LMFDB - Embedded newform 1755.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1755,2,Mod(1,1755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1755, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1755.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1755 = 3^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1755.a (trivial)

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:	yes
Analytic conductor:	$14.0137455547$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 12x^{5} + 9x^{4} + 39x^{3} - 16x^{2} - 30x + 12 $ x^7 - x^6 - 12x^5 + 9x^4 + 39x^3 - 16x^2 - 30*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.36513$ of defining polynomial
Character	$\chi$	$=$	1755.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36513 q^{2} +3.59384 q^{4} -1.00000 q^{5} +3.86012 q^{7} -3.76963 q^{8} +O(q^{10})$	\(q-2.36513 q^{2} +3.59384 q^{4} -1.00000 q^{5} +3.86012 q^{7} -3.76963 q^{8} +2.36513 q^{10} +1.19506 q^{11} +1.00000 q^{13} -9.12968 q^{14} +1.72800 q^{16} +0.670784 q^{17} +3.86584 q^{19} -3.59384 q^{20} -2.82647 q^{22} +2.76229 q^{23} +1.00000 q^{25} -2.36513 q^{26} +13.8726 q^{28} +6.30321 q^{29} +2.29289 q^{31} +3.45233 q^{32} -1.58649 q^{34} -3.86012 q^{35} -10.5157 q^{37} -9.14322 q^{38} +3.76963 q^{40} +4.66913 q^{41} +5.49420 q^{43} +4.29485 q^{44} -6.53317 q^{46} -1.57295 q^{47} +7.90051 q^{49} -2.36513 q^{50} +3.59384 q^{52} -13.8753 q^{53} -1.19506 q^{55} -14.5512 q^{56} -14.9079 q^{58} +3.75031 q^{59} +5.60206 q^{61} -5.42298 q^{62} -11.6212 q^{64} -1.00000 q^{65} -1.82749 q^{67} +2.41069 q^{68} +9.12968 q^{70} +0.529999 q^{71} -7.01516 q^{73} +24.8710 q^{74} +13.8932 q^{76} +4.61307 q^{77} +5.49993 q^{79} -1.72800 q^{80} -11.0431 q^{82} -7.05352 q^{83} -0.670784 q^{85} -12.9945 q^{86} -4.50493 q^{88} -9.58442 q^{89} +3.86012 q^{91} +9.92721 q^{92} +3.72024 q^{94} -3.86584 q^{95} -9.33419 q^{97} -18.6857 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) $ 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8	$ 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8} + q^{10} - q^{11} + 7 q^{13} + 12 q^{14} + 23 q^{16} - 11 q^{17} + 2 q^{19} - 11 q^{20} + 16 q^{22} + q^{23} + 7 q^{25} - q^{26} + 10 q^{28} + 4 q^{29} - 13 q^{32} + q^{34} - 2 q^{35} + 23 q^{37} + 15 q^{38} + 6 q^{40} - 2 q^{41} + 8 q^{43} - 10 q^{44} + 37 q^{46} - 2 q^{47} + 43 q^{49} - q^{50} + 11 q^{52} - 10 q^{53} + q^{55} + 68 q^{56} - 26 q^{58} + 13 q^{59} + 21 q^{61} - 9 q^{62} + 46 q^{64} - 7 q^{65} + 21 q^{67} - 53 q^{68} - 12 q^{70} + 10 q^{71} + 13 q^{73} + 68 q^{74} - 41 q^{76} - 6 q^{77} + 8 q^{79} - 23 q^{80} - 26 q^{82} + 4 q^{83} + 11 q^{85} + 12 q^{86} + 44 q^{88} + 27 q^{89} + 2 q^{91} + 9 q^{92} - 24 q^{94} - 2 q^{95} - 15 q^{97} - 7 q^{98}+O(q^{100}) $ 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8 + q^10 - q^11 + 7 * q^13 + 12 * q^14 + 23 * q^16 - 11 * q^17 + 2 * q^19 - 11 * q^20 + 16 * q^22 + q^23 + 7 * q^25 - q^26 + 10 * q^28 + 4 * q^29 - 13 * q^32 + q^34 - 2 * q^35 + 23 * q^37 + 15 * q^38 + 6 * q^40 - 2 * q^41 + 8 * q^43 - 10 * q^44 + 37 * q^46 - 2 * q^47 + 43 * q^49 - q^50 + 11 * q^52 - 10 * q^53 + q^55 + 68 * q^56 - 26 * q^58 + 13 * q^59 + 21 * q^61 - 9 * q^62 + 46 * q^64 - 7 * q^65 + 21 * q^67 - 53 * q^68 - 12 * q^70 + 10 * q^71 + 13 * q^73 + 68 * q^74 - 41 * q^76 - 6 * q^77 + 8 * q^79 - 23 * q^80 - 26 * q^82 + 4 * q^83 + 11 * q^85 + 12 * q^86 + 44 * q^88 + 27 * q^89 + 2 * q^91 + 9 * q^92 - 24 * q^94 - 2 * q^95 - 15 * q^97 - 7 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.36513	−1.67240	−0.836200	−	0.548425i	$-0.815228\pi$
$2$	−2.36513	−1.67240	−0.836200	+	0.548425i	$0.815228\pi$
$3$	0	0
$3$	0	0
$4$	3.59384	1.79692
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.86012	1.45899	0.729494	−	0.683988i	$-0.239756\pi$
$7$	3.86012	1.45899	0.729494	+	0.683988i	$0.239756\pi$
$8$	−3.76963	−1.33277
$9$	0	0
$10$	2.36513	0.747920
$11$	1.19506	0.360324	0.180162	−	0.983637i	$-0.442338\pi$
$11$	1.19506	0.360324	0.180162	+	0.983637i	$0.442338\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−9.12968	−2.44001
$15$	0	0
$16$	1.72800	0.431999
$17$	0.670784	0.162689	0.0813445	−	0.996686i	$-0.474079\pi$
$17$	0.670784	0.162689	0.0813445	+	0.996686i	$0.474079\pi$
$18$	0	0
$19$	3.86584	0.886885	0.443443	−	0.896303i	$-0.353757\pi$
$19$	3.86584	0.886885	0.443443	+	0.896303i	$0.353757\pi$
$20$	−3.59384	−0.803607
$21$	0	0
$22$	−2.82647	−0.602605
$23$	2.76229	0.575977	0.287988	−	0.957634i	$-0.407014\pi$
$23$	2.76229	0.575977	0.287988	+	0.957634i	$0.407014\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.36513	−0.463840
$27$	0	0
$28$	13.8726	2.62168
$29$	6.30321	1.17048	0.585238	−	0.810861i	$-0.301001\pi$
$29$	6.30321	1.17048	0.585238	+	0.810861i	$0.301001\pi$
$30$	0	0
$31$	2.29289	0.411815	0.205908	−	0.978571i	$-0.433985\pi$
$31$	2.29289	0.411815	0.205908	+	0.978571i	$0.433985\pi$
$32$	3.45233	0.610292
$33$	0	0
$34$	−1.58649	−0.272081
$35$	−3.86012	−0.652479
$36$	0	0
$37$	−10.5157	−1.72877	−0.864385	−	0.502831i	$-0.832292\pi$
$37$	−10.5157	−1.72877	−0.864385	+	0.502831i	$0.832292\pi$
$38$	−9.14322	−1.48323
$39$	0	0
$40$	3.76963	0.596031
$41$	4.66913	0.729195	0.364597	−	0.931165i	$-0.381207\pi$
$41$	4.66913	0.729195	0.364597	+	0.931165i	$0.381207\pi$
$42$	0	0
