LMFDB - Embedded newform 1755.2.a.p.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.8957.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + x + 1 $ x^4 - x^3 - 5*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.542055$ 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)

$f(q)$	$=$	$q-0.542055 q^{2} -1.70618 q^{4} -1.00000 q^{5} +2.84483 q^{7} +2.00895 q^{8} +O(q^{10})$	\(q-0.542055 q^{2} -1.70618 q^{4} -1.00000 q^{5} +2.84483 q^{7} +2.00895 q^{8} +0.542055 q^{10} -2.44143 q^{11} -1.00000 q^{13} -1.54206 q^{14} +2.32339 q^{16} -0.218665 q^{17} -3.38689 q^{19} +1.70618 q^{20} +1.32339 q^{22} -5.00895 q^{23} +1.00000 q^{25} +0.542055 q^{26} -4.85378 q^{28} +5.24067 q^{29} +9.08550 q^{31} -5.27731 q^{32} +0.118529 q^{34} -2.84483 q^{35} -1.28626 q^{37} +1.83588 q^{38} -2.00895 q^{40} -5.05503 q^{41} +0.495974 q^{43} +4.16551 q^{44} +2.71513 q^{46} +5.34129 q^{47} +1.09306 q^{49} -0.542055 q^{50} +1.70618 q^{52} -8.87440 q^{53} +2.44143 q^{55} +5.71513 q^{56} -2.84073 q^{58} -6.17307 q^{59} -0.432478 q^{61} -4.92484 q^{62} -1.78619 q^{64} +1.00000 q^{65} -1.66010 q^{67} +0.373082 q^{68} +1.54206 q^{70} -7.34886 q^{71} +0.735254 q^{73} +0.697224 q^{74} +5.77862 q^{76} -6.94546 q^{77} +5.53449 q^{79} -2.32339 q^{80} +2.74011 q^{82} -15.2537 q^{83} +0.218665 q^{85} -0.268846 q^{86} -4.90472 q^{88} -2.44899 q^{89} -2.84483 q^{91} +8.54615 q^{92} -2.89528 q^{94} +3.38689 q^{95} -6.22415 q^{97} -0.592500 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 3 q^{4} - 4 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q - q^2 + 3 * q^4 - 4 * q^5 + 3 * q^7 - 9 * q^8	$ 4 q - q^{2} + 3 q^{4} - 4 q^{5} + 3 q^{7} - 9 q^{8} + q^{10} - 4 q^{11} - 4 q^{13} - 5 q^{14} + 13 q^{16} + 4 q^{17} - 4 q^{19} - 3 q^{20} + 9 q^{22} - 3 q^{23} + 4 q^{25} + q^{26} + 6 q^{28} - 14 q^{29} - 7 q^{31} - 24 q^{32} - 29 q^{34} - 3 q^{35} + 9 q^{37} + 16 q^{38} + 9 q^{40} - 4 q^{41} - 33 q^{44} - 16 q^{46} - 9 q^{47} - 15 q^{49} - q^{50} - 3 q^{52} - 21 q^{53} + 4 q^{55} - 4 q^{56} + q^{58} + q^{59} - 13 q^{61} - 5 q^{62} + 9 q^{64} + 4 q^{65} + 4 q^{67} + 30 q^{68} + 5 q^{70} - 23 q^{71} + 7 q^{73} + 10 q^{74} - 17 q^{76} - 24 q^{77} - 3 q^{79} - 13 q^{80} - 6 q^{82} - 13 q^{83} - 4 q^{85} - q^{86} + 26 q^{88} - 28 q^{89} - 3 q^{91} + 37 q^{92} - 3 q^{94} + 4 q^{95} + 17 q^{97} - q^{98}+O(q^{100}) $ 4 * q - q^2 + 3 * q^4 - 4 * q^5 + 3 * q^7 - 9 * q^8 + q^10 - 4 * q^11 - 4 * q^13 - 5 * q^14 + 13 * q^16 + 4 * q^17 - 4 * q^19 - 3 * q^20 + 9 * q^22 - 3 * q^23 + 4 * q^25 + q^26 + 6 * q^28 - 14 * q^29 - 7 * q^31 - 24 * q^32 - 29 * q^34 - 3 * q^35 + 9 * q^37 + 16 * q^38 + 9 * q^40 - 4 * q^41 - 33 * q^44 - 16 * q^46 - 9 * q^47 - 15 * q^49 - q^50 - 3 * q^52 - 21 * q^53 + 4 * q^55 - 4 * q^56 + q^58 + q^59 - 13 * q^61 - 5 * q^62 + 9 * q^64 + 4 * q^65 + 4 * q^67 + 30 * q^68 + 5 * q^70 - 23 * q^71 + 7 * q^73 + 10 * q^74 - 17 * q^76 - 24 * q^77 - 3 * q^79 - 13 * q^80 - 6 * q^82 - 13 * q^83 - 4 * q^85 - q^86 + 26 * q^88 - 28 * q^89 - 3 * q^91 + 37 * q^92 - 3 * q^94 + 4 * q^95 + 17 * q^97 - 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 $

$2$	−0.542055	−0.383291	−0.191645	−	0.981464i	$-0.561382\pi$
$2$	−0.542055	−0.383291	−0.191645	+	0.981464i	$0.561382\pi$
$3$	0	0
$3$	0	0
$4$	−1.70618	−0.853088
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.84483	1.07524	0.537622	−	0.843186i	$-0.319323\pi$
$7$	2.84483	1.07524	0.537622	+	0.843186i	$0.319323\pi$
$8$	2.00895	0.710272
$9$	0	0
$10$	0.542055	0.171413
$11$	−2.44143	−0.736119	−0.368059	−	0.929802i	$-0.619978\pi$
$11$	−2.44143	−0.736119	−0.368059	+	0.929802i	$0.619978\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.54206	−0.412132
$15$	0	0
$16$	2.32339	0.580847
$17$	−0.218665	−0.0530341	−0.0265171	−	0.999648i	$-0.508442\pi$
$17$	−0.218665	−0.0530341	−0.0265171	+	0.999648i	$0.508442\pi$
$18$	0	0
$19$	−3.38689	−0.777005	−0.388502	−	0.921448i	$-0.627008\pi$
$19$	−3.38689	−0.777005	−0.388502	+	0.921448i	$0.627008\pi$
$20$	1.70618	0.381513
$21$	0	0
$22$	1.32339	0.282148
$23$	−5.00895	−1.04444	−0.522219	−	0.852811i	$-0.674896\pi$
$23$	−5.00895	−1.04444	−0.522219	+	0.852811i	$0.674896\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.542055	0.106306
$27$	0	0
$28$	−4.85378	−0.917279
$29$	5.24067	0.973168	0.486584	−	0.873634i	$-0.338243\pi$
$29$	5.24067	0.973168	0.486584	+	0.873634i	$0.338243\pi$
$30$	0	0
$31$	9.08550	1.63180	0.815902	−	0.578190i	$-0.196241\pi$
$31$	9.08550	1.63180	0.815902	+	0.578190i	$0.196241\pi$
$32$	−5.27731	−0.932905
$33$	0	0
$34$	0.118529	0.0203275
$35$	−2.84483	−0.480864
$36$	0	0
$37$	−1.28626	−0.211460	−0.105730	−	0.994395i	$-0.533718\pi$
$37$	−1.28626	−0.211460	−0.105730	+	0.994395i	$0.533718\pi$
$38$	1.83588	0.297819
$39$	0	0
$40$	−2.00895	−0.317643
$41$	−5.05503	−0.789463	−0.394732	−	0.918796i	$-0.629162\pi$
$41$	−5.05503	−0.789463	−0.394732	+	0.918796i	$0.629162\pi$
$42$	0	0
$43$	0.495974	0.0756354	0.0378177	−	0.999285i	$-0.487959\pi$
$43$	0.495974	0.0756354	0.0378177	+	0.999285i	$0.487959\pi$
$44$	4.16551	0.627974
$45$	0	0
$46$	2.71513	0.400324
$47$	5.34129	0.779108	0.389554	−	0.921004i	$-0.372629\pi$
$47$	5.34129	0.779108	0.389554	+	0.921004i	$0.372629\pi$
$48$	0	0
$49$	1.09306	0.156152
$50$	−0.542055	−0.0766582
$51$	0	0
