5.92442			0.249684			0.124842	−	0.992177i	$-0.460158\pi$
$563$	5.92442			0.249684			0.124842	+	0.992177i	$0.460158\pi$
$564$		0			0
$565$		−	9.73087i		−	0.409381i
$566$		0			0
$567$			5.16275i			0.216815i
$568$		0			0
$569$			13.0381i			0.546586i	0.961931	+	0.273293i	$0.0881128\pi$
$569$			13.0381i			0.546586i	−0.961931	+	0.273293i	$0.911887\pi$
$570$		0			0
$571$		−	6.36120i		−	0.266208i	−0.991102	−	0.133104i	$-0.957506\pi$
$571$		−	6.36120i		−	0.266208i	0.991102	−	0.133104i	$-0.0424944\pi$
$572$		0			0
$573$			6.85573i			0.286402i
$574$		0			0
$575$		−	2.60109i		−	0.108473i
$576$		0			0
$577$	−14.5592			−0.606107			−0.303053	−	0.952974i	$-0.598006\pi$
$577$	−14.5592			−0.606107			−0.303053	+	0.952974i	$0.598006\pi$
$578$		0			0
$579$			20.6986i			0.860206i
$580$		0			0
$581$	73.8763			3.06490
$582$		0			0
$583$	−24.3604			−1.00890
$584$		0			0
$585$		−	1.15254i		−	0.0476518i
$586$		0			0
$587$		−	43.2481i		−	1.78504i	−0.451008	−	0.892520i	$-0.648935\pi$
$587$		−	43.2481i		−	1.78504i	0.451008	−	0.892520i	$-0.351065\pi$
$588$		0			0
$589$	7.20518	+	5.95447i	0.296884	+	0.245350i
$590$		0			0
$591$	5.77774			0.237665
$592$		0			0
$593$	41.4861			1.70363			0.851815	−	0.523842i	$-0.175502\pi$
$593$	41.4861			1.70363			0.851815	+	0.523842i	$0.175502\pi$
$594$		0			0
$595$		−	5.13999i		−	0.210719i
$596$		0			0
$597$			6.45943i			0.264367i
$598$		0			0
$599$	0.196021			0.00800921			0.00400461	−	0.999992i	$-0.498725\pi$
$599$	0.196021			0.00800921			0.00400461	+	0.999992i	$0.498725\pi$
$600$		0			0
$601$		−	14.9285i		−	0.608946i	−0.952521	−	0.304473i	$-0.901520\pi$
$601$		−	14.9285i		−	0.608946i	0.952521	−	0.304473i	$-0.0984804\pi$
$602$		0			0
$603$	−8.46269			−0.344627
$604$		0			0
$605$	9.09241			0.369659
$606$		0			0
$607$	−3.83703			−0.155740			−0.0778702	−	0.996964i	$-0.524812\pi$
$607$	−3.83703			−0.155740			−0.0778702	+	0.996964i	$0.524812\pi$
$608$		0			0
$609$	51.3061			2.07903
$610$		0			0
$611$	−5.76780			−0.233340
$612$		0			0
$613$	24.6488			0.995556			0.497778	−	0.867304i	$-0.334149\pi$
$613$	24.6488			0.995556			0.497778	+	0.867304i	$0.334149\pi$
$614$		0			0
$615$			11.4810i			0.462960i
$616$		0			0
$617$	27.7863			1.11864			0.559318	−	0.828953i	$-0.311063\pi$
$617$	27.7863			1.11864			0.559318	+	0.828953i	$0.311063\pi$
$618$		0			0
$619$			49.1726i			1.97641i	0.153123	+	0.988207i	$0.451067\pi$
$619$			49.1726i			1.97641i	−0.153123	+	0.988207i	$0.548933\pi$
$620$		0			0
$621$			2.60109i			0.104378i
$622$		0			0
$623$	51.6623			2.06981
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	12.4467	−	15.0611i	0.497073	−	0.601481i