$43$	5.49420	0.837858	0.418929	−	0.908019i	$-0.362406\pi$
$43$	5.49420	0.837858	0.418929	+	0.908019i	$0.362406\pi$
$44$	4.29485	0.647472
$45$	0	0
$46$	−6.53317	−0.963263
$47$	−1.57295	−0.229439	−0.114719	−	0.993398i	$-0.536597\pi$
$47$	−1.57295	−0.229439	−0.114719	+	0.993398i	$0.536597\pi$
$48$	0	0
$49$	7.90051	1.12864
$50$	−2.36513	−0.334480
$51$	0	0
$52$	3.59384	0.498376
$53$	−13.8753	−1.90592	−0.952958	−	0.303102i	$-0.901978\pi$
$53$	−13.8753	−1.90592	−0.952958	+	0.303102i	$0.901978\pi$
$54$	0	0
$55$	−1.19506	−0.161142
$56$	−14.5512	−1.94449
$57$	0	0
$58$	−14.9079	−1.95750
$59$	3.75031	0.488248	0.244124	−	0.969744i	$-0.421500\pi$
$59$	3.75031	0.488248	0.244124	+	0.969744i	$0.421500\pi$
$60$	0	0
$61$	5.60206	0.717270	0.358635	−	0.933478i	$-0.383242\pi$
$61$	5.60206	0.717270	0.358635	+	0.933478i	$0.383242\pi$
$62$	−5.42298	−0.688719
$63$	0	0
$64$	−11.6212	−1.45265
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−1.82749	−0.223263	−0.111631	−	0.993750i	$-0.535608\pi$
$67$	−1.82749	−0.223263	−0.111631	+	0.993750i	$0.535608\pi$
$68$	2.41069	0.292339
$69$	0	0
$70$	9.12968	1.09121
$71$	0.529999	0.0628993	0.0314496	−	0.999505i	$-0.489988\pi$
$71$	0.529999	0.0628993	0.0314496	+	0.999505i	$0.489988\pi$
$72$	0	0
$73$	−7.01516	−0.821063	−0.410531	−	0.911847i	$-0.634657\pi$
$73$	−7.01516	−0.821063	−0.410531	+	0.911847i	$0.634657\pi$
$74$	24.8710	2.89119
$75$	0	0
$76$	13.8932	1.59366
$77$	4.61307	0.525708
$78$	0	0
$79$	5.49993	0.618790	0.309395	−	0.950934i	$-0.399873\pi$
$79$	5.49993	0.618790	0.309395	+	0.950934i	$0.399873\pi$
$80$	−1.72800	−0.193196
$81$	0	0
$82$	−11.0431	−1.21951
$83$	−7.05352	−0.774224	−0.387112	−	0.922033i	$-0.626527\pi$
$83$	−7.05352	−0.774224	−0.387112	+	0.922033i	$0.626527\pi$
$84$	0	0
$85$	−0.670784	−0.0727567
$86$	−12.9945	−1.40123
$87$	0	0
$88$	−4.50493	−0.480227
$89$	−9.58442	−1.01595	−0.507973	−	0.861373i	$-0.669605\pi$
$89$	−9.58442	−1.01595	−0.507973	+	0.861373i	$0.669605\pi$
$90$	0	0
$91$	3.86012	0.404650
$92$	9.92721	1.03498
$93$	0	0
$94$	3.72024	0.383713
$95$	−3.86584	−0.396627
$96$	0	0
$97$	−9.33419	−0.947743	−0.473871	−	0.880594i	$-0.657144\pi$
$97$	−9.33419	−0.947743	−0.473871	+	0.880594i	$0.657144\pi$
$98$	−18.6857	−1.88754
$99$	0	0
$100$	3.59384	0.359384
$101$	14.4505	1.43788	0.718939	−	0.695073i	$-0.244628\pi$
$101$	14.4505	1.43788	0.718939	+	0.695073i	$0.244628\pi$
$102$	0	0
$103$	−1.61472	−0.159103	−0.0795517	−	0.996831i	$-0.525349\pi$
$103$	−1.61472	−0.159103	−0.0795517	+	0.996831i	$0.525349\pi$
$104$	−3.76963	−0.369643
$105$	0	0
$106$	32.8168	3.18745
$107$	18.5203	1.79042	0.895212	−	0.445640i	$-0.147024\pi$
$107$	18.5203	1.79042	0.895212	+	0.445640i	$0.147024\pi$
$108$	0	0
$109$	1.81262	0.173618	0.0868088	−	0.996225i	$-0.472333\pi$
$109$	1.81262	0.173618	0.0868088	+	0.996225i	$0.472333\pi$
$110$	2.82647	0.269493
$111$	0	0
$112$	6.67027	0.630281
$113$	10.2476	0.964014	0.482007	−	0.876167i	$-0.339908\pi$
$113$	10.2476	0.964014	0.482007	+	0.876167i	$0.339908\pi$
$114$	0	0
$115$	−2.76229	−0.257585
$116$	22.6527	2.10325
$117$	0	0
$118$	−8.86996	−0.816546
$119$	2.58930	0.237361
$120$	0	0
$121$	−9.57184	−0.870167
$122$	−13.2496	−1.19956
$123$	0	0
$124$	8.24028	0.739999
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.7233	−0.951542	−0.475771	−	0.879569i	$-0.657831\pi$
$127$	−10.7233	−0.951542	−0.475771	+	0.879569i	$0.657831\pi$
$128$	20.5810	1.81912
$129$	0	0
$130$	2.36513	0.207436
$131$	1.56150	0.136429	0.0682146	−	0.997671i	$-0.478270\pi$
$131$	1.56150	0.136429	0.0682146	+	0.997671i	$0.478270\pi$
$132$	0	0
$133$	14.9226	1.29395
$134$	4.32224	0.373385
$135$	0	0
$136$	−2.52861	−0.216826
$137$	−21.5155	−1.83819	−0.919095	−	0.394036i	$-0.871079\pi$
$137$	−21.5155	−1.83819	−0.919095	+	0.394036i	$0.871079\pi$
$138$	0	0
$139$	−4.68111	−0.397046	−0.198523	−	0.980096i	$-0.563614\pi$
$139$	−4.68111	−0.397046	−0.198523	+	0.980096i	$0.563614\pi$
$140$	−13.8726	−1.17245
$141$	0	0
$142$	−1.25352	−0.105193
$143$	1.19506	0.0999358
$144$	0	0
$145$	−6.30321	−0.523453
$146$	16.5918	1.37314
$147$	0	0
$148$	−37.7917	−3.10646
$149$	−14.2429	−1.16682	−0.583410	−	0.812178i	$-0.698282\pi$
$149$	−14.2429	−1.16682	−0.583410	+	0.812178i	$0.698282\pi$
$150$	0	0
$151$	22.2963	1.81445	0.907223	−	0.420650i	$-0.138198\pi$
$151$	22.2963	1.81445	0.907223	+	0.420650i	$0.138198\pi$
$152$	−14.5728	−1.18201
$153$	0	0
$154$	−10.9105	−0.879193
$155$	−2.29289	−0.184169
$156$	0	0
$157$	11.6262	0.927870	0.463935	−	0.885869i	$-0.346437\pi$
$157$	11.6262	0.927870	0.463935	+	0.885869i	$0.346437\pi$
$158$	−13.0080	−1.03486
$159$	0	0
$160$	−3.45233	−0.272931
$161$	10.6628	0.840342
$162$	0	0
$163$	7.33419	0.574458	0.287229	−	0.957862i	$-0.407266\pi$
$163$	7.33419	0.574458	0.287229	+	0.957862i	$0.407266\pi$
$164$	16.7801	1.31030
$165$	0	0
$166$	16.6825	1.29481
$167$	16.9715	1.31329	0.656645	−	0.754200i	$-0.271975\pi$
$167$	16.9715	1.31329	0.656645	+	0.754200i	$0.271975\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	1.58649	0.121678
$171$	0	0
$172$	19.7453	1.50556
$173$	17.6579	1.34251	0.671253	−	0.741228i	$-0.265756\pi$
$173$	17.6579	1.34251	0.671253	+	0.741228i	$0.265756\pi$
$174$	0	0
$175$	3.86012	0.291797
$176$	2.06506	0.155659
$177$	0	0
$178$	22.6684	1.69907
$179$	0.354907	0.0265270	0.0132635	−	0.999912i	$-0.495778\pi$