$52$	1.70618	0.236604
$53$	−8.87440	−1.21899	−0.609496	−	0.792789i	$-0.708628\pi$
$53$	−8.87440	−1.21899	−0.609496	+	0.792789i	$0.708628\pi$
$54$	0	0
$55$	2.44143	0.329202
$56$	5.71513	0.763716
$57$	0	0
$58$	−2.84073	−0.373006
$59$	−6.17307	−0.803666	−0.401833	−	0.915713i	$-0.631627\pi$
$59$	−6.17307	−0.803666	−0.401833	+	0.915713i	$0.631627\pi$
$60$	0	0
$61$	−0.432478	−0.0553732	−0.0276866	−	0.999617i	$-0.508814\pi$
$61$	−0.432478	−0.0553732	−0.0276866	+	0.999617i	$0.508814\pi$
$62$	−4.92484	−0.625456
$63$	0	0
$64$	−1.78619	−0.223273
$65$	1.00000	0.124035
$66$	0	0
$67$	−1.66010	−0.202813	−0.101406	−	0.994845i	$-0.532334\pi$
$67$	−1.66010	−0.202813	−0.101406	+	0.994845i	$0.532334\pi$
$68$	0.373082	0.0452428
$69$	0	0
$70$	1.54206	0.184311
$71$	−7.34886	−0.872149	−0.436074	−	0.899911i	$-0.643632\pi$
$71$	−7.34886	−0.872149	−0.436074	+	0.899911i	$0.643632\pi$
$72$	0	0
$73$	0.735254	0.0860550	0.0430275	−	0.999074i	$-0.486300\pi$
$73$	0.735254	0.0860550	0.0430275	+	0.999074i	$0.486300\pi$
$74$	0.697224	0.0810507
$75$	0	0
$76$	5.77862	0.662854
$77$	−6.94546	−0.791508
$78$	0	0
$79$	5.53449	0.622679	0.311340	−	0.950299i	$-0.399222\pi$
$79$	5.53449	0.622679	0.311340	+	0.950299i	$0.399222\pi$
$80$	−2.32339	−0.259763
$81$	0	0
$82$	2.74011	0.302594
$83$	−15.2537	−1.67431	−0.837157	−	0.546963i	$-0.815784\pi$
$83$	−15.2537	−1.67431	−0.837157	+	0.546963i	$0.815784\pi$
$84$	0	0
$85$	0.218665	0.0237176
$86$	−0.268846	−0.0289904
$87$	0	0
$88$	−4.90472	−0.522844
$89$	−2.44899	−0.259593	−0.129796	−	0.991541i	$-0.541432\pi$
$89$	−2.44899	−0.259593	−0.129796	+	0.991541i	$0.541432\pi$
$90$	0	0
$91$	−2.84483	−0.298219
$92$	8.54615	0.890998
$93$	0	0
$94$	−2.89528	−0.298625
$95$	3.38689	0.347487
$96$	0	0
$97$	−6.22415	−0.631967	−0.315984	−	0.948765i	$-0.602334\pi$
$97$	−6.22415	−0.631967	−0.315984	+	0.948765i	$0.602334\pi$
$98$	−0.592500	−0.0598515
$99$	0	0
$100$	−1.70618	−0.170618
$101$	−15.4813	−1.54045	−0.770225	−	0.637772i	$-0.779856\pi$
$101$	−15.4813	−1.54045	−0.770225	+	0.637772i	$0.779856\pi$
$102$	0	0
$103$	−12.5228	−1.23391	−0.616956	−	0.786998i	$-0.711634\pi$
$103$	−12.5228	−1.23391	−0.616956	+	0.786998i	$0.711634\pi$
$104$	−2.00895	−0.196994
$105$	0	0
$106$	4.81041	0.467229
$107$	−18.2661	−1.76585	−0.882927	−	0.469510i	$-0.844430\pi$
$107$	−18.2661	−1.76585	−0.882927	+	0.469510i	$0.844430\pi$
$108$	0	0
$109$	−4.51708	−0.432657	−0.216329	−	0.976321i	$-0.569408\pi$
$109$	−4.51708	−0.432657	−0.216329	+	0.976321i	$0.569408\pi$
$110$	−1.32339	−0.126180
$111$	0	0
$112$	6.60965	0.624553
$113$	−4.96148	−0.466737	−0.233368	−	0.972388i	$-0.574975\pi$
$113$	−4.96148	−0.466737	−0.233368	+	0.972388i	$0.574975\pi$
$114$	0	0
$115$	5.00895	0.467087
$116$	−8.94150	−0.830198
$117$	0	0
$118$	3.34615	0.308038
$119$	−0.622066	−0.0570247
$120$	0	0
$121$	−5.03942	−0.458129
$122$	0.234427	0.0212240
$123$	0	0
$124$	−15.5015	−1.39207
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	11.2911	1.00192	0.500962	−	0.865469i	$-0.332980\pi$
$127$	11.2911	1.00192	0.500962	+	0.865469i	$0.332980\pi$
$128$	11.5228	1.01848
$129$	0	0
$130$	−0.542055	−0.0475414
$131$	−3.91153	−0.341752	−0.170876	−	0.985293i	$-0.554660\pi$
$131$	−3.91153	−0.341752	−0.170876	+	0.985293i	$0.554660\pi$
$132$	0	0
$133$	−9.63512	−0.835471
$134$	0.899863	0.0777364
$135$	0	0
$136$	−0.439288	−0.0376687
$137$	7.79119	0.665646	0.332823	−	0.942989i	$-0.391999\pi$
$137$	7.79119	0.665646	0.332823	+	0.942989i	$0.391999\pi$
$138$	0	0
$139$	−8.45933	−0.717511	−0.358756	−	0.933432i	$-0.616799\pi$
$139$	−8.45933	−0.717511	−0.358756	+	0.933432i	$0.616799\pi$
$140$	4.85378	0.410220
$141$	0	0
$142$	3.98349	0.334287
$143$	2.44143	0.204163
$144$	0	0
$145$	−5.24067	−0.435214
$146$	−0.398548	−0.0329841
$147$	0	0
$148$	2.19459	0.180394
$149$	−17.6615	−1.44689	−0.723443	−	0.690385i	$-0.757441\pi$
$149$	−17.6615	−1.44689	−0.723443	+	0.690385i	$0.757441\pi$
$150$	0	0
$151$	−10.2111	−0.830968	−0.415484	−	0.909601i	$-0.636388\pi$
$151$	−10.2111	−0.830968	−0.415484	+	0.909601i	$0.636388\pi$
$152$	−6.80409	−0.551885
$153$	0	0
$154$	3.76482	0.303378
$155$	−9.08550	−0.729765
$156$	0	0
$157$	16.9897	1.35593	0.677964	−	0.735095i	$-0.262863\pi$
$157$	16.9897	1.35593	0.677964	+	0.735095i	$0.262863\pi$
$158$	−3.00000	−0.238667
$159$	0	0
$160$	5.27731	0.417208
$161$	−14.2496	−1.12303
$162$	0	0
$163$	−3.59314	−0.281436	−0.140718	−	0.990050i	$-0.544941\pi$
$163$	−3.59314	−0.281436	−0.140718	+	0.990050i	$0.544941\pi$
$164$	8.62478	0.673482
$165$	0	0
$166$	8.26836	0.641749
$167$	14.2881	1.10565	0.552825	−	0.833298i	$-0.313550\pi$
$167$	14.2881	1.10565	0.552825	+	0.833298i	$0.313550\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−0.118529	−0.00909074
$171$	0	0
$172$	−0.846220	−0.0645237
$173$	14.6611	1.11466	0.557330	−	0.830291i	$-0.311826\pi$
$173$	14.6611	1.11466	0.557330	+	0.830291i	$0.311826\pi$
$174$	0	0
$175$	2.84483	0.215049
$176$	−5.67239	−0.427573
$177$	0	0
$178$	1.32749	0.0994995
$179$	−14.1820	−1.06001	−0.530007	−	0.847993i	$-0.677811\pi$
$179$	−14.1820	−1.06001	−0.530007	+	0.847993i	$0.677811\pi$
$180$	0	0
$181$	−11.5551	−0.858884	−0.429442	−	0.903094i	$-0.641290\pi$
$181$	−11.5551	−0.858884	−0.429442	+	0.903094i	$0.641290\pi$
$182$	1.54206	0.114305
$183$	0	0
$184$	−10.0627	−0.741835