$628$		0			0
$629$		−	6.99040i		−	0.278725i
$630$		0			0
$631$		−	14.4304i		−	0.574464i	−0.957861	−	0.287232i	$-0.907265\pi$
$631$		−	14.4304i		−	0.574464i	0.957861	−	0.287232i	$-0.0927350\pi$
$632$		0			0
$633$	18.4870			0.734792
$634$		0			0
$635$	−3.70373			−0.146978
$636$		0			0
$637$		−	22.6521i		−	0.897509i
$638$		0			0
$639$	6.99040			0.276536
$640$		0			0
$641$			23.5888i			0.931700i	0.884864	+	0.465850i	$0.154251\pi$
$641$			23.5888i			0.931700i	−0.884864	+	0.465850i	$0.845749\pi$
$642$		0			0
$643$		−	26.4984i		−	1.04499i	−0.852641	−	0.522497i	$-0.825001\pi$
$643$		−	26.4984i		−	1.04499i	0.852641	−	0.522497i	$-0.174999\pi$
$644$		0			0
$645$		−	3.55725i		−	0.140067i
$646$		0			0
$647$		−	10.5619i		−	0.415229i	−0.978211	−	0.207615i	$-0.933430\pi$
$647$		−	10.5619i		−	0.415229i	0.978211	−	0.207615i	$-0.0665700\pi$
$648$		0			0
$649$			21.3902i			0.839640i
$650$		0			0
$651$			11.0710i			0.433906i
$652$		0			0
$653$	−11.3807			−0.445362			−0.222681	−	0.974891i	$-0.571481\pi$
$653$	−11.3807			−0.445362			−0.222681	+	0.974891i	$0.571481\pi$
$654$		0			0
$655$		−	2.37756i		−	0.0928988i
$656$		0			0
$657$	3.86881			0.150937
$658$		0			0
$659$	39.8293			1.55153			0.775765	−	0.631022i	$-0.217364\pi$
$659$	39.8293			1.55153			0.775765	+	0.631022i	$0.217364\pi$
$660$		0			0
$661$		−	46.0820i		−	1.79238i	−0.443670	−	0.896190i	$-0.646324\pi$
$661$		−	46.0820i		−	1.79238i	0.443670	−	0.896190i	$-0.353676\pi$
$662$		0			0
$663$		−	1.14746i		−	0.0445637i
$664$		0			0
$665$	−14.3357	+	17.3469i	−0.555915	+	0.672682i
$666$		0			0
$667$	25.8490			1.00088
$668$		0			0
$669$	−12.8114			−0.495319
$670$		0			0
$671$			13.1121i			0.506187i
$672$		0			0
$673$		−	2.48259i		−	0.0956967i	−0.998855	−	0.0478483i	$-0.984764\pi$
$673$		−	2.48259i		−	0.0956967i	0.998855	−	0.0478483i	$-0.0152364\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$			43.1610i			1.65881i	0.558645	+	0.829407i	$0.311321\pi$
$677$			43.1610i			1.65881i	−0.558645	+	0.829407i	$0.688679\pi$
$678$		0			0
$679$	82.5331			3.16733
$680$		0			0
$681$	−26.8133			−1.02749
$682$		0			0
$683$	−0.447502			−0.0171232			−0.00856159	−	0.999963i	$-0.502725\pi$
$683$	−0.447502			−0.0171232			−0.00856159	+	0.999963i	$0.502725\pi$
$684$		0			0
$685$	−9.36587			−0.357851
$686$		0			0
$687$	6.85061			0.261367
$688$		0			0
$689$	6.26361			0.238625
$690$		0			0
$691$			4.36401i			0.166015i	0.996549	+	0.0830075i	$0.0264525\pi$
$691$			4.36401i			0.166015i	−0.996549	+	0.0830075i	$0.973547\pi$
$692$		0			0
$693$	23.1418			0.879085
$694$		0			0
$695$			15.7808i			0.598599i