$179$	0.354907	0.0265270	0.0132635	+	0.999912i	$0.495778\pi$
$180$	0	0
$181$	24.9284	1.85291	0.926455	−	0.376405i	$-0.122840\pi$
$181$	24.9284	1.85291	0.926455	+	0.376405i	$0.122840\pi$
$182$	−9.12968	−0.676737
$183$	0	0
$184$	−10.4128	−0.767642
$185$	10.5157	0.773129
$186$	0	0
$187$	0.801626	0.0586207
$188$	−5.65294	−0.412283
$189$	0	0
$190$	9.14322	0.663319
$191$	12.4316	0.899517	0.449759	−	0.893150i	$-0.351510\pi$
$191$	12.4316	0.899517	0.449759	+	0.893150i	$0.351510\pi$
$192$	0	0
$193$	−9.31158	−0.670262	−0.335131	−	0.942172i	$-0.608781\pi$
$193$	−9.31158	−0.670262	−0.335131	+	0.942172i	$0.608781\pi$
$194$	22.0766	1.58500
$195$	0	0
$196$	28.3932	2.02808
$197$	14.0118	0.998303	0.499151	−	0.866515i	$-0.333645\pi$
$197$	14.0118	0.998303	0.499151	+	0.866515i	$0.333645\pi$
$198$	0	0
$199$	−25.9645	−1.84058	−0.920288	−	0.391241i	$-0.872046\pi$
$199$	−25.9645	−1.84058	−0.920288	+	0.391241i	$0.872046\pi$
$200$	−3.76963	−0.266553
$201$	0	0
$202$	−34.1773	−2.40471
$203$	24.3311	1.70771
$204$	0	0
$205$	−4.66913	−0.326106
$206$	3.81903	0.266085
$207$	0	0
$208$	1.72800	0.119815
$209$	4.61991	0.319566
$210$	0	0
$211$	25.2831	1.74056	0.870281	−	0.492556i	$-0.163937\pi$
$211$	25.2831	1.74056	0.870281	+	0.492556i	$0.163937\pi$
$212$	−49.8655	−3.42478
$213$	0	0
$214$	−43.8029	−2.99430
$215$	−5.49420	−0.374702
$216$	0	0
$217$	8.85083	0.600833
$218$	−4.28709	−0.290358
$219$	0	0
$220$	−4.29485	−0.289558
$221$	0.670784	0.0451218
$222$	0	0
$223$	26.4191	1.76916	0.884578	−	0.466392i	$-0.154446\pi$
$223$	26.4191	1.76916	0.884578	+	0.466392i	$0.154446\pi$
$224$	13.3264	0.890408
$225$	0	0
$226$	−24.2369	−1.61222
$227$	11.0051	0.730437	0.365218	−	0.930922i	$-0.380994\pi$
$227$	11.0051	0.730437	0.365218	+	0.930922i	$0.380994\pi$
$228$	0	0
$229$	2.75691	0.182182	0.0910910	−	0.995843i	$-0.470965\pi$
$229$	2.75691	0.182182	0.0910910	+	0.995843i	$0.470965\pi$
$230$	6.53317	0.430784
$231$	0	0
$232$	−23.7608	−1.55997
$233$	−5.47484	−0.358669	−0.179334	−	0.983788i	$-0.557394\pi$
$233$	−5.47484	−0.358669	−0.179334	+	0.983788i	$0.557394\pi$
$234$	0	0
$235$	1.57295	0.102608
$236$	13.4780	0.877343
$237$	0	0
$238$	−6.12404	−0.396963
$239$	−3.46262	−0.223978	−0.111989	−	0.993709i	$-0.535722\pi$
$239$	−3.46262	−0.223978	−0.111989	+	0.993709i	$0.535722\pi$
$240$	0	0
$241$	5.69303	0.366720	0.183360	−	0.983046i	$-0.441303\pi$
$241$	5.69303	0.366720	0.183360	+	0.983046i	$0.441303\pi$
$242$	22.6386	1.45527
$243$	0	0
$244$	20.1329	1.28888
$245$	−7.90051	−0.504745
$246$	0	0
$247$	3.86584	0.245978
$248$	−8.64335	−0.548854
$249$	0	0
$250$	2.36513	0.149584
$251$	15.1122	0.953875	0.476937	−	0.878937i	$-0.341747\pi$
$251$	15.1122	0.953875	0.476937	+	0.878937i	$0.341747\pi$
$252$	0	0
$253$	3.30109	0.207538
$254$	25.3621	1.59136
$255$	0	0
$256$	−25.4343	−1.58964
$257$	−19.1391	−1.19387	−0.596933	−	0.802291i	$-0.703614\pi$
$257$	−19.1391	−1.19387	−0.596933	+	0.802291i	$0.703614\pi$
$258$	0	0
$259$	−40.5918	−2.52225
$260$	−3.59384	−0.222880
$261$	0	0
$262$	−3.69316	−0.228164
$263$	5.18581	0.319771	0.159885	−	0.987136i	$-0.448888\pi$
$263$	5.18581	0.319771	0.159885	+	0.987136i	$0.448888\pi$
$264$	0	0
$265$	13.8753	0.852352
$266$	−35.2939	−2.16401
$267$	0	0
$268$	−6.56769	−0.401185
$269$	26.0751	1.58983	0.794913	−	0.606723i	$-0.207516\pi$
$269$	26.0751	1.58983	0.794913	+	0.606723i	$0.207516\pi$
$270$	0	0
$271$	−30.2294	−1.83631	−0.918154	−	0.396225i	$-0.870320\pi$
$271$	−30.2294	−1.83631	−0.918154	+	0.396225i	$0.870320\pi$
$272$	1.15911	0.0702815
$273$	0	0
$274$	50.8869	3.07419
$275$	1.19506	0.0720647
$276$	0	0
$277$	0.438496	0.0263467	0.0131733	−	0.999913i	$-0.495807\pi$
$277$	0.438496	0.0263467	0.0131733	+	0.999913i	$0.495807\pi$
$278$	11.0714	0.664020
$279$	0	0
$280$	14.5512	0.869602
$281$	21.6180	1.28962	0.644810	−	0.764343i	$-0.276936\pi$
$281$	21.6180	1.28962	0.644810	+	0.764343i	$0.276936\pi$
$282$	0	0
$283$	30.0718	1.78758	0.893791	−	0.448484i	$-0.148036\pi$
$283$	30.0718	1.78758	0.893791	+	0.448484i	$0.148036\pi$
$284$	1.90473	0.113025
$285$	0	0
$286$	−2.82647	−0.167133
$287$	18.0234	1.06389
$288$	0	0
$289$	−16.5500	−0.973532
$290$	14.9079	0.875423
$291$	0	0
$292$	−25.2114	−1.47538
$293$	−15.6492	−0.914235	−0.457117	−	0.889406i	$-0.651118\pi$
$293$	−15.6492	−0.914235	−0.457117	+	0.889406i	$0.651118\pi$
$294$	0	0
$295$	−3.75031	−0.218351
$296$	39.6403	2.30405
$297$	0	0
$298$	33.6862	1.95139
$299$	2.76229	0.159747
$300$	0	0
$301$	21.2083	1.22242
$302$	−52.7336	−3.03448
$303$	0	0
$304$	6.68016	0.383133
$305$	−5.60206	−0.320773
$306$	0	0
$307$	−27.7801	−1.58549	−0.792747	−	0.609551i	$-0.791350\pi$
$307$	−27.7801	−1.58549	−0.792747	+	0.609551i	$0.791350\pi$
$308$	16.5786	0.944654
$309$	0	0
$310$	5.42298	0.308005
$311$	4.06730	0.230635	0.115318	−	0.993329i	$-0.463211\pi$
$311$	4.06730	0.230635	0.115318	+	0.993329i	$0.463211\pi$
$312$	0	0
$313$	−9.68498	−0.547427	−0.273714	−	0.961811i	$-0.588252\pi$
$313$	−9.68498	−0.547427	−0.273714	+	0.961811i	$0.588252\pi$
$314$	−27.4974	−1.55177
$315$	0	0
$316$	19.7658	1.11192
$317$	−16.5798	−0.931213	−0.465606	−	0.884992i	$-0.654164\pi$
$317$	−16.5798	−0.931213	−0.465606	+	0.884992i	$0.654164\pi$
$318$	0	0
$319$	7.53271	0.421751
$320$	11.6212	0.649645
$321$	0	0
$322$	−25.2188	−1.40539