$185$	1.28626	0.0945678
$186$	0	0
$187$	0.533856	0.0390394
$188$	−9.11319	−0.664648
$189$	0	0
$190$	−1.83588	−0.133189
$191$	17.2035	1.24480	0.622402	−	0.782698i	$-0.286157\pi$
$191$	17.2035	1.24480	0.622402	+	0.782698i	$0.286157\pi$
$192$	0	0
$193$	−9.06052	−0.652191	−0.326095	−	0.945337i	$-0.605733\pi$
$193$	−9.06052	−0.652191	−0.326095	+	0.945337i	$0.605733\pi$
$194$	3.37383	0.242227
$195$	0	0
$196$	−1.86496	−0.133211
$197$	17.3213	1.23409	0.617046	−	0.786927i	$-0.288329\pi$
$197$	17.3213	1.23409	0.617046	+	0.786927i	$0.288329\pi$
$198$	0	0
$199$	−15.5480	−1.10217	−0.551085	−	0.834449i	$-0.685786\pi$
$199$	−15.5480	−1.10217	−0.551085	+	0.834449i	$0.685786\pi$
$200$	2.00895	0.142054
$201$	0	0
$202$	8.39174	0.590441
$203$	14.9088	1.04639
$204$	0	0
$205$	5.05503	0.353059
$206$	6.78806	0.472947
$207$	0	0
$208$	−2.32339	−0.161098
$209$	8.26885	0.571968
$210$	0	0
$211$	26.2095	1.80434	0.902169	−	0.431383i	$-0.141974\pi$
$211$	26.2095	1.80434	0.902169	+	0.431383i	$0.141974\pi$
$212$	15.1413	1.03991
$213$	0	0
$214$	9.90125	0.676836
$215$	−0.495974	−0.0338252
$216$	0	0
$217$	25.8467	1.75459
$218$	2.44850	0.165834
$219$	0	0
$220$	−4.16551	−0.280839
$221$	0.218665	0.0147090
$222$	0	0
$223$	−9.05067	−0.606078	−0.303039	−	0.952978i	$-0.598001\pi$
$223$	−9.05067	−0.606078	−0.303039	+	0.952978i	$0.598001\pi$
$224$	−15.0131	−1.00310
$225$	0	0
$226$	2.68940	0.178896
$227$	14.4259	0.957480	0.478740	−	0.877957i	$-0.341094\pi$
$227$	14.4259	0.957480	0.478740	+	0.877957i	$0.341094\pi$
$228$	0	0
$229$	9.57850	0.632965	0.316483	−	0.948598i	$-0.397498\pi$
$229$	9.57850	0.632965	0.316483	+	0.948598i	$0.397498\pi$
$230$	−2.71513	−0.179030
$231$	0	0
$232$	10.5283	0.691214
$233$	4.97182	0.325715	0.162857	−	0.986650i	$-0.447929\pi$
$233$	4.97182	0.325715	0.162857	+	0.986650i	$0.447929\pi$
$234$	0	0
$235$	−5.34129	−0.348428
$236$	10.5324	0.685598
$237$	0	0
$238$	0.337194	0.0218570
$239$	13.5785	0.878320	0.439160	−	0.898409i	$-0.355276\pi$
$239$	13.5785	0.878320	0.439160	+	0.898409i	$0.355276\pi$
$240$	0	0
$241$	−12.7859	−0.823614	−0.411807	−	0.911271i	$-0.635102\pi$
$241$	−12.7859	−0.823614	−0.411807	+	0.911271i	$0.635102\pi$
$242$	2.73164	0.175597
$243$	0	0
$244$	0.737884	0.0472382
$245$	−1.09306	−0.0698332
$246$	0	0
$247$	3.38689	0.215502
$248$	18.2523	1.15902
$249$	0	0
$250$	0.542055	0.0342826
$251$	11.6649	0.736285	0.368142	−	0.929769i	$-0.379994\pi$
$251$	11.6649	0.736285	0.368142	+	0.929769i	$0.379994\pi$
$252$	0	0
$253$	12.2290	0.768831
$254$	−6.12041	−0.384028
$255$	0	0
$256$	−2.67364	−0.167102
$257$	0.500827	0.0312407	0.0156204	−	0.999878i	$-0.495028\pi$
$257$	0.500827	0.0312407	0.0156204	+	0.999878i	$0.495028\pi$
$258$	0	0
$259$	−3.65919	−0.227371
$260$	−1.70618	−0.105813
$261$	0	0
$262$	2.12026	0.130990
$263$	−6.65573	−0.410410	−0.205205	−	0.978719i	$-0.565786\pi$
$263$	−6.65573	−0.410410	−0.205205	+	0.978719i	$0.565786\pi$
$264$	0	0
$265$	8.87440	0.545150
$266$	5.22276	0.320228
$267$	0	0
$268$	2.83242	0.173017
$269$	−3.80285	−0.231864	−0.115932	−	0.993257i	$-0.536985\pi$
$269$	−3.80285	−0.231864	−0.115932	+	0.993257i	$0.536985\pi$
$270$	0	0
$271$	−32.0413	−1.94637	−0.973185	−	0.230024i	$-0.926120\pi$
$271$	−32.0413	−1.94637	−0.973185	+	0.230024i	$0.926120\pi$
$272$	−0.508045	−0.0308047
$273$	0	0
$274$	−4.22325	−0.255136
$275$	−2.44143	−0.147224
$276$	0	0
$277$	−4.89230	−0.293950	−0.146975	−	0.989140i	$-0.546954\pi$
$277$	−4.89230	−0.293950	−0.146975	+	0.989140i	$0.546954\pi$
$278$	4.58543	0.275016
$279$	0	0
$280$	−5.71513	−0.341544
$281$	26.1956	1.56270	0.781348	−	0.624095i	$-0.214532\pi$
$281$	26.1956	1.56270	0.781348	+	0.624095i	$0.214532\pi$
$282$	0	0
$283$	−25.4027	−1.51003	−0.755017	−	0.655705i	$-0.772372\pi$
$283$	−25.4027	−1.51003	−0.755017	+	0.655705i	$0.772372\pi$
$284$	12.5384	0.744020
$285$	0	0
$286$	−1.32339	−0.0782537
$287$	−14.3807	−0.848867
$288$	0	0
$289$	−16.9522	−0.997187
$290$	2.84073	0.166813
$291$	0	0
$292$	−1.25447	−0.0734125
$293$	−9.18910	−0.536833	−0.268416	−	0.963303i	$-0.586500\pi$
$293$	−9.18910	−0.536833	−0.268416	+	0.963303i	$0.586500\pi$
$294$	0	0
$295$	6.17307	0.359410
$296$	−2.58404	−0.150194
$297$	0	0
$298$	9.57350	0.554578
$299$	5.00895	0.289675
$300$	0	0
$301$	1.41096	0.0813266
$302$	5.53498	0.318502
$303$	0	0
$304$	−7.86906	−0.451321
$305$	0.432478	0.0247636
$306$	0	0
$307$	16.3920	0.935539	0.467769	−	0.883851i	$-0.345058\pi$
$307$	16.3920	0.935539	0.467769	+	0.883851i	$0.345058\pi$
$308$	11.8502	0.675226
$309$	0	0
$310$	4.92484	0.279712
$311$	6.60055	0.374283	0.187141	−	0.982333i	$-0.440078\pi$
$311$	6.60055	0.374283	0.187141	+	0.982333i	$0.440078\pi$
$312$	0	0
$313$	−0.160922	−0.00909587	−0.00454794	−	0.999990i	$-0.501448\pi$
$313$	−0.160922	−0.00909587	−0.00454794	+	0.999990i	$0.501448\pi$
$314$	−9.20937	−0.519715
$315$	0	0
$316$	−9.44282	−0.531200
$317$	−11.5205	−0.647058	−0.323529	−	0.946218i	$-0.604869\pi$
$317$	−11.5205	−0.647058	−0.323529	+	0.946218i	$0.604869\pi$
$318$	0	0
$319$	−12.7947	−0.716367
$320$	1.78619	0.0998509
$321$	0	0
$322$	7.72408	0.430446
$323$	0.740595	0.0412078
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	1.94768	0.107872
$327$	0	0
$328$	−10.1553	−0.560734
$329$	15.1951	0.837732
$330$	0	0
$331$	−22.7534	−1.25064	−0.625319	−	0.780369i	$-0.715031\pi$