$696$		0			0
$697$			11.4304i			0.432958i
$698$		0			0
$699$	6.33196			0.239497
$700$		0			0
$701$	27.2254			1.02829			0.514145	−	0.857703i	$-0.328109\pi$
$701$	27.2254			1.02829			0.514145	+	0.857703i	$0.328109\pi$
$702$		0			0
$703$	−19.4966	+	23.5918i	−0.735328	+	0.889780i
$704$		0			0
$705$			5.00441i			0.188477i
$706$		0			0
$707$			70.9625i			2.66882i
$708$		0			0
$709$	−36.9476			−1.38760			−0.693798	−	0.720169i	$-0.744064\pi$
$709$	−36.9476			−1.38760			−0.693798	+	0.720169i	$0.744064\pi$
$710$		0			0
$711$	−12.8235			−0.480919
$712$		0			0
$713$			5.57777i			0.208889i
$714$		0			0
$715$	−5.16622			−0.193206
$716$		0			0
$717$			13.8565i			0.517481i
$718$		0			0
$719$			23.2685i			0.867769i	0.900969	+	0.433884i	$0.142857\pi$
$719$			23.2685i			0.867769i	−0.900969	+	0.433884i	$0.857143\pi$
$720$		0			0
$721$		−	3.74336i		−	0.139410i
$722$		0			0
$723$			5.30041i			0.197124i
$724$		0			0
$725$			9.93774i			0.369078i
$726$		0			0
$727$			29.4621i			1.09269i	0.837561	+	0.546344i	$0.183981\pi$
$727$			29.4621i			1.09269i	−0.837561	+	0.546344i	$0.816019\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	3.54157i		−	0.130990i
$732$		0			0
$733$	27.5024			1.01582			0.507912	−	0.861409i	$-0.330417\pi$
$733$	27.5024			1.01582			0.507912	+	0.861409i	$0.330417\pi$
$734$		0			0
$735$	−19.6540			−0.724949
$736$		0			0
$737$			37.9336i			1.39730i
$738$		0			0
$739$			34.8909i			1.28348i	0.766921	+	0.641741i	$0.221788\pi$
$739$			34.8909i			1.28348i	−0.766921	+	0.641741i	$0.778212\pi$
$740$		0			0
$741$	−3.20033	+	3.87255i	−0.117567	+	0.142262i
$742$		0			0
$743$	10.4237			0.382410			0.191205	−	0.981550i	$-0.438761\pi$
$743$	10.4237			0.382410			0.191205	+	0.981550i	$0.438761\pi$
$744$		0			0
$745$	5.43760			0.199218
$746$		0			0
$747$		−	14.3095i		−	0.523556i
$748$		0			0
$749$		−	15.6633i		−	0.572325i
$750$		0			0
$751$	−45.4633			−1.65898			−0.829489	−	0.558523i	$-0.811368\pi$
$751$	−45.4633			−1.65898			−0.829489	+	0.558523i	$0.811368\pi$
$752$		0			0
$753$			14.4249i			0.525673i
$754$		0			0
$755$	5.41557			0.197093
$756$		0			0
$757$	3.18584			0.115791			0.0578957	−	0.998323i	$-0.481561\pi$
$757$	3.18584			0.115791			0.0578957	+	0.998323i	$0.481561\pi$
$758$		0			0
$759$	11.6593			0.423205
$760$		0			0
$761$	9.31177			0.337552			0.168776	−	0.985654i	$-0.446019\pi$
$761$	9.31177			0.337552			0.168776	+	0.985654i	$0.446019\pi$
$762$		0			0
$763$	−87.5401			−3.16916
$764$		0			0
$765$	−0.995591			−0.0359957
$766$		0			0
$767$		−	5.49992i		−	0.198591i
$768$		0			0
$769$	0.315250			0.0113682			0.00568410	−	0.999984i	$-0.498191\pi$