$323$	2.59314	0.144286
$324$	0	0
$325$	1.00000	0.0554700
$326$	−17.3463	−0.960723
$327$	0	0
$328$	−17.6009	−0.971847
$329$	−6.07178	−0.334748
$330$	0	0
$331$	13.6300	0.749175	0.374587	−	0.927192i	$-0.377784\pi$
$331$	13.6300	0.749175	0.374587	+	0.927192i	$0.377784\pi$
$332$	−25.3492	−1.39122
$333$	0	0
$334$	−40.1397	−2.19635
$335$	1.82749	0.0998462
$336$	0	0
$337$	23.7355	1.29296	0.646478	−	0.762932i	$-0.276241\pi$
$337$	23.7355	1.29296	0.646478	+	0.762932i	$0.276241\pi$
$338$	−2.36513	−0.128646
$339$	0	0
$340$	−2.41069	−0.130738
$341$	2.74014	0.148387
$342$	0	0
$343$	3.47608	0.187690
$344$	−20.7111	−1.11667
$345$	0	0
$346$	−41.7632	−2.24521
$347$	4.14781	0.222666	0.111333	−	0.993783i	$-0.464488\pi$
$347$	4.14781	0.222666	0.111333	+	0.993783i	$0.464488\pi$
$348$	0	0
$349$	−15.8403	−0.847913	−0.423956	−	0.905683i	$-0.639359\pi$
$349$	−15.8403	−0.847913	−0.423956	+	0.905683i	$0.639359\pi$
$350$	−9.12968	−0.488002
$351$	0	0
$352$	4.12574	0.219903
$353$	8.27444	0.440404	0.220202	−	0.975454i	$-0.429328\pi$
$353$	8.27444	0.440404	0.220202	+	0.975454i	$0.429328\pi$
$354$	0	0
$355$	−0.529999	−0.0281294
$356$	−34.4449	−1.82557
$357$	0	0
$358$	−0.839401	−0.0443637
$359$	−2.23778	−0.118105	−0.0590527	−	0.998255i	$-0.518808\pi$
$359$	−2.23778	−0.118105	−0.0590527	+	0.998255i	$0.518808\pi$
$360$	0	0
$361$	−4.05526	−0.213435
$362$	−58.9588	−3.09881
$363$	0	0
$364$	13.8726	0.727124
$365$	7.01516	0.367190
$366$	0	0
$367$	−29.1886	−1.52363	−0.761816	−	0.647793i	$-0.775692\pi$
$367$	−29.1886	−1.52363	−0.761816	+	0.647793i	$0.775692\pi$
$368$	4.77322	0.248821
$369$	0	0
$370$	−24.8710	−1.29298
$371$	−53.5602	−2.78071
$372$	0	0
$373$	0.522237	0.0270404	0.0135202	−	0.999909i	$-0.495696\pi$
$373$	0.522237	0.0270404	0.0135202	+	0.999909i	$0.495696\pi$
$374$	−1.89595	−0.0980372
$375$	0	0
$376$	5.92945	0.305788
$377$	6.30321	0.324632
$378$	0	0
$379$	10.0249	0.514944	0.257472	−	0.966286i	$-0.417111\pi$
$379$	10.0249	0.514944	0.257472	+	0.966286i	$0.417111\pi$
$380$	−13.8932	−0.712707
$381$	0	0
$382$	−29.4023	−1.50435
$383$	−18.6781	−0.954407	−0.477204	−	0.878793i	$-0.658350\pi$
$383$	−18.6781	−0.954407	−0.477204	+	0.878793i	$0.658350\pi$
$384$	0	0
$385$	−4.61307	−0.235104
$386$	22.0231	1.12095
$387$	0	0
$388$	−33.5456	−1.70302
$389$	−25.6703	−1.30153	−0.650767	−	0.759278i	$-0.725553\pi$
$389$	−25.6703	−1.30153	−0.650767	+	0.759278i	$0.725553\pi$
$390$	0	0
$391$	1.85290	0.0937050
$392$	−29.7820	−1.50422
$393$	0	0
$394$	−33.1398	−1.66956
$395$	−5.49993	−0.276731
$396$	0	0
$397$	−27.7041	−1.39043	−0.695215	−	0.718802i	$-0.744691\pi$
$397$	−27.7041	−1.39043	−0.695215	+	0.718802i	$0.744691\pi$
$398$	61.4095	3.07818
$399$	0	0
$400$	1.72800	0.0863998
$401$	−18.8910	−0.943370	−0.471685	−	0.881767i	$-0.656354\pi$
$401$	−18.8910	−0.943370	−0.471685	+	0.881767i	$0.656354\pi$
$402$	0	0
$403$	2.29289	0.114217
$404$	51.9327	2.58375
$405$	0	0
$406$	−57.5463	−2.85597
$407$	−12.5669	−0.622917
$408$	0	0
$409$	1.12082	0.0554209	0.0277105	−	0.999616i	$-0.491178\pi$
$409$	1.12082	0.0554209	0.0277105	+	0.999616i	$0.491178\pi$
$410$	11.0431	0.545379
$411$	0	0
$412$	−5.80306	−0.285896
$413$	14.4766	0.712348
$414$	0	0
$415$	7.05352	0.346244
$416$	3.45233	0.169265
$417$	0	0
$418$	−10.9267	−0.534441
$419$	21.5209	1.05136	0.525682	−	0.850681i	$-0.323810\pi$
$419$	21.5209	1.05136	0.525682	+	0.850681i	$0.323810\pi$
$420$	0	0
$421$	−8.74842	−0.426372	−0.213186	−	0.977012i	$-0.568384\pi$
$421$	−8.74842	−0.426372	−0.213186	+	0.977012i	$0.568384\pi$
$422$	−59.7979	−2.91091
$423$	0	0
$424$	52.3047	2.54014
$425$	0.670784	0.0325378
$426$	0	0
$427$	21.6246	1.04649
$428$	66.5589	3.21725
$429$	0	0
$430$	12.9945	0.626651
$431$	−22.1809	−1.06842	−0.534208	−	0.845353i	$-0.679390\pi$
$431$	−22.1809	−1.06842	−0.534208	+	0.845353i	$0.679390\pi$
$432$	0	0
$433$	20.5358	0.986889	0.493445	−	0.869777i	$-0.335738\pi$
$433$	20.5358	0.986889	0.493445	+	0.869777i	$0.335738\pi$
$434$	−20.9333	−1.00483
$435$	0	0
$436$	6.51427	0.311977
$437$	10.6786	0.510825
$438$	0	0
$439$	−0.221594	−0.0105761	−0.00528805	−	0.999986i	$-0.501683\pi$
$439$	−0.221594	−0.0105761	−0.00528805	+	0.999986i	$0.501683\pi$
$440$	4.50493	0.214764
$441$	0	0
$442$	−1.58649	−0.0754617
$443$	−10.1764	−0.483495	−0.241747	−	0.970339i	$-0.577721\pi$
$443$	−10.1764	−0.483495	−0.241747	+	0.970339i	$0.577721\pi$
$444$	0	0
$445$	9.58442	0.454345
$446$	−62.4847	−2.95874
$447$	0	0
$448$	−44.8592	−2.11940
$449$	14.0102	0.661184	0.330592	−	0.943774i	$-0.392752\pi$
$449$	14.0102	0.661184	0.330592	+	0.943774i	$0.392752\pi$
$450$	0	0
$451$	5.57988	0.262746
$452$	36.8283	1.73226
$453$	0	0
$454$	−26.0286	−1.22158
$455$	−3.86012	−0.180965
$456$	0	0
$457$	34.9568	1.63521	0.817604	−	0.575780i	$-0.195302\pi$
$457$	34.9568	1.63521	0.817604	+	0.575780i	$0.195302\pi$
$458$	−6.52046	−0.304681
$459$	0	0
$460$	−9.92721	−0.462859
$461$	0.192022	0.00894336	0.00447168	−	0.999990i	$-0.498577\pi$
$461$	0.192022	0.00894336	0.00447168	+	0.999990i	$0.498577\pi$
$462$	0	0
$463$	−17.3818	−0.807802	−0.403901	−	0.914803i	$-0.632346\pi$
$463$	−17.3818	−0.807802	−0.403901	+	0.914803i	$0.632346\pi$
$464$	10.8919	0.505645
$465$	0	0
$466$	12.9487	0.599837
$467$	−18.2343	−0.843781	−0.421890	−	0.906647i	$-0.638633\pi$