$331$	−22.7534	−1.25064	−0.625319	+	0.780369i	$0.715031\pi$
$332$	26.0255	1.42834
$333$	0	0
$334$	−7.74496	−0.423785
$335$	1.66010	0.0907007
$336$	0	0
$337$	1.71923	0.0936523	0.0468261	−	0.998903i	$-0.485089\pi$
$337$	1.71923	0.0936523	0.0468261	+	0.998903i	$0.485089\pi$
$338$	−0.542055	−0.0294839
$339$	0	0
$340$	−0.373082	−0.0202332
$341$	−22.1816	−1.20120
$342$	0	0
$343$	−16.8042	−0.907344
$344$	0.996389	0.0537217
$345$	0	0
$346$	−7.94711	−0.427239
$347$	−28.2524	−1.51667	−0.758334	−	0.651866i	$-0.773986\pi$
$347$	−28.2524	−1.51667	−0.758334	+	0.651866i	$0.773986\pi$
$348$	0	0
$349$	13.2506	0.709288	0.354644	−	0.935001i	$-0.384602\pi$
$349$	13.2506	0.709288	0.354644	+	0.935001i	$0.384602\pi$
$350$	−1.54206	−0.0824263
$351$	0	0
$352$	12.8842	0.686729
$353$	−1.57211	−0.0836749	−0.0418375	−	0.999124i	$-0.513321\pi$
$353$	−1.57211	−0.0836749	−0.0418375	+	0.999124i	$0.513321\pi$
$354$	0	0
$355$	7.34886	0.390037
$356$	4.17841	0.221455
$357$	0	0
$358$	7.68744	0.406294
$359$	32.0736	1.69278	0.846389	−	0.532565i	$-0.178772\pi$
$359$	32.0736	1.69278	0.846389	+	0.532565i	$0.178772\pi$
$360$	0	0
$361$	−7.52900	−0.396263
$362$	6.26350	0.329203
$363$	0	0
$364$	4.85378	0.254407
$365$	−0.735254	−0.0384850
$366$	0	0
$367$	17.7172	0.924828	0.462414	−	0.886664i	$-0.346983\pi$
$367$	17.7172	0.924828	0.462414	+	0.886664i	$0.346983\pi$
$368$	−11.6377	−0.606660
$369$	0	0
$370$	−0.697224	−0.0362470
$371$	−25.2462	−1.31072
$372$	0	0
$373$	24.2177	1.25395	0.626973	−	0.779041i	$-0.284294\pi$
$373$	24.2177	1.25395	0.626973	+	0.779041i	$0.284294\pi$
$374$	−0.289380	−0.0149635
$375$	0	0
$376$	10.7304	0.553378
$377$	−5.24067	−0.269908
$378$	0	0
$379$	−6.26426	−0.321773	−0.160887	−	0.986973i	$-0.551435\pi$
$379$	−6.26426	−0.321773	−0.160887	+	0.986973i	$0.551435\pi$
$380$	−5.77862	−0.296437
$381$	0	0
$382$	−9.32527	−0.477122
$383$	−11.1570	−0.570098	−0.285049	−	0.958513i	$-0.592010\pi$
$383$	−11.1570	−0.570098	−0.285049	+	0.958513i	$0.592010\pi$
$384$	0	0
$385$	6.94546	0.353973
$386$	4.91130	0.249979
$387$	0	0
$388$	10.6195	0.539124
$389$	24.3360	1.23389	0.616943	−	0.787008i	$-0.288371\pi$
$389$	24.3360	1.23389	0.616943	+	0.787008i	$0.288371\pi$
$390$	0	0
$391$	1.09528	0.0553909
$392$	2.19591	0.110910
$393$	0	0
$394$	−9.38911	−0.473016
$395$	−5.53449	−0.278471
$396$	0	0
$397$	25.3319	1.27137	0.635685	−	0.771949i	$-0.280718\pi$
$397$	25.3319	1.27137	0.635685	+	0.771949i	$0.280718\pi$
$398$	8.42789	0.422452
$399$	0	0
$400$	2.32339	0.116169
$401$	−36.8616	−1.84078	−0.920389	−	0.391003i	$-0.872128\pi$
$401$	−36.8616	−1.84078	−0.920389	+	0.391003i	$0.872128\pi$
$402$	0	0
$403$	−9.08550	−0.452581
$404$	26.4139	1.31414
$405$	0	0
$406$	−8.08140	−0.401073
$407$	3.14032	0.155660
$408$	0	0
$409$	−33.1715	−1.64022	−0.820112	−	0.572203i	$-0.806089\pi$
$409$	−33.1715	−1.64022	−0.820112	+	0.572203i	$0.806089\pi$
$410$	−2.74011	−0.135324
$411$	0	0
$412$	21.3662	1.05263
$413$	−17.5613	−0.864137
$414$	0	0
$415$	15.2537	0.748776
$416$	5.27731	0.258741
$417$	0	0
$418$	−4.48217	−0.219230
$419$	−15.9943	−0.781373	−0.390687	−	0.920524i	$-0.627762\pi$
$419$	−15.9943	−0.781373	−0.390687	+	0.920524i	$0.627762\pi$
$420$	0	0
$421$	−23.7931	−1.15960	−0.579802	−	0.814758i	$-0.696870\pi$
$421$	−23.7931	−1.15960	−0.579802	+	0.814758i	$0.696870\pi$
$422$	−14.2070	−0.691586
$423$	0	0
$424$	−17.8282	−0.865816
$425$	−0.218665	−0.0106068
$426$	0	0
$427$	−1.23033	−0.0595397
$428$	31.1652	1.50643
$429$	0	0
$430$	0.268846	0.0129649
$431$	7.05503	0.339829	0.169914	−	0.985459i	$-0.445651\pi$
$431$	7.05503	0.339829	0.169914	+	0.985459i	$0.445651\pi$
$432$	0	0
$433$	39.1460	1.88124	0.940619	−	0.339465i	$-0.110246\pi$
$433$	39.1460	1.88124	0.940619	+	0.339465i	$0.110246\pi$
$434$	−14.0103	−0.672518
$435$	0	0
$436$	7.70693	0.369095
$437$	16.9647	0.811534
$438$	0	0
$439$	9.55011	0.455802	0.227901	−	0.973684i	$-0.426814\pi$
$439$	9.55011	0.455802	0.227901	+	0.973684i	$0.426814\pi$
$440$	4.90472	0.233823
$441$	0	0
$442$	−0.118529	−0.00563783
$443$	−7.01993	−0.333527	−0.166763	−	0.985997i	$-0.553332\pi$
$443$	−7.01993	−0.333527	−0.166763	+	0.985997i	$0.553332\pi$
$444$	0	0
$445$	2.44899	0.116093
$446$	4.90596	0.232304
$447$	0	0
$448$	−5.08140	−0.240074
$449$	2.09355	0.0988008	0.0494004	−	0.998779i	$-0.484269\pi$
$449$	2.09355	0.0988008	0.0494004	+	0.998779i	$0.484269\pi$
$450$	0	0
$451$	12.3415	0.581139
$452$	8.46516	0.398168
$453$	0	0
$454$	−7.81963	−0.366993
$455$	2.84483	0.133368
$456$	0	0
$457$	5.98883	0.280145	0.140073	−	0.990141i	$-0.455266\pi$
$457$	5.98883	0.280145	0.140073	+	0.990141i	$0.455266\pi$
$458$	−5.19207	−0.242610
$459$	0	0
$460$	−8.54615	−0.398467
$461$	25.8240	1.20274	0.601372	−	0.798969i	$-0.294621\pi$
$461$	25.8240	1.20274	0.601372	+	0.798969i	$0.294621\pi$
$462$	0	0
$463$	32.4963	1.51023	0.755115	−	0.655593i	$-0.227581\pi$
$463$	32.4963	1.51023	0.755115	+	0.655593i	$0.227581\pi$
$464$	12.1761	0.565262
$465$	0	0
$466$	−2.69500	−0.124844
$467$	35.7722	1.65534	0.827670	−	0.561216i	$-0.189666\pi$
$467$	35.7722	1.65534	0.827670	+	0.561216i	$0.189666\pi$
$468$	0	0
$469$	−4.72269	−0.218074
$470$	2.89528	0.133549
$471$	0	0
$472$	−12.4014	−0.570821
$473$	−1.21089	−0.0556766
$474$	0	0
$475$	−3.38689	−0.155401
$476$	1.06135	0.0486471