$769$	0.315250			0.0113682			0.00568410	+	0.999984i	$0.498191\pi$
$770$		0			0
$771$			18.7940i			0.676850i
$772$		0			0
$773$		−	43.7998i		−	1.57537i	−0.616080	−	0.787684i	$-0.711280\pi$
$773$		−	43.7998i		−	1.57537i	0.616080	−	0.787684i	$-0.288720\pi$
$774$		0			0
$775$	−2.14440			−0.0770290
$776$		0			0
$777$	−36.2495			−1.30044
$778$		0			0
$779$	31.8801	−	38.5763i	1.14222	−	1.38214i
$780$		0			0
$781$		−	31.3341i		−	1.12122i
$782$		0			0
$783$		−	9.93774i		−	0.355146i
$784$		0			0
$785$	−20.4933			−0.731438
$786$		0			0
$787$	−13.6022			−0.484866			−0.242433	−	0.970168i	$-0.577946\pi$
$787$	−13.6022			−0.484866			−0.242433	+	0.970168i	$0.577946\pi$
$788$		0			0
$789$		−	7.48658i		−	0.266529i
$790$		0			0
$791$	−50.2381			−1.78626
$792$		0			0
$793$		−	3.37142i		−	0.119723i
$794$		0			0
$795$		−	5.43460i		−	0.192745i
$796$		0			0
$797$			30.0201i			1.06337i	0.846943	+	0.531683i	$0.178440\pi$
$797$			30.0201i			1.06337i	−0.846943	+	0.531683i	$0.821560\pi$
$798$		0			0
$799$			4.98234i			0.176263i
$800$		0			0
$801$		−	10.0067i		−	0.353571i
$802$		0			0
$803$		−	17.3418i		−	0.611978i
$804$		0			0
$805$	−13.4288			−0.473302
$806$		0			0
$807$			7.70524i			0.271237i
$808$		0			0
$809$	19.8299			0.697180			0.348590	−	0.937275i	$-0.386660\pi$
$809$	19.8299			0.697180			0.348590	+	0.937275i	$0.386660\pi$
$810$		0			0
$811$	−37.7040			−1.32397			−0.661983	−	0.749519i	$-0.730285\pi$
$811$	−37.7040			−1.32397			−0.661983	+	0.749519i	$0.730285\pi$
$812$		0			0
$813$			16.1502i			0.566411i
$814$		0			0
$815$			1.99765i			0.0699748i
$816$		0			0
$817$	−9.87763	+	11.9524i	−0.345575	+	0.418161i
$818$		0			0
$819$	−5.95029			−0.207920
$820$		0			0
$821$	12.3399			0.430666			0.215333	−	0.976541i	$-0.430916\pi$
$821$	12.3399			0.430666			0.215333	+	0.976541i	$0.430916\pi$
$822$		0			0
$823$			15.2739i			0.532416i	0.963916	+	0.266208i	$0.0857709\pi$
$823$			15.2739i			0.532416i	−0.963916	+	0.266208i	$0.914229\pi$
$824$		0			0
$825$			4.48246i			0.156059i
$826$		0			0
$827$	−36.8582			−1.28168			−0.640842	−	0.767673i	$-0.721415\pi$
$827$	−36.8582			−1.28168			−0.640842	+	0.767673i	$0.721415\pi$
$828$		0			0
$829$			27.0555i			0.939678i	0.882752	+	0.469839i	$0.155688\pi$
$829$			27.0555i			0.939678i	−0.882752	+	0.469839i	$0.844312\pi$
$830$		0			0
$831$	17.0926			0.592934
$832$		0			0
$833$	−19.5673			−0.677968
$834$		0			0
$835$	6.57140			0.227413
$836$		0			0
$837$	2.14440			0.0741212
$838$		0			0
$839$	1.19899			0.0413937			0.0206968	−	0.999786i	$-0.493412\pi$
$839$	1.19899			0.0413937			0.0206968	+	0.999786i	$0.493412\pi$