$467$	−18.2343	−0.843781	−0.421890	+	0.906647i	$0.638633\pi$
$468$	0	0
$469$	−7.05431	−0.325738
$470$	−3.72024	−0.171602
$471$	0	0
$472$	−14.1373	−0.650721
$473$	6.56589	0.301900
$474$	0	0
$475$	3.86584	0.177377
$476$	9.30554	0.426519
$477$	0	0
$478$	8.18954	0.374581
$479$	−25.8133	−1.17944	−0.589719	−	0.807608i	$-0.700762\pi$
$479$	−25.8133	−1.17944	−0.589719	+	0.807608i	$0.700762\pi$
$480$	0	0
$481$	−10.5157	−0.479474
$482$	−13.4648	−0.613303
$483$	0	0
$484$	−34.3996	−1.56362
$485$	9.33419	0.423844
$486$	0	0
$487$	36.2195	1.64126	0.820632	−	0.571457i	$-0.193622\pi$
$487$	36.2195	1.64126	0.820632	+	0.571457i	$0.193622\pi$
$488$	−21.1177	−0.955954
$489$	0	0
$490$	18.6857	0.844135
$491$	−8.97004	−0.404812	−0.202406	−	0.979302i	$-0.564876\pi$
$491$	−8.97004	−0.404812	−0.202406	+	0.979302i	$0.564876\pi$
$492$	0	0
$493$	4.22809	0.190424
$494$	−9.14322	−0.411373
$495$	0	0
$496$	3.96210	0.177904
$497$	2.04586	0.0917692
$498$	0	0
$499$	−2.05214	−0.0918663	−0.0459332	−	0.998945i	$-0.514626\pi$
$499$	−2.05214	−0.0918663	−0.0459332	+	0.998945i	$0.514626\pi$
$500$	−3.59384	−0.160721
$501$	0	0
$502$	−35.7424	−1.59526
$503$	−10.2625	−0.457583	−0.228792	−	0.973475i	$-0.573477\pi$
$503$	−10.2625	−0.457583	−0.228792	+	0.973475i	$0.573477\pi$
$504$	0	0
$505$	−14.4505	−0.643039
$506$	−7.80751	−0.347086
$507$	0	0
$508$	−38.5379	−1.70984
$509$	−25.9824	−1.15165	−0.575824	−	0.817573i	$-0.695319\pi$
$509$	−25.9824	−1.15165	−0.575824	+	0.817573i	$0.695319\pi$
$510$	0	0
$511$	−27.0794	−1.19792
$512$	18.9934	0.839399
$513$	0	0
$514$	45.2665	1.99662
$515$	1.61472	0.0711532
$516$	0	0
$517$	−1.87977	−0.0826722
$518$	96.0049	4.21821
$519$	0	0
$520$	3.76963	0.165309
$521$	19.9636	0.874622	0.437311	−	0.899310i	$-0.355931\pi$
$521$	19.9636	0.874622	0.437311	+	0.899310i	$0.355931\pi$
$522$	0	0
$523$	−7.05305	−0.308408	−0.154204	−	0.988039i	$-0.549281\pi$
$523$	−7.05305	−0.308408	−0.154204	+	0.988039i	$0.549281\pi$
$524$	5.61179	0.245152
$525$	0	0
$526$	−12.2651	−0.534784
$527$	1.53803	0.0669978
$528$	0	0
$529$	−15.3698	−0.668251
$530$	−32.8168	−1.42547
$531$	0	0
$532$	53.6294	2.32513
$533$	4.66913	0.202242
$534$	0	0
$535$	−18.5203	−0.800702
$536$	6.88895	0.297557
$537$	0	0
$538$	−61.6710	−2.65882
$539$	9.44157	0.406677
$540$	0	0
$541$	34.5109	1.48374	0.741870	−	0.670544i	$-0.233939\pi$
$541$	34.5109	1.48374	0.741870	+	0.670544i	$0.233939\pi$
$542$	71.4965	3.07104
$543$	0	0
$544$	2.31577	0.0992878
$545$	−1.81262	−0.0776442
$546$	0	0
$547$	10.5764	0.452213	0.226106	−	0.974103i	$-0.427400\pi$
$547$	10.5764	0.452213	0.226106	+	0.974103i	$0.427400\pi$
$548$	−77.3231	−3.30308
$549$	0	0
$550$	−2.82647	−0.120521
$551$	24.3672	1.03808
$552$	0	0
$553$	21.2304	0.902807
$554$	−1.03710	−0.0440622
$555$	0	0
$556$	−16.8231	−0.713460
$557$	−3.91135	−0.165729	−0.0828646	−	0.996561i	$-0.526407\pi$
$557$	−3.91135	−0.165729	−0.0828646	+	0.996561i	$0.526407\pi$
$558$	0	0
$559$	5.49420	0.232380
$560$	−6.67027	−0.281870
$561$	0	0
$562$	−51.1293	−2.15676
$563$	−23.5913	−0.994255	−0.497127	−	0.867678i	$-0.665612\pi$
$563$	−23.5913	−0.994255	−0.497127	+	0.867678i	$0.665612\pi$
$564$	0	0
$565$	−10.2476	−0.431120
$566$	−71.1237	−2.98955
$567$	0	0
$568$	−1.99790	−0.0838301
$569$	−47.2640	−1.98141	−0.990705	−	0.136030i	$-0.956565\pi$
$569$	−47.2640	−1.98141	−0.990705	+	0.136030i	$0.956565\pi$
$570$	0	0
$571$	5.76696	0.241340	0.120670	−	0.992693i	$-0.461496\pi$
$571$	5.76696	0.241340	0.120670	+	0.992693i	$0.461496\pi$
$572$	4.29485	0.179577
$573$	0	0
$574$	−42.6276	−1.77924
$575$	2.76229	0.115195
$576$	0	0
$577$	40.6992	1.69433	0.847164	−	0.531331i	$-0.178308\pi$
$577$	40.6992	1.69433	0.847164	+	0.531331i	$0.178308\pi$
$578$	39.1430	1.62813
$579$	0	0
$580$	−22.6527	−0.940603
$581$	−27.2274	−1.12958
$582$	0	0
$583$	−16.5818	−0.686747
$584$	26.4446	1.09428
$585$	0	0
$586$	37.0123	1.52897
$587$	−37.7546	−1.55830	−0.779150	−	0.626837i	$-0.784349\pi$
$587$	−37.7546	−1.55830	−0.779150	+	0.626837i	$0.784349\pi$
$588$	0	0
$589$	8.86395	0.365233
$590$	8.86996	0.365171
$591$	0	0
$592$	−18.1711	−0.746827
$593$	26.4555	1.08640	0.543198	−	0.839605i	$-0.317213\pi$
$593$	26.4555	1.08640	0.543198	+	0.839605i	$0.317213\pi$
$594$	0	0
$595$	−2.58930	−0.106151
$596$	−51.1865	−2.09668
$597$	0	0
$598$	−6.53317	−0.267161
$599$	20.0635	0.819774	0.409887	−	0.912136i	$-0.365568\pi$
$599$	20.0635	0.819774	0.409887	+	0.912136i	$0.365568\pi$
$600$	0	0
$601$	37.5017	1.52973	0.764864	−	0.644192i	$-0.222806\pi$
$601$	37.5017	1.52973	0.764864	+	0.644192i	$0.222806\pi$
$602$	−50.1603	−2.04438
$603$	0	0
$604$	80.1293	3.26041
$605$	9.57184	0.389150
$606$	0	0
$607$	24.2199	0.983055	0.491528	−	0.870862i	$-0.336439\pi$
$607$	24.2199	0.983055	0.491528	+	0.870862i	$0.336439\pi$
$608$	13.3462	0.541259
$609$	0	0
$610$	13.2496	0.536460
$611$	−1.57295	−0.0636348
$612$	0	0
$613$	−2.42842	−0.0980829	−0.0490414	−	0.998797i	$-0.515617\pi$
$613$	−2.42842	−0.0980829	−0.0490414	+	0.998797i	$0.515617\pi$
$614$	65.7035	2.65158
$615$	0	0
$616$	−17.3896	−0.700646
$617$	−24.3569	−0.980572	−0.490286	−	0.871562i	$-0.663108\pi$
$617$	−24.3569	−0.980572	−0.490286	+	0.871562i	$0.663108\pi$
$618$	0	0
$619$	45.9886	1.84844	0.924220	−	0.381861i	$-0.124717\pi$
$619$	45.9886	1.84844	0.924220	+	0.381861i	$0.124717\pi$
$620$	−8.24028	−0.330937
$621$	0	0