$477$	0	0
$478$	−7.36029	−0.336652
$479$	−3.79424	−0.173363	−0.0866816	−	0.996236i	$-0.527626\pi$
$479$	−3.79424	−0.173363	−0.0866816	+	0.996236i	$0.527626\pi$
$480$	0	0
$481$	1.28626	0.0586485
$482$	6.93067	0.315684
$483$	0	0
$484$	8.59814	0.390824
$485$	6.22415	0.282624
$486$	0	0
$487$	16.6324	0.753686	0.376843	−	0.926277i	$-0.377010\pi$
$487$	16.6324	0.753686	0.376843	+	0.926277i	$0.377010\pi$
$488$	−0.868828	−0.0393300
$489$	0	0
$490$	0.592500	0.0267664
$491$	15.0518	0.679280	0.339640	−	0.940556i	$-0.389695\pi$
$491$	15.0518	0.679280	0.339640	+	0.940556i	$0.389695\pi$
$492$	0	0
$493$	−1.14595	−0.0516111
$494$	−1.83588	−0.0826001
$495$	0	0
$496$	21.1092	0.947829
$497$	−20.9063	−0.937774
$498$	0	0
$499$	−9.54732	−0.427397	−0.213698	−	0.976900i	$-0.568551\pi$
$499$	−9.54732	−0.427397	−0.213698	+	0.976900i	$0.568551\pi$
$500$	1.70618	0.0763025
$501$	0	0
$502$	−6.32304	−0.282211
$503$	36.7416	1.63823	0.819114	−	0.573630i	$-0.194465\pi$
$503$	36.7416	1.63823	0.819114	+	0.573630i	$0.194465\pi$
$504$	0	0
$505$	15.4813	0.688910
$506$	−6.62880	−0.294686
$507$	0	0
$508$	−19.2646	−0.854730
$509$	−41.4559	−1.83750	−0.918750	−	0.394841i	$-0.870800\pi$
$509$	−41.4559	−1.83750	−0.918750	+	0.394841i	$0.870800\pi$
$510$	0	0
$511$	2.09167	0.0925302
$512$	−21.5964	−0.954435
$513$	0	0
$514$	−0.271476	−0.0119743
$515$	12.5228	0.551822
$516$	0	0
$517$	−13.0404	−0.573516
$518$	1.98349	0.0871493
$519$	0	0
$520$	2.00895	0.0880984
$521$	30.4809	1.33539	0.667697	−	0.744433i	$-0.267280\pi$
$521$	30.4809	1.33539	0.667697	+	0.744433i	$0.267280\pi$
$522$	0	0
$523$	29.3402	1.28296	0.641479	−	0.767141i	$-0.278321\pi$
$523$	29.3402	1.28296	0.641479	+	0.767141i	$0.278321\pi$
$524$	6.67375	0.291544
$525$	0	0
$526$	3.60777	0.157306
$527$	−1.98668	−0.0865413
$528$	0	0
$529$	2.08960	0.0908521
$530$	−4.81041	−0.208951
$531$	0	0
$532$	16.4392	0.712730
$533$	5.05503	0.218958
$534$	0	0
$535$	18.2661	0.789714
$536$	−3.33505	−0.144052
$537$	0	0
$538$	2.06135	0.0888713
$539$	−2.66863	−0.114946
$540$	0	0
$541$	−23.5137	−1.01093	−0.505467	−	0.862846i	$-0.668680\pi$
$541$	−23.5137	−1.01093	−0.505467	+	0.862846i	$0.668680\pi$
$542$	17.3681	0.746026
$543$	0	0
$544$	1.15396	0.0494758
$545$	4.51708	0.193490
$546$	0	0
$547$	−21.4684	−0.917924	−0.458962	−	0.888456i	$-0.651779\pi$
$547$	−21.4684	−0.917924	−0.458962	+	0.888456i	$0.651779\pi$
$548$	−13.2931	−0.567855
$549$	0	0
$550$	1.32339	0.0564295
$551$	−17.7495	−0.756156
$552$	0	0
$553$	15.7447	0.669532
$554$	2.65190	0.112668
$555$	0	0
$556$	14.4331	0.612100
$557$	−25.3282	−1.07319	−0.536596	−	0.843839i	$-0.680290\pi$
$557$	−25.3282	−1.07319	−0.536596	+	0.843839i	$0.680290\pi$
$558$	0	0
$559$	−0.495974	−0.0209775
$560$	−6.60965	−0.279309
$561$	0	0
$562$	−14.1994	−0.598967
$563$	27.6666	1.16601	0.583003	−	0.812470i	$-0.301877\pi$
$563$	27.6666	1.16601	0.583003	+	0.812470i	$0.301877\pi$
$564$	0	0
$565$	4.96148	0.208731
$566$	13.7697	0.578782
$567$	0	0
$568$	−14.7635	−0.619463
$569$	2.22596	0.0933172	0.0466586	−	0.998911i	$-0.485143\pi$
$569$	2.22596	0.0933172	0.0466586	+	0.998911i	$0.485143\pi$
$570$	0	0
$571$	33.7756	1.41347	0.706733	−	0.707480i	$-0.250168\pi$
$571$	33.7756	1.41347	0.706733	+	0.707480i	$0.250168\pi$
$572$	−4.16551	−0.174169
$573$	0	0
$574$	7.79514	0.325363
$575$	−5.00895	−0.208888
$576$	0	0
$577$	30.9379	1.28796	0.643981	−	0.765042i	$-0.277282\pi$
$577$	30.9379	1.28796	0.643981	+	0.765042i	$0.277282\pi$
$578$	9.18902	0.382213
$579$	0	0
$580$	8.94150	0.371276
$581$	−43.3943	−1.80030
$582$	0	0
$583$	21.6662	0.897323
$584$	1.47709	0.0611224
$585$	0	0
$586$	4.98100	0.205763
$587$	−8.35247	−0.344743	−0.172372	−	0.985032i	$-0.555143\pi$
$587$	−8.35247	−0.344743	−0.172372	+	0.985032i	$0.555143\pi$
$588$	0	0
$589$	−30.7715	−1.26792
$590$	−3.34615	−0.137759
$591$	0	0
$592$	−2.98849	−0.122826
$593$	−24.6605	−1.01269	−0.506343	−	0.862332i	$-0.669003\pi$
$593$	−24.6605	−1.01269	−0.506343	+	0.862332i	$0.669003\pi$
$594$	0	0
$595$	0.622066	0.0255022
$596$	30.1336	1.23432
$597$	0	0
$598$	−2.71513	−0.111030
$599$	27.0949	1.10707	0.553535	−	0.832826i	$-0.313279\pi$
$599$	27.0949	1.10707	0.553535	+	0.832826i	$0.313279\pi$
$600$	0	0
$601$	6.21479	0.253507	0.126753	−	0.991934i	$-0.459544\pi$
$601$	6.21479	0.253507	0.126753	+	0.991934i	$0.459544\pi$
$602$	−0.764820	−0.0311717
$603$	0	0
$604$	17.4219	0.708889
$605$	5.03942	0.204882
$606$	0	0
$607$	−37.2825	−1.51325	−0.756624	−	0.653850i	$-0.773153\pi$
$607$	−37.2825	−1.51325	−0.756624	+	0.653850i	$0.773153\pi$
$608$	17.8736	0.724872
$609$	0	0
$610$	−0.234427	−0.00949168
$611$	−5.34129	−0.216086
$612$	0	0
$613$	−35.2788	−1.42490	−0.712448	−	0.701725i	$-0.752414\pi$
$613$	−35.2788	−1.42490	−0.712448	+	0.701725i	$0.752414\pi$
$614$	−8.88535	−0.358583
$615$	0	0
$616$	−13.9531	−0.562186
$617$	−29.6232	−1.19258	−0.596292	−	0.802767i	$-0.703360\pi$
$617$	−29.6232	−1.19258	−0.596292	+	0.802767i	$0.703360\pi$
$618$	0	0
$619$	4.29778	0.172742	0.0863711	−	0.996263i	$-0.472473\pi$
$619$	4.29778	0.172742	0.0863711	+	0.996263i	$0.472473\pi$
$620$	15.5015	0.622554
$621$	0	0
$622$	−3.57786	−0.143459
$623$	−6.96697	−0.279126
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.0872288	0.00348636
$627$	0	0
$628$	−28.9875	−1.15673
$629$	0.281261	0.0112146