$840$		0			0
$841$	−69.7587			−2.40547
$842$		0			0
$843$		−	14.3099i		−	0.492861i
$844$		0			0
$845$	−11.6716			−0.401517
$846$		0			0
$847$		−	46.9419i		−	1.61294i
$848$		0			0
$849$			18.0771i			0.620403i
$850$		0			0
$851$	−18.2632			−0.626054
$852$		0			0
$853$	−24.9746			−0.855113			−0.427556	−	0.903989i	$-0.640625\pi$
$853$	−24.9746			−0.855113			−0.427556	+	0.903989i	$0.640625\pi$
$854$		0			0
$855$	3.36000	+	2.77676i	0.114910	+	0.0949631i
$856$		0			0
$857$		−	39.2060i		−	1.33925i	−0.742698	−	0.669626i	$-0.766454\pi$
$857$		−	39.2060i		−	1.33925i	0.742698	−	0.669626i	$-0.233546\pi$
$858$		0			0
$859$			7.14872i			0.243911i	0.992536	+	0.121956i	$0.0389165\pi$
$859$			7.14872i			0.243911i	−0.992536	+	0.121956i	$0.961083\pi$
$860$		0			0
$861$	59.2738			2.02005
$862$		0			0
$863$	37.1320			1.26399			0.631993	−	0.774974i	$-0.282237\pi$
$863$	37.1320			1.26399			0.631993	+	0.774974i	$0.282237\pi$
$864$		0			0
$865$			20.5228i			0.697797i
$866$		0			0
$867$	16.0088			0.543687
$868$		0			0
$869$			57.4808i			1.94990i
$870$		0			0
$871$		−	9.75362i		−	0.330489i
$872$		0			0
$873$		−	15.9863i		−	0.541053i
$874$		0			0
$875$		−	5.16275i		−	0.174533i
$876$		0			0
$877$			24.3162i			0.821100i	0.911838	+	0.410550i	$0.134663\pi$
$877$			24.3162i			0.821100i	−0.911838	+	0.410550i	$0.865337\pi$
$878$		0			0
$879$		−	5.33637i		−	0.179991i
$880$		0			0
$881$	−49.0694			−1.65319			−0.826595	−	0.562797i	$-0.809725\pi$
$881$	−49.0694			−1.65319			−0.826595	+	0.562797i	$0.809725\pi$
$882$		0			0
$883$			5.11482i			0.172127i	0.996290	+	0.0860636i	$0.0274288\pi$
$883$			5.11482i			0.172127i	−0.996290	+	0.0860636i	$0.972571\pi$
$884$		0			0
$885$	−4.77199			−0.160409
$886$		0			0
$887$	−2.37877			−0.0798713			−0.0399356	−	0.999202i	$-0.512715\pi$
$887$	−2.37877			−0.0798713			−0.0399356	+	0.999202i	$0.512715\pi$
$888$		0			0
$889$			19.1215i			0.641313i
$890$		0			0
$891$		−	4.48246i		−	0.150168i
$892$		0			0
$893$	13.8960	−	16.8148i	0.465013	−	0.562687i
$894$		0			0
$895$	12.7469			0.426081
$896$		0			0
$897$	−2.99787			−0.100096
$898$		0			0
$899$		−	21.3105i		−	0.710744i
$900$		0			0
$901$		−	5.41064i		−	0.180255i
$902$		0			0
$903$	−18.3652			−0.611156
$904$		0			0
$905$			20.4121i			0.678521i
$906$		0			0
$907$	−21.0637			−0.699408			−0.349704	−	0.936860i	$-0.613718\pi$
$907$	−21.0637			−0.699408			−0.349704	+	0.936860i	$0.613718\pi$
$908$		0			0
$909$	13.7451			0.455896
$910$		0			0
$911$	22.9681			0.760967			0.380483	−	0.924788i	$-0.375758\pi$
$911$	22.9681			0.760967			0.380483	+	0.924788i	$0.375758\pi$
$912$		0			0