$622$	−9.61969	−0.385715
$623$	−36.9970	−1.48225
$624$	0	0
$625$	1.00000	0.0400000
$626$	22.9062	0.915517
$627$	0	0
$628$	41.7826	1.66731
$629$	−7.05376	−0.281252
$630$	0	0
$631$	−23.7080	−0.943802	−0.471901	−	0.881652i	$-0.656432\pi$
$631$	−23.7080	−0.943802	−0.471901	+	0.881652i	$0.656432\pi$
$632$	−20.7327	−0.824703
$633$	0	0
$634$	39.2133	1.55736
$635$	10.7233	0.425543
$636$	0	0
$637$	7.90051	0.313030
$638$	−17.8158	−0.705335
$639$	0	0
$640$	−20.5810	−0.813535
$641$	−10.7341	−0.423972	−0.211986	−	0.977273i	$-0.567993\pi$
$641$	−10.7341	−0.423972	−0.211986	+	0.977273i	$0.567993\pi$
$642$	0	0
$643$	−21.0474	−0.830027	−0.415014	−	0.909815i	$-0.636223\pi$
$643$	−21.0474	−0.830027	−0.415014	+	0.909815i	$0.636223\pi$
$644$	38.3202	1.51003
$645$	0	0
$646$	−6.13312	−0.241304
$647$	14.9485	0.587687	0.293844	−	0.955853i	$-0.405065\pi$
$647$	14.9485	0.587687	0.293844	+	0.955853i	$0.405065\pi$
$648$	0	0
$649$	4.48184	0.175927
$650$	−2.36513	−0.0927680
$651$	0	0
$652$	26.3579	1.03225
$653$	−38.2073	−1.49517	−0.747584	−	0.664168i	$-0.768786\pi$
$653$	−38.2073	−1.49517	−0.747584	+	0.664168i	$0.768786\pi$
$654$	0	0
$655$	−1.56150	−0.0610130
$656$	8.06823	0.315011
$657$	0	0
$658$	14.3605	0.559832
$659$	−8.15217	−0.317563	−0.158782	−	0.987314i	$-0.550757\pi$
$659$	−8.15217	−0.317563	−0.158782	+	0.987314i	$0.550757\pi$
$660$	0	0
$661$	7.44191	0.289457	0.144728	−	0.989471i	$-0.453769\pi$
$661$	7.44191	0.289457	0.144728	+	0.989471i	$0.453769\pi$
$662$	−32.2368	−1.25292
$663$	0	0
$664$	26.5892	1.03186
$665$	−14.9226	−0.578674
$666$	0	0
$667$	17.4113	0.674167
$668$	60.9927	2.35988
$669$	0	0
$670$	−4.32224	−0.166983
$671$	6.69479	0.258449
$672$	0	0
$673$	48.6849	1.87667	0.938333	−	0.345733i	$-0.112370\pi$
$673$	48.6849	1.87667	0.938333	+	0.345733i	$0.112370\pi$
$674$	−56.1376	−2.16234
$675$	0	0
$676$	3.59384	0.138225
$677$	−36.1346	−1.38876	−0.694382	−	0.719606i	$-0.744322\pi$
$677$	−36.1346	−1.38876	−0.694382	+	0.719606i	$0.744322\pi$
$678$	0	0
$679$	−36.0311	−1.38275
$680$	2.52861	0.0969677
$681$	0	0
$682$	−6.48078	−0.248162
$683$	−42.7590	−1.63613	−0.818064	−	0.575128i	$-0.804952\pi$
$683$	−42.7590	−1.63613	−0.818064	+	0.575128i	$0.804952\pi$
$684$	0	0
$685$	21.5155	0.822064
$686$	−8.22137	−0.313893
$687$	0	0
$688$	9.49396	0.361954
$689$	−13.8753	−0.528606
$690$	0	0
$691$	−20.4346	−0.777369	−0.388684	−	0.921371i	$-0.627070\pi$
$691$	−20.4346	−0.777369	−0.388684	+	0.921371i	$0.627070\pi$
$692$	63.4597	2.41237
$693$	0	0
$694$	−9.81012	−0.372387
$695$	4.68111	0.177564
$696$	0	0
$697$	3.13197	0.118632
$698$	37.4644	1.41805
$699$	0	0
$700$	13.8726	0.524336
$701$	−12.6314	−0.477081	−0.238541	−	0.971133i	$-0.576669\pi$
$701$	−12.6314	−0.477081	−0.238541	+	0.971133i	$0.576669\pi$
$702$	0	0
$703$	−40.6520	−1.53322
$704$	−13.8880	−0.523424
$705$	0	0
$706$	−19.5701	−0.736531
$707$	55.7806	2.09785
$708$	0	0
$709$	3.98824	0.149781	0.0748907	−	0.997192i	$-0.476139\pi$
$709$	3.98824	0.149781	0.0748907	+	0.997192i	$0.476139\pi$
$710$	1.25352	0.0470436
$711$	0	0
$712$	36.1298	1.35402
$713$	6.33362	0.237196
$714$	0	0
$715$	−1.19506	−0.0446926
$716$	1.27548	0.0476669
$717$	0	0
$718$	5.29264	0.197519
$719$	−2.24736	−0.0838124	−0.0419062	−	0.999122i	$-0.513343\pi$
$719$	−2.24736	−0.0838124	−0.0419062	+	0.999122i	$0.513343\pi$
$720$	0	0
$721$	−6.23302	−0.232130
$722$	9.59123	0.356948
$723$	0	0
$724$	89.5885	3.32953
$725$	6.30321	0.234095
$726$	0	0
$727$	−16.9214	−0.627581	−0.313790	−	0.949492i	$-0.601599\pi$
$727$	−16.9214	−0.627581	−0.313790	+	0.949492i	$0.601599\pi$
$728$	−14.5512	−0.539304
$729$	0	0
$730$	−16.5918	−0.614089
$731$	3.68542	0.136310
$732$	0	0
$733$	32.5692	1.20297	0.601486	−	0.798883i	$-0.294576\pi$
$733$	32.5692	1.20297	0.601486	+	0.798883i	$0.294576\pi$
$734$	69.0348	2.54812
$735$	0	0
$736$	9.53633	0.351514
$737$	−2.18395	−0.0804469
$738$	0	0
$739$	−32.5810	−1.19851	−0.599256	−	0.800558i	$-0.704537\pi$
$739$	−32.5810	−1.19851	−0.599256	+	0.800558i	$0.704537\pi$
$740$	37.7917	1.38925
$741$	0	0
$742$	126.677	4.65045
$743$	36.1625	1.32667	0.663337	−	0.748321i	$-0.269139\pi$
$743$	36.1625	1.32667	0.663337	+	0.748321i	$0.269139\pi$
$744$	0	0
$745$	14.2429	0.521818
$746$	−1.23516	−0.0452224
$747$	0	0
$748$	2.88091	0.105337
$749$	71.4905	2.61221
$750$	0	0
$751$	−29.9492	−1.09286	−0.546432	−	0.837504i	$-0.684014\pi$
$751$	−29.9492	−1.09286	−0.546432	+	0.837504i	$0.684014\pi$
$752$	−2.71805	−0.0991173
$753$	0	0
$754$	−14.9079	−0.542914
$755$	−22.2963	−0.811445
$756$	0	0
$757$	32.1200	1.16742	0.583711	−	0.811961i	$-0.301600\pi$
$757$	32.1200	1.16742	0.583711	+	0.811961i	$0.301600\pi$
$758$	−23.7102	−0.861192
$759$	0	0
$760$	14.5728	0.528611
$761$	30.7235	1.11373	0.556863	−	0.830605i	$-0.312005\pi$
$761$	30.7235	1.11373	0.556863	+	0.830605i	$0.312005\pi$
$762$	0	0
$763$	6.99693	0.253306
$764$	44.6771	1.61636
$765$	0	0
$766$	44.1762	1.59615
$767$	3.75031	0.135416
$768$	0	0
$769$	−13.9999	−0.504850	−0.252425	−	0.967616i	$-0.581228\pi$
$769$	−13.9999	−0.504850	−0.252425	+	0.967616i	$0.581228\pi$
$770$	10.9105	0.393187
$771$	0	0
$772$	−33.4643	−1.20441
$773$	−9.11105	−0.327702	−0.163851	−	0.986485i	$-0.552392\pi$
$773$	−9.11105	−0.327702	−0.163851	+	0.986485i	$0.552392\pi$
$774$	0	0
$775$	2.29289	0.0823630
$776$	35.1865	1.26312