$630$	0	0
$631$	−25.2326	−1.00449	−0.502247	−	0.864724i	$-0.667493\pi$
$631$	−25.2326	−1.00449	−0.502247	+	0.864724i	$0.667493\pi$
$632$	11.1185	0.442271
$633$	0	0
$634$	6.24477	0.248011
$635$	−11.2911	−0.448074
$636$	0	0
$637$	−1.09306	−0.0433087
$638$	6.93545	0.274577
$639$	0	0
$640$	−11.5228	−0.455480
$641$	32.2909	1.27541	0.637707	−	0.770279i	$-0.279883\pi$
$641$	32.2909	1.27541	0.637707	+	0.770279i	$0.279883\pi$
$642$	0	0
$643$	−1.98349	−0.0782210	−0.0391105	−	0.999235i	$-0.512452\pi$
$643$	−1.98349	−0.0782210	−0.0391105	+	0.999235i	$0.512452\pi$
$644$	24.3124	0.958041
$645$	0	0
$646$	−0.401443	−0.0157946
$647$	33.0972	1.30119	0.650593	−	0.759426i	$-0.274520\pi$
$647$	33.0972	1.30119	0.650593	+	0.759426i	$0.274520\pi$
$648$	0	0
$649$	15.0711	0.591593
$650$	0.542055	0.0212612
$651$	0	0
$652$	6.13052	0.240090
$653$	−23.1091	−0.904329	−0.452164	−	0.891935i	$-0.649348\pi$
$653$	−23.1091	−0.904329	−0.452164	+	0.891935i	$0.649348\pi$
$654$	0	0
$655$	3.91153	0.152836
$656$	−11.7448	−0.458558
$657$	0	0
$658$	−8.23657	−0.321095
$659$	30.0195	1.16939	0.584697	−	0.811252i	$-0.301213\pi$
$659$	30.0195	1.16939	0.584697	+	0.811252i	$0.301213\pi$
$660$	0	0
$661$	−23.6287	−0.919049	−0.459525	−	0.888165i	$-0.651980\pi$
$661$	−23.6287	−0.919049	−0.459525	+	0.888165i	$0.651980\pi$
$662$	12.3336	0.479358
$663$	0	0
$664$	−30.6440	−1.18922
$665$	9.63512	0.373634
$666$	0	0
$667$	−26.2503	−1.01641
$668$	−24.3781	−0.943216
$669$	0	0
$670$	−0.899863	−0.0347648
$671$	1.05587	0.0407612
$672$	0	0
$673$	−20.7426	−0.799569	−0.399785	−	0.916609i	$-0.630915\pi$
$673$	−20.7426	−0.799569	−0.399785	+	0.916609i	$0.630915\pi$
$674$	−0.931916	−0.0358961
$675$	0	0
$676$	−1.70618	−0.0656222
$677$	20.6423	0.793346	0.396673	−	0.917960i	$-0.370165\pi$
$677$	20.6423	0.793346	0.396673	+	0.917960i	$0.370165\pi$
$678$	0	0
$679$	−17.7067	−0.679519
$680$	0.439288	0.0168459
$681$	0	0
$682$	12.0237	0.460410
$683$	18.9500	0.725103	0.362552	−	0.931964i	$-0.381906\pi$
$683$	18.9500	0.725103	0.362552	+	0.931964i	$0.381906\pi$
$684$	0	0
$685$	−7.79119	−0.297686
$686$	9.10882	0.347777
$687$	0	0
$688$	1.15234	0.0439326
$689$	8.87440	0.338088
$690$	0	0
$691$	49.9358	1.89965	0.949824	−	0.312785i	$-0.101262\pi$
$691$	49.9358	1.89965	0.949824	+	0.312785i	$0.101262\pi$
$692$	−25.0144	−0.950904
$693$	0	0
$694$	15.3144	0.581325
$695$	8.45933	0.320881
$696$	0	0
$697$	1.10536	0.0418685
$698$	−7.18255	−0.271864
$699$	0	0
$700$	−4.85378	−0.183456
$701$	−5.34122	−0.201735	−0.100868	−	0.994900i	$-0.532162\pi$
$701$	−5.34122	−0.201735	−0.100868	+	0.994900i	$0.532162\pi$
$702$	0	0
$703$	4.35642	0.164305
$704$	4.36085	0.164356
$705$	0	0
$706$	0.852170	0.0320718
$707$	−44.0418	−1.65636
$708$	0	0
$709$	−47.7620	−1.79374	−0.896870	−	0.442295i	$-0.854164\pi$
$709$	−47.7620	−1.79374	−0.896870	+	0.442295i	$0.854164\pi$
$710$	−3.98349	−0.149498
$711$	0	0
$712$	−4.91991	−0.184381
$713$	−45.5088	−1.70432
$714$	0	0
$715$	−2.44143	−0.0913043
$716$	24.1970	0.904286
$717$	0	0
$718$	−17.3856	−0.648826
$719$	−21.1515	−0.788818	−0.394409	−	0.918935i	$-0.629051\pi$
$719$	−21.1515	−0.788818	−0.394409	+	0.918935i	$0.629051\pi$
$720$	0	0
$721$	−35.6253	−1.32676
$722$	4.08114	0.151884
$723$	0	0
$724$	19.7150	0.732704
$725$	5.24067	0.194634
$726$	0	0
$727$	−32.6558	−1.21114	−0.605568	−	0.795793i	$-0.707054\pi$
$727$	−32.6558	−1.21114	−0.605568	+	0.795793i	$0.707054\pi$
$728$	−5.71513	−0.211817
$729$	0	0
$730$	0.398548	0.0147509
$731$	−0.108452	−0.00401126
$732$	0	0
$733$	7.21423	0.266463	0.133232	−	0.991085i	$-0.457465\pi$
$733$	7.21423	0.266463	0.133232	+	0.991085i	$0.457465\pi$
$734$	−9.60367	−0.354478
$735$	0	0
$736$	26.4338	0.974362
$737$	4.05301	0.149294
$738$	0	0
$739$	18.9126	0.695713	0.347857	−	0.937548i	$-0.386910\pi$
$739$	18.9126	0.695713	0.347857	+	0.937548i	$0.386910\pi$
$740$	−2.19459	−0.0806747
$741$	0	0
$742$	13.6848	0.502385
$743$	4.06252	0.149039	0.0745197	−	0.997220i	$-0.476258\pi$
$743$	4.06252	0.149039	0.0745197	+	0.997220i	$0.476258\pi$
$744$	0	0
$745$	17.6615	0.647067
$746$	−13.1273	−0.480626
$747$	0	0
$748$	−0.910853	−0.0333041
$749$	−51.9641	−1.89873
$750$	0	0
$751$	12.6448	0.461416	0.230708	−	0.973023i	$-0.425896\pi$
$751$	12.6448	0.461416	0.230708	+	0.973023i	$0.425896\pi$
$752$	12.4099	0.452543
$753$	0	0
$754$	2.84073	0.103453
$755$	10.2111	0.371620
$756$	0	0
$757$	54.5093	1.98117	0.990587	−	0.136882i	$-0.0437080\pi$
$757$	54.5093	1.98117	0.990587	+	0.136882i	$0.0437080\pi$
$758$	3.39557	0.123333
$759$	0	0
$760$	6.80409	0.246810
$761$	−26.3323	−0.954547	−0.477273	−	0.878755i	$-0.658375\pi$
$761$	−26.3323	−0.954547	−0.477273	+	0.878755i	$0.658375\pi$
$762$	0	0
$763$	−12.8503	−0.465213
$764$	−29.3523	−1.06193
$765$	0	0
$766$	6.04773	0.218514
$767$	6.17307	0.222897
$768$	0	0
$769$	−11.0236	−0.397521	−0.198760	−	0.980048i	$-0.563692\pi$
$769$	−11.0236	−0.397521	−0.198760	+	0.980048i	$0.563692\pi$
$770$	−3.76482	−0.135675
$771$	0	0
$772$	15.4588	0.556376
$773$	−39.3321	−1.41468	−0.707339	−	0.706874i	$-0.750105\pi$
$773$	−39.3321	−1.41468	−0.707339	+	0.706874i	$0.750105\pi$
$774$	0	0
$775$	9.08550	0.326361
$776$	−12.5040	−0.448868
$777$	0	0
$778$	−13.1915	−0.472937
$779$	17.1208	0.613417
$780$	0	0
$781$	17.9417	0.642005
$782$	−0.593704	−0.0212308