$913$	−64.1416			−2.12278
$914$		0			0
$915$	−2.92520			−0.0967042
$916$		0			0
$917$	−12.2747			−0.405347
$918$		0			0
$919$		−	52.8892i		−	1.74465i	−0.488924	−	0.872326i	$-0.662611\pi$
$919$		−	52.8892i		−	1.74465i	0.488924	−	0.872326i	$-0.337389\pi$
$920$		0			0
$921$	4.91660			0.162008
$922$		0			0
$923$			8.05673i			0.265191i
$924$		0			0
$925$		−	7.02135i		−	0.230861i
$926$		0			0
$927$	−0.725070			−0.0238144
$928$		0			0
$929$	−59.5452			−1.95361			−0.976807	−	0.214122i	$-0.931311\pi$
$929$	−59.5452			−1.95361			−0.976807	+	0.214122i	$0.931311\pi$
$930$		0			0
$931$	66.0375	+	54.5744i	2.16429	+	1.78860i
$932$		0			0
$933$			15.3870i			0.503749i
$934$		0			0
$935$			4.46269i			0.145946i
$936$		0			0
$937$	−21.8671			−0.714366			−0.357183	−	0.934034i	$-0.616263\pi$
$937$	−21.8671			−0.714366			−0.357183	+	0.934034i	$0.616263\pi$
$938$		0			0
$939$	32.3545			1.05585
$940$		0			0
$941$			18.7261i			0.610452i	0.952280	+	0.305226i	$0.0987321\pi$
$941$			18.7261i			0.610452i	−0.952280	+	0.305226i	$0.901268\pi$
$942$		0			0
$943$	29.8632			0.972481
$944$		0			0
$945$			5.16275i			0.167944i
$946$		0			0
$947$		−	50.3189i		−	1.63514i	−0.575826	−	0.817572i	$-0.695320\pi$
$947$		−	50.3189i		−	1.63514i	0.575826	−	0.817572i	$-0.304680\pi$
$948$		0			0
$949$			4.45897i			0.144744i
$950$		0			0
$951$			3.44342i			0.111661i
$952$		0			0
$953$			30.9031i			1.00105i	0.865722	+	0.500524i	$0.166859\pi$
$953$			30.9031i			1.00105i	−0.865722	+	0.500524i	$0.833141\pi$
$954$		0			0
$955$			6.85573i			0.221846i
$956$		0			0
$957$	−44.5455			−1.43995
$958$		0			0
$959$			48.3537i			1.56142i
$960$		0			0
$961$	−26.4016			−0.851663
$962$		0			0
$963$	−3.03391			−0.0977664
$964$		0			0
$965$			20.6986i			0.666313i
$966$		0			0
$967$			34.6337i			1.11375i	0.830598	+	0.556873i	$0.187999\pi$
$967$			34.6337i			1.11375i	−0.830598	+	0.556873i	$0.812001\pi$
$968$		0			0
$969$	3.34519	+	2.76451i	0.107463	+	0.0888090i
$970$		0			0
$971$	20.2909			0.651166			0.325583	−	0.945514i	$-0.394439\pi$
$971$	20.2909			0.651166			0.325583	+	0.945514i	$0.394439\pi$
$972$		0			0
$973$	81.4723			2.61188
$974$		0			0
$975$		−	1.15254i		−	0.0369109i
$976$		0			0
$977$			5.84740i			0.187075i	0.995616	+	0.0935374i	$0.0298175\pi$
$977$			5.84740i			0.187075i	−0.995616	+	0.0935374i	$0.970183\pi$
$978$		0			0
$979$	−44.8548			−1.43356
$980$		0			0
$981$			16.9561i			0.541366i
$982$		0			0
$983$	−19.5915			−0.624873			−0.312436	−	0.949939i	$-0.601145\pi$
$983$	−19.5915			−0.624873			−0.312436	+	0.949939i	$0.601145\pi$
$984$		0			0
$985$	5.77774			0.184094