$777$	0	0
$778$	60.7135	2.17668
$779$	18.0501	0.646712
$780$	0	0
$781$	0.633379	0.0226641
$782$	−4.38234	−0.156712
$783$	0	0
$784$	13.6520	0.487573
$785$	−11.6262	−0.414956
$786$	0	0
$787$	−18.0382	−0.642993	−0.321496	−	0.946911i	$-0.604186\pi$
$787$	−18.0382	−0.642993	−0.321496	+	0.946911i	$0.604186\pi$
$788$	50.3563	1.79387
$789$	0	0
$790$	13.0080	0.462805
$791$	39.5570	1.40648
$792$	0	0
$793$	5.60206	0.198935
$794$	65.5238	2.32535
$795$	0	0
$796$	−93.3123	−3.30737
$797$	−7.68414	−0.272186	−0.136093	−	0.990696i	$-0.543455\pi$
$797$	−7.68414	−0.272186	−0.136093	+	0.990696i	$0.543455\pi$
$798$	0	0
$799$	−1.05511	−0.0373271
$800$	3.45233	0.122058
$801$	0	0
$802$	44.6796	1.57769
$803$	−8.38353	−0.295848
$804$	0	0
$805$	−10.6628	−0.375813
$806$	−5.42298	−0.191016
$807$	0	0
$808$	−54.4731	−1.91636
$809$	28.2891	0.994593	0.497296	−	0.867581i	$-0.334326\pi$
$809$	28.2891	0.994593	0.497296	+	0.867581i	$0.334326\pi$
$810$	0	0
$811$	−14.2745	−0.501245	−0.250622	−	0.968085i	$-0.580635\pi$
$811$	−14.2745	−0.501245	−0.250622	+	0.968085i	$0.580635\pi$
$812$	87.4422	3.06862
$813$	0	0
$814$	29.7223	1.04177
$815$	−7.33419	−0.256905
$816$	0	0
$817$	21.2397	0.743084
$818$	−2.65088	−0.0926859
$819$	0	0
$820$	−16.7801	−0.585986
$821$	39.0397	1.36249	0.681247	−	0.732054i	$-0.261438\pi$
$821$	39.0397	1.36249	0.681247	+	0.732054i	$0.261438\pi$
$822$	0	0
$823$	26.2592	0.915339	0.457670	−	0.889122i	$-0.348684\pi$
$823$	26.2592	0.915339	0.457670	+	0.889122i	$0.348684\pi$
$824$	6.08692	0.212048
$825$	0	0
$826$	−34.2391	−1.19133
$827$	−8.96895	−0.311881	−0.155941	−	0.987766i	$-0.549841\pi$
$827$	−8.96895	−0.311881	−0.155941	+	0.987766i	$0.549841\pi$
$828$	0	0
$829$	13.2015	0.458506	0.229253	−	0.973367i	$-0.426372\pi$
$829$	13.2015	0.458506	0.229253	+	0.973367i	$0.426372\pi$
$830$	−16.6825	−0.579058
$831$	0	0
$832$	−11.6212	−0.402893
$833$	5.29953	0.183618
$834$	0	0
$835$	−16.9715	−0.587321
$836$	16.6032	0.574234
$837$	0	0
$838$	−50.8997	−1.75830
$839$	9.36783	0.323413	0.161707	−	0.986839i	$-0.448300\pi$
$839$	9.36783	0.323413	0.161707	+	0.986839i	$0.448300\pi$
$840$	0	0
$841$	10.7305	0.370016
$842$	20.6912	0.713064
$843$	0	0
$844$	90.8634	3.12765
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−36.9484	−1.26956
$848$	−23.9764	−0.823354
$849$	0	0
$850$	−1.58649	−0.0544162
$851$	−29.0474	−0.995731
$852$	0	0
$853$	43.4180	1.48660	0.743302	−	0.668956i	$-0.233259\pi$
$853$	43.4180	1.48660	0.743302	+	0.668956i	$0.233259\pi$
$854$	−51.1450	−1.75015
$855$	0	0
$856$	−69.8147	−2.38622
$857$	−17.9545	−0.613314	−0.306657	−	0.951820i	$-0.599210\pi$
$857$	−17.9545	−0.613314	−0.306657	+	0.951820i	$0.599210\pi$
$858$	0	0
$859$	−28.3248	−0.966431	−0.483216	−	0.875501i	$-0.660531\pi$
$859$	−28.3248	−0.966431	−0.483216	+	0.875501i	$0.660531\pi$
$860$	−19.7453	−0.673308
$861$	0	0
$862$	52.4607	1.78682
$863$	20.7858	0.707557	0.353778	−	0.935329i	$-0.384897\pi$
$863$	20.7858	0.707557	0.353778	+	0.935329i	$0.384897\pi$
$864$	0	0
$865$	−17.6579	−0.600387
$866$	−48.5699	−1.65047
$867$	0	0
$868$	31.8084	1.07965
$869$	6.57273	0.222965
$870$	0	0
$871$	−1.82749	−0.0619220
$872$	−6.83292	−0.231392
$873$	0	0
$874$	−25.2562	−0.854303
$875$	−3.86012	−0.130496
$876$	0	0
$877$	10.8276	0.365623	0.182812	−	0.983148i	$-0.441480\pi$
$877$	10.8276	0.365623	0.182812	+	0.983148i	$0.441480\pi$
$878$	0.524099	0.0176875
$879$	0	0
$880$	−2.06506	−0.0696130
$881$	19.1403	0.644852	0.322426	−	0.946595i	$-0.395502\pi$
$881$	19.1403	0.644852	0.322426	+	0.946595i	$0.395502\pi$
$882$	0	0
$883$	32.2361	1.08483	0.542416	−	0.840110i	$-0.317510\pi$
$883$	32.2361	1.08483	0.542416	+	0.840110i	$0.317510\pi$
$884$	2.41069	0.0810802
$885$	0	0
$886$	24.0685	0.808596
$887$	56.8223	1.90791	0.953953	−	0.299957i	$-0.0969724\pi$
$887$	56.8223	1.90791	0.953953	+	0.299957i	$0.0969724\pi$
$888$	0	0
$889$	−41.3933	−1.38829
$890$	−22.6684	−0.759846
$891$	0	0
$892$	94.9461	3.17903
$893$	−6.08078	−0.203486
$894$	0	0
$895$	−0.354907	−0.0118632
$896$	79.4451	2.65407
$897$	0	0
$898$	−33.1360	−1.10576
$899$	14.4526	0.482020
$900$	0	0
$901$	−9.30731	−0.310072
$902$	−13.1971	−0.439416
$903$	0	0
$904$	−38.6297	−1.28481
$905$	−24.9284	−0.828647
$906$	0	0
$907$	−46.3228	−1.53812	−0.769062	−	0.639174i	$-0.779276\pi$
$907$	−46.3228	−1.53812	−0.769062	+	0.639174i	$0.779276\pi$
$908$	39.5507	1.31254
$909$	0	0
$910$	9.12968	0.302646
$911$	−25.0668	−0.830502	−0.415251	−	0.909707i	$-0.636306\pi$
$911$	−25.0668	−0.830502	−0.415251	+	0.909707i	$0.636306\pi$
$912$	0	0
$913$	−8.42937	−0.278971
$914$	−82.6773	−2.73472
$915$	0	0
$916$	9.90791	0.327366
$917$	6.02759	0.199049
$918$	0	0
$919$	−45.8297	−1.51178	−0.755891	−	0.654697i	$-0.772796\pi$
$919$	−45.8297	−1.51178	−0.755891	+	0.654697i	$0.772796\pi$
$920$	10.4128	0.343300
$921$	0	0
$922$	−0.454157	−0.0149569
$923$	0.529999	0.0174451
$924$	0	0
$925$	−10.5157	−0.345754
$926$	41.1103	1.35097
$927$	0	0
$928$	21.7608	0.714333
$929$	−49.6577	−1.62922	−0.814608	−	0.580012i	$-0.803048\pi$
$929$	−49.6577	−1.62922	−0.814608	+	0.580012i	$0.803048\pi$
$930$	0	0
$931$	30.5421	1.00098
$932$	−19.6757	−0.644499
$933$	0	0
$934$	43.1264	1.41114
$935$	−0.801626	−0.0262160
$936$	0	0
$937$	−27.4158	−0.895636	−0.447818	−	0.894125i	$-0.647799\pi$