$783$	0	0
$784$	2.53961	0.0907003
$785$	−16.9897	−0.606389
$786$	0	0
$787$	−27.9452	−0.996138	−0.498069	−	0.867137i	$-0.665957\pi$
$787$	−27.9452	−0.996138	−0.498069	+	0.867137i	$0.665957\pi$
$788$	−29.5532	−1.05279
$789$	0	0
$790$	3.00000	0.106735
$791$	−14.1146	−0.501857
$792$	0	0
$793$	0.432478	0.0153578
$794$	−13.7313	−0.487304
$795$	0	0
$796$	26.5277	0.940249
$797$	9.34900	0.331159	0.165579	−	0.986196i	$-0.447051\pi$
$797$	9.34900	0.331159	0.165579	+	0.986196i	$0.447051\pi$
$798$	0	0
$799$	−1.16796	−0.0413193
$800$	−5.27731	−0.186581
$801$	0	0
$802$	19.9810	0.705554
$803$	−1.79507	−0.0633467
$804$	0	0
$805$	14.2496	0.502233
$806$	4.92484	0.173470
$807$	0	0
$808$	−31.1013	−1.09414
$809$	−46.2859	−1.62733	−0.813663	−	0.581337i	$-0.802530\pi$
$809$	−46.2859	−1.62733	−0.813663	+	0.581337i	$0.802530\pi$
$810$	0	0
$811$	−34.9701	−1.22797	−0.613983	−	0.789319i	$-0.710434\pi$
$811$	−34.9701	−1.22797	−0.613983	+	0.789319i	$0.710434\pi$
$812$	−25.4371	−0.892666
$813$	0	0
$814$	−1.70222	−0.0596629
$815$	3.59314	0.125862
$816$	0	0
$817$	−1.67981	−0.0587691
$818$	17.9808	0.628683
$819$	0	0
$820$	−8.62478	−0.301190
$821$	−44.9996	−1.57050	−0.785248	−	0.619181i	$-0.787465\pi$
$821$	−44.9996	−1.57050	−0.785248	+	0.619181i	$0.787465\pi$
$822$	0	0
$823$	44.1176	1.53784	0.768922	−	0.639343i	$-0.220794\pi$
$823$	44.1176	1.53784	0.768922	+	0.639343i	$0.220794\pi$
$824$	−25.1578	−0.876412
$825$	0	0
$826$	9.51922	0.331216
$827$	50.5802	1.75884	0.879422	−	0.476043i	$-0.157929\pi$
$827$	50.5802	1.75884	0.879422	+	0.476043i	$0.157929\pi$
$828$	0	0
$829$	3.24323	0.112642	0.0563210	−	0.998413i	$-0.482063\pi$
$829$	3.24323	0.112642	0.0563210	+	0.998413i	$0.482063\pi$
$830$	−8.26836	−0.286999
$831$	0	0
$832$	1.78619	0.0619249
$833$	−0.239015	−0.00828137
$834$	0	0
$835$	−14.2881	−0.494461
$836$	−14.1081	−0.487939
$837$	0	0
$838$	8.66980	0.299493
$839$	−15.2875	−0.527783	−0.263892	−	0.964552i	$-0.585006\pi$
$839$	−15.2875	−0.527783	−0.263892	+	0.964552i	$0.585006\pi$
$840$	0	0
$841$	−1.53539	−0.0529446
$842$	12.8972	0.444465
$843$	0	0
$844$	−44.7181	−1.53926
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−14.3363	−0.492601
$848$	−20.6187	−0.708048
$849$	0	0
$850$	0.118529	0.00406550
$851$	6.44282	0.220857
$852$	0	0
$853$	12.1156	0.414829	0.207414	−	0.978253i	$-0.433495\pi$
$853$	12.1156	0.414829	0.207414	+	0.978253i	$0.433495\pi$
$854$	0.666905	0.0228210
$855$	0	0
$856$	−36.6958	−1.25424
$857$	−31.2525	−1.06756	−0.533782	−	0.845622i	$-0.679230\pi$
$857$	−31.2525	−1.06756	−0.533782	+	0.845622i	$0.679230\pi$
$858$	0	0
$859$	−53.8976	−1.83896	−0.919481	−	0.393134i	$-0.871391\pi$
$859$	−53.8976	−1.83896	−0.919481	+	0.393134i	$0.871391\pi$
$860$	0.846220	0.0288559
$861$	0	0
$862$	−3.82422	−0.130253
$863$	42.7960	1.45679	0.728396	−	0.685156i	$-0.240266\pi$
$863$	42.7960	1.45679	0.728396	+	0.685156i	$0.240266\pi$
$864$	0	0
$865$	−14.6611	−0.498491
$866$	−21.2193	−0.721061
$867$	0	0
$868$	−44.0990	−1.49682
$869$	−13.5121	−0.458366
$870$	0	0
$871$	1.66010	0.0562502
$872$	−9.07459	−0.307304
$873$	0	0
$874$	−9.19583	−0.311054
$875$	−2.84483	−0.0961728
$876$	0	0
$877$	33.1314	1.11877	0.559384	−	0.828909i	$-0.311038\pi$
$877$	33.1314	1.11877	0.559384	+	0.828909i	$0.311038\pi$
$878$	−5.17668	−0.174705
$879$	0	0
$880$	5.67239	0.191216
$881$	11.7742	0.396682	0.198341	−	0.980133i	$-0.436445\pi$
$881$	11.7742	0.396682	0.198341	+	0.980133i	$0.436445\pi$
$882$	0	0
$883$	−6.89008	−0.231870	−0.115935	−	0.993257i	$-0.536986\pi$
$883$	−6.89008	−0.231870	−0.115935	+	0.993257i	$0.536986\pi$
$884$	−0.373082	−0.0125481
$885$	0	0
$886$	3.80519	0.127838
$887$	−32.5217	−1.09197	−0.545986	−	0.837794i	$-0.683845\pi$
$887$	−32.5217	−1.09197	−0.545986	+	0.837794i	$0.683845\pi$
$888$	0	0
$889$	32.1213	1.07731
$890$	−1.32749	−0.0444975
$891$	0	0
$892$	15.4420	0.517038
$893$	−18.0904	−0.605371
$894$	0	0
$895$	14.1820	0.474053
$896$	32.7805	1.09512
$897$	0	0
$898$	−1.13482	−0.0378694
$899$	47.6141	1.58802
$900$	0	0
$901$	1.94052	0.0646482
$902$	−6.68978	−0.222745
$903$	0	0
$904$	−9.96738	−0.331510
$905$	11.5551	0.384105
$906$	0	0
$907$	27.3151	0.906984	0.453492	−	0.891260i	$-0.350178\pi$
$907$	27.3151	0.906984	0.453492	+	0.891260i	$0.350178\pi$
$908$	−24.6131	−0.816815
$909$	0	0
$910$	−1.54206	−0.0511186
$911$	31.3328	1.03810	0.519051	−	0.854743i	$-0.326286\pi$
$911$	31.3328	1.03810	0.519051	+	0.854743i	$0.326286\pi$
$912$	0	0
$913$	37.2409	1.23249
$914$	−3.24627	−0.107377
$915$	0	0
$916$	−16.3426	−0.539975
$917$	−11.1276	−0.367467
$918$	0	0
$919$	3.95365	0.130419	0.0652095	−	0.997872i	$-0.479228\pi$
$919$	3.95365	0.130419	0.0652095	+	0.997872i	$0.479228\pi$
$920$	10.0627	0.331759
$921$	0	0
$922$	−13.9980	−0.461001
$923$	7.34886	0.241891
$924$	0	0
$925$	−1.28626	−0.0422920
$926$	−17.6148	−0.578857
$927$	0	0
$928$	−27.6566	−0.907873
$929$	18.5702	0.609268	0.304634	−	0.952469i	$-0.401466\pi$
$929$	18.5702	0.609268	0.304634	+	0.952469i	$0.401466\pi$
$930$	0	0
$931$	−3.70208	−0.121331
$932$	−8.48281	−0.277864
$933$	0	0
$934$	−19.3905	−0.634476
$935$	−0.533856	−0.0174590
$936$	0	0
$937$	35.8014	1.16958	0.584790	−	0.811185i	$-0.301177\pi$
$937$	35.8014	1.16958	0.584790	+	0.811185i	$0.301177\pi$
$938$	2.55996	0.0835856
$939$	0	0
$940$	9.11319	0.297239