$986$		0			0
$987$	25.8365			0.822386
$988$		0			0
$989$	−9.25273			−0.294220
$990$		0			0
$991$	26.2215			0.832954			0.416477	−	0.909146i	$-0.363265\pi$
$991$	26.2215			0.832954			0.416477	+	0.909146i	$0.363265\pi$
$992$		0			0
$993$	22.6438			0.718581
$994$		0			0
$995$			6.45943i			0.204778i
$996$		0			0
$997$	−59.3206			−1.87870			−0.939352	−	0.342955i	$-0.888572\pi$
$997$	−59.3206			−1.87870			−0.939352	+	0.342955i	$0.888572\pi$
$998$		0			0
$999$			7.02135i			0.222146i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.i.2431.12	yes	12
4.3	odd	2		4560.2.d.k.2431.1	yes	12
19.18	odd	2		4560.2.d.k.2431.12	yes	12
76.75	even	2	inner	4560.2.d.i.2431.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.i.2431.1	✓	12	76.75	even	2	inner
4560.2.d.i.2431.12	yes	12	1.1	even	1	trivial
4560.2.d.k.2431.1	yes	12	4.3	odd	2
4560.2.d.k.2431.12	yes	12	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +5.16275i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +5.16275i q^{7} +1.00000 q^{9} -4.48246i q^{11} +1.15254i q^{13} +1.00000 q^{15} +0.995591 q^{17} +(-3.36000 - 2.77676i) q^{19} -5.16275i q^{21} -2.60109i q^{23} +1.00000 q^{25} -1.00000 q^{27} +9.93774i q^{29} -2.14440 q^{31} +4.48246i q^{33} -5.16275i q^{35} -7.02135i q^{37} -1.15254i q^{39} +11.4810i q^{41} -3.55725i q^{43} -1.00000 q^{45} +5.00441i q^{47} -19.6540 q^{49} -0.995591 q^{51} -5.43460i q^{53} +4.48246i q^{55} +(3.36000 + 2.77676i) q^{57} -4.77199 q^{59} -2.92520 q^{61} +5.16275i q^{63} -1.15254i q^{65} -8.46269 q^{67} +2.60109i q^{69} +6.99040 q^{71} +3.86881 q^{73} -1.00000 q^{75} +23.1418 q^{77} -12.8235 q^{79} +1.00000 q^{81} -14.3095i q^{83} -0.995591 q^{85} -9.93774i q^{87} -10.0067i q^{89} -5.95029 q^{91} +2.14440 q^{93} +(3.36000 + 2.77676i) q^{95} -15.9863i q^{97} -4.48246i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 12 q^{5} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 12 * q^5 + 12 * q^9	\( 12 q - 12 q^{3} - 12 q^{5} + 12 q^{9} + 12 q^{15} - 8 q^{17} - 8 q^{19} + 12 q^{25} - 12 q^{27} - 12 q^{45} - 20 q^{49} + 8 q^{51} + 8 q^{57} - 8 q^{59} + 8 q^{67} - 32 q^{71} - 24 q^{73} - 12 q^{75} + 56 q^{77} - 16 q^{79} + 12 q^{81} + 8 q^{85} + 16 q^{91} + 8 q^{95}+O(q^{100}) \) 12 * q - 12 * q^3 - 12 * q^5 + 12 * q^9 + 12 * q^15 - 8 * q^17 - 8 * q^19 + 12 * q^25 - 12 * q^27 - 12 * q^45 - 20 * q^49 + 8 * q^51 + 8 * q^57 - 8 * q^59 + 8 * q^67 - 32 * q^71 - 24 * q^73 - 12 * q^75 + 56 * q^77 - 16 * q^79 + 12 * q^81 + 8 * q^85 + 16 * q^91 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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