$937$	−27.4158	−0.895636	−0.447818	+	0.894125i	$0.647799\pi$
$938$	16.6844	0.544764
$939$	0	0
$940$	5.65294	0.184378
$941$	−11.3525	−0.370080	−0.185040	−	0.982731i	$-0.559241\pi$
$941$	−11.3525	−0.370080	−0.185040	+	0.982731i	$0.559241\pi$
$942$	0	0
$943$	12.8975	0.419999
$944$	6.48051	0.210923
$945$	0	0
$946$	−15.5292	−0.504897
$947$	36.6104	1.18968	0.594839	−	0.803845i	$-0.297216\pi$
$947$	36.6104	1.18968	0.594839	+	0.803845i	$0.297216\pi$
$948$	0	0
$949$	−7.01516	−0.227722
$950$	−9.14322	−0.296645
$951$	0	0
$952$	−9.76073	−0.316347
$953$	−41.4047	−1.34123	−0.670615	−	0.741806i	$-0.733970\pi$
$953$	−41.4047	−1.34123	−0.670615	+	0.741806i	$0.733970\pi$
$954$	0	0
$955$	−12.4316	−0.402276
$956$	−12.4441	−0.402471
$957$	0	0
$958$	61.0517	1.97249
$959$	−83.0522	−2.68190
$960$	0	0
$961$	−25.7427	−0.830408
$962$	24.8710	0.801873
$963$	0	0
$964$	20.4598	0.658967
$965$	9.31158	0.299750
$966$	0	0
$967$	−1.38137	−0.0444220	−0.0222110	−	0.999753i	$-0.507071\pi$
$967$	−1.38137	−0.0444220	−0.0222110	+	0.999753i	$0.507071\pi$
$968$	36.0823	1.15973
$969$	0	0
$970$	−22.0766	−0.708836
$971$	24.5710	0.788520	0.394260	−	0.918999i	$-0.371001\pi$
$971$	24.5710	0.788520	0.394260	+	0.918999i	$0.371001\pi$
$972$	0	0
$973$	−18.0696	−0.579285
$974$	−85.6639	−2.74485
$975$	0	0
$976$	9.68033	0.309860
$977$	−22.0289	−0.704768	−0.352384	−	0.935856i	$-0.614629\pi$
$977$	−22.0289	−0.704768	−0.352384	+	0.935856i	$0.614629\pi$
$978$	0	0
$979$	−11.4539	−0.366070
$980$	−28.3932	−0.906986
$981$	0	0
$982$	21.2153	0.677007
$983$	−29.1751	−0.930543	−0.465271	−	0.885168i	$-0.654043\pi$
$983$	−29.1751	−0.930543	−0.465271	+	0.885168i	$0.654043\pi$
$984$	0	0
$985$	−14.0118	−0.446455
$986$	−9.99999	−0.318464
$987$	0	0
$988$	13.8932	0.442002
$989$	15.1766	0.482587
$990$	0	0
$991$	−24.5338	−0.779341	−0.389671	−	0.920954i	$-0.627411\pi$
$991$	−24.5338	−0.779341	−0.389671	+	0.920954i	$0.627411\pi$
$992$	7.91582	0.251328
$993$	0	0
$994$	−4.83872	−0.153475
$995$	25.9645	0.823131
$996$	0	0
$997$	−62.4572	−1.97804	−0.989019	−	0.147787i	$-0.952785\pi$
$997$	−62.4572	−1.97804	−0.989019	+	0.147787i	$0.952785\pi$
$998$	4.85357	0.153637
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.u.1.2	✓	7
3.2	odd	2		1755.2.a.v.1.6	yes	7
5.4	even	2		8775.2.a.bx.1.6		7
15.14	odd	2		8775.2.a.bw.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.u.1.2	✓	7	1.1	even	1	trivial
1755.2.a.v.1.6	yes	7	3.2	odd	2
8775.2.a.bw.1.2		7	15.14	odd	2
8775.2.a.bx.1.6		7	5.4	even	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:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36513 q^{2} +3.59384 q^{4} -1.00000 q^{5} +3.86012 q^{7} -3.76963 q^{8} +O(q^{10})\)	\(q-2.36513 q^{2} +3.59384 q^{4} -1.00000 q^{5} +3.86012 q^{7} -3.76963 q^{8} +2.36513 q^{10} +1.19506 q^{11} +1.00000 q^{13} -9.12968 q^{14} +1.72800 q^{16} +0.670784 q^{17} +3.86584 q^{19} -3.59384 q^{20} -2.82647 q^{22} +2.76229 q^{23} +1.00000 q^{25} -2.36513 q^{26} +13.8726 q^{28} +6.30321 q^{29} +2.29289 q^{31} +3.45233 q^{32} -1.58649 q^{34} -3.86012 q^{35} -10.5157 q^{37} -9.14322 q^{38} +3.76963 q^{40} +4.66913 q^{41} +5.49420 q^{43} +4.29485 q^{44} -6.53317 q^{46} -1.57295 q^{47} +7.90051 q^{49} -2.36513 q^{50} +3.59384 q^{52} -13.8753 q^{53} -1.19506 q^{55} -14.5512 q^{56} -14.9079 q^{58} +3.75031 q^{59} +5.60206 q^{61} -5.42298 q^{62} -11.6212 q^{64} -1.00000 q^{65} -1.82749 q^{67} +2.41069 q^{68} +9.12968 q^{70} +0.529999 q^{71} -7.01516 q^{73} +24.8710 q^{74} +13.8932 q^{76} +4.61307 q^{77} +5.49993 q^{79} -1.72800 q^{80} -11.0431 q^{82} -7.05352 q^{83} -0.670784 q^{85} -12.9945 q^{86} -4.50493 q^{88} -9.58442 q^{89} +3.86012 q^{91} +9.92721 q^{92} +3.72024 q^{94} -3.86584 q^{95} -9.33419 q^{97} -18.6857 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8	\( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8} + q^{10} - q^{11} + 7 q^{13} + 12 q^{14} + 23 q^{16} - 11 q^{17} + 2 q^{19} - 11 q^{20} + 16 q^{22} + q^{23} + 7 q^{25} - q^{26} + 10 q^{28} + 4 q^{29} - 13 q^{32} + q^{34} - 2 q^{35} + 23 q^{37} + 15 q^{38} + 6 q^{40} - 2 q^{41} + 8 q^{43} - 10 q^{44} + 37 q^{46} - 2 q^{47} + 43 q^{49} - q^{50} + 11 q^{52} - 10 q^{53} + q^{55} + 68 q^{56} - 26 q^{58} + 13 q^{59} + 21 q^{61} - 9 q^{62} + 46 q^{64} - 7 q^{65} + 21 q^{67} - 53 q^{68} - 12 q^{70} + 10 q^{71} + 13 q^{73} + 68 q^{74} - 41 q^{76} - 6 q^{77} + 8 q^{79} - 23 q^{80} - 26 q^{82} + 4 q^{83} + 11 q^{85} + 12 q^{86} + 44 q^{88} + 27 q^{89} + 2 q^{91} + 9 q^{92} - 24 q^{94} - 2 q^{95} - 15 q^{97} - 7 q^{98}+O(q^{100}) \) 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8 + q^10 - q^11 + 7 * q^13 + 12 * q^14 + 23 * q^16 - 11 * q^17 + 2 * q^19 - 11 * q^20 + 16 * q^22 + q^23 + 7 * q^25 - q^26 + 10 * q^28 + 4 * q^29 - 13 * q^32 + q^34 - 2 * q^35 + 23 * q^37 + 15 * q^38 + 6 * q^40 - 2 * q^41 + 8 * q^43 - 10 * q^44 + 37 * q^46 - 2 * q^47 + 43 * q^49 - q^50 + 11 * q^52 - 10 * q^53 + q^55 + 68 * q^56 - 26 * q^58 + 13 * q^59 + 21 * q^61 - 9 * q^62 + 46 * q^64 - 7 * q^65 + 21 * q^67 - 53 * q^68 - 12 * q^70 + 10 * q^71 + 13 * q^73 + 68 * q^74 - 41 * q^76 - 6 * q^77 + 8 * q^79 - 23 * q^80 - 26 * q^82 + 4 * q^83 + 11 * q^85 + 12 * q^86 + 44 * q^88 + 27 * q^89 + 2 * q^91 + 9 * q^92 - 24 * q^94 - 2 * q^95 - 15 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more