$941$	−41.5474	−1.35441	−0.677203	−	0.735796i	$-0.736808\pi$
$941$	−41.5474	−1.35441	−0.677203	+	0.735796i	$0.736808\pi$
$942$	0	0
$943$	25.3204	0.824546
$944$	−14.3425	−0.466807
$945$	0	0
$946$	0.656368	0.0213403
$947$	−53.9671	−1.75370	−0.876848	−	0.480768i	$-0.840358\pi$
$947$	−53.9671	−1.75370	−0.876848	+	0.480768i	$0.840358\pi$
$948$	0	0
$949$	−0.735254	−0.0238674
$950$	1.83588	0.0595638
$951$	0	0
$952$	−1.24970	−0.0405030
$953$	24.6617	0.798872	0.399436	−	0.916761i	$-0.369206\pi$
$953$	24.6617	0.798872	0.399436	+	0.916761i	$0.369206\pi$
$954$	0	0
$955$	−17.2035	−0.556693
$956$	−23.1673	−0.749284
$957$	0	0
$958$	2.05669	0.0664485
$959$	22.1646	0.715733
$960$	0	0
$961$	51.5463	1.66278
$962$	−0.697224	−0.0224794
$963$	0	0
$964$	21.8150	0.702615
$965$	9.06052	0.291668
$966$	0	0
$967$	9.60987	0.309033	0.154516	−	0.987990i	$-0.450618\pi$
$967$	9.60987	0.309033	0.154516	+	0.987990i	$0.450618\pi$
$968$	−10.1239	−0.325396
$969$	0	0
$970$	−3.37383	−0.108327
$971$	58.8598	1.88890	0.944450	−	0.328656i	$-0.106596\pi$
$971$	58.8598	1.88890	0.944450	+	0.328656i	$0.106596\pi$
$972$	0	0
$973$	−24.0654	−0.771500
$974$	−9.01568	−0.288881
$975$	0	0
$976$	−1.00482	−0.0321634
$977$	−31.7374	−1.01537	−0.507685	−	0.861543i	$-0.669498\pi$
$977$	−31.7374	−1.01537	−0.507685	+	0.861543i	$0.669498\pi$
$978$	0	0
$979$	5.97905	0.191091
$980$	1.86496	0.0595739
$981$	0	0
$982$	−8.15892	−0.260362
$983$	30.8824	0.984997	0.492498	−	0.870313i	$-0.336084\pi$
$983$	30.8824	0.984997	0.492498	+	0.870313i	$0.336084\pi$
$984$	0	0
$985$	−17.3213	−0.551903
$986$	0.621170	0.0197821
$987$	0	0
$988$	−5.77862	−0.183843
$989$	−2.48431	−0.0789965
$990$	0	0
$991$	−5.76121	−0.183011	−0.0915054	−	0.995805i	$-0.529168\pi$
$991$	−5.76121	−0.183011	−0.0915054	+	0.995805i	$0.529168\pi$
$992$	−47.9470	−1.52232
$993$	0	0
$994$	11.3323	0.359440
$995$	15.5480	0.492906
$996$	0	0
$997$	−25.8954	−0.820116	−0.410058	−	0.912059i	$-0.634492\pi$
$997$	−25.8954	−0.820116	−0.410058	+	0.912059i	$0.634492\pi$
$998$	5.17517	0.163817
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.p.1.2	✓	4
3.2	odd	2		1755.2.a.r.1.3	yes	4
5.4	even	2		8775.2.a.bq.1.3		4
15.14	odd	2		8775.2.a.bi.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.p.1.2	✓	4	1.1	even	1	trivial
1755.2.a.r.1.3	yes	4	3.2	odd	2
8775.2.a.bi.1.2		4	15.14	odd	2
8775.2.a.bq.1.3		4	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)

\(f(q)\)	\(=\)	\(q-0.542055 q^{2} -1.70618 q^{4} -1.00000 q^{5} +2.84483 q^{7} +2.00895 q^{8} +O(q^{10})\)	\(q-0.542055 q^{2} -1.70618 q^{4} -1.00000 q^{5} +2.84483 q^{7} +2.00895 q^{8} +0.542055 q^{10} -2.44143 q^{11} -1.00000 q^{13} -1.54206 q^{14} +2.32339 q^{16} -0.218665 q^{17} -3.38689 q^{19} +1.70618 q^{20} +1.32339 q^{22} -5.00895 q^{23} +1.00000 q^{25} +0.542055 q^{26} -4.85378 q^{28} +5.24067 q^{29} +9.08550 q^{31} -5.27731 q^{32} +0.118529 q^{34} -2.84483 q^{35} -1.28626 q^{37} +1.83588 q^{38} -2.00895 q^{40} -5.05503 q^{41} +0.495974 q^{43} +4.16551 q^{44} +2.71513 q^{46} +5.34129 q^{47} +1.09306 q^{49} -0.542055 q^{50} +1.70618 q^{52} -8.87440 q^{53} +2.44143 q^{55} +5.71513 q^{56} -2.84073 q^{58} -6.17307 q^{59} -0.432478 q^{61} -4.92484 q^{62} -1.78619 q^{64} +1.00000 q^{65} -1.66010 q^{67} +0.373082 q^{68} +1.54206 q^{70} -7.34886 q^{71} +0.735254 q^{73} +0.697224 q^{74} +5.77862 q^{76} -6.94546 q^{77} +5.53449 q^{79} -2.32339 q^{80} +2.74011 q^{82} -15.2537 q^{83} +0.218665 q^{85} -0.268846 q^{86} -4.90472 q^{88} -2.44899 q^{89} -2.84483 q^{91} +8.54615 q^{92} -2.89528 q^{94} +3.38689 q^{95} -6.22415 q^{97} -0.592500 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 3 q^{4} - 4 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q - q^2 + 3 * q^4 - 4 * q^5 + 3 * q^7 - 9 * q^8	\( 4 q - q^{2} + 3 q^{4} - 4 q^{5} + 3 q^{7} - 9 q^{8} + q^{10} - 4 q^{11} - 4 q^{13} - 5 q^{14} + 13 q^{16} + 4 q^{17} - 4 q^{19} - 3 q^{20} + 9 q^{22} - 3 q^{23} + 4 q^{25} + q^{26} + 6 q^{28} - 14 q^{29} - 7 q^{31} - 24 q^{32} - 29 q^{34} - 3 q^{35} + 9 q^{37} + 16 q^{38} + 9 q^{40} - 4 q^{41} - 33 q^{44} - 16 q^{46} - 9 q^{47} - 15 q^{49} - q^{50} - 3 q^{52} - 21 q^{53} + 4 q^{55} - 4 q^{56} + q^{58} + q^{59} - 13 q^{61} - 5 q^{62} + 9 q^{64} + 4 q^{65} + 4 q^{67} + 30 q^{68} + 5 q^{70} - 23 q^{71} + 7 q^{73} + 10 q^{74} - 17 q^{76} - 24 q^{77} - 3 q^{79} - 13 q^{80} - 6 q^{82} - 13 q^{83} - 4 q^{85} - q^{86} + 26 q^{88} - 28 q^{89} - 3 q^{91} + 37 q^{92} - 3 q^{94} + 4 q^{95} + 17 q^{97} - q^{98}+O(q^{100}) \) 4 * q - q^2 + 3 * q^4 - 4 * q^5 + 3 * q^7 - 9 * q^8 + q^10 - 4 * q^11 - 4 * q^13 - 5 * q^14 + 13 * q^16 + 4 * q^17 - 4 * q^19 - 3 * q^20 + 9 * q^22 - 3 * q^23 + 4 * q^25 + q^26 + 6 * q^28 - 14 * q^29 - 7 * q^31 - 24 * q^32 - 29 * q^34 - 3 * q^35 + 9 * q^37 + 16 * q^38 + 9 * q^40 - 4 * q^41 - 33 * q^44 - 16 * q^46 - 9 * q^47 - 15 * q^49 - q^50 - 3 * q^52 - 21 * q^53 + 4 * q^55 - 4 * q^56 + q^58 + q^59 - 13 * q^61 - 5 * q^62 + 9 * q^64 + 4 * q^65 + 4 * q^67 + 30 * q^68 + 5 * q^70 - 23 * q^71 + 7 * q^73 + 10 * q^74 - 17 * q^76 - 24 * q^77 - 3 * q^79 - 13 * q^80 - 6 * q^82 - 13 * q^83 - 4 * q^85 - q^86 + 26 * q^88 - 28 * q^89 - 3 * q^91 + 37 * q^92 - 3 * q^94 + 4 * q^95 + 17 * q^97 - q^98

Introduction

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