LMFDB - Embedded newform 1755.2.a.u.1.7

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			1.7
Root			$-2.65299$ 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)

$f(q)$	$=$	$q+2.65299 q^{2} +5.03838 q^{4} -1.00000 q^{5} +3.85102 q^{7} +8.06081 q^{8} +O(q^{10})$	\(q+2.65299 q^{2} +5.03838 q^{4} -1.00000 q^{5} +3.85102 q^{7} +8.06081 q^{8} -2.65299 q^{10} +1.32504 q^{11} +1.00000 q^{13} +10.2167 q^{14} +11.3085 q^{16} -7.59518 q^{17} -4.27014 q^{19} -5.03838 q^{20} +3.51532 q^{22} +8.05078 q^{23} +1.00000 q^{25} +2.65299 q^{26} +19.4029 q^{28} -6.70141 q^{29} -2.87472 q^{31} +13.8798 q^{32} -20.1500 q^{34} -3.85102 q^{35} +9.43592 q^{37} -11.3287 q^{38} -8.06081 q^{40} -8.96461 q^{41} +6.11422 q^{43} +6.67605 q^{44} +21.3587 q^{46} +1.39542 q^{47} +7.83033 q^{49} +2.65299 q^{50} +5.03838 q^{52} -12.1466 q^{53} -1.32504 q^{55} +31.0423 q^{56} -17.7788 q^{58} +11.5080 q^{59} -9.09356 q^{61} -7.62662 q^{62} +14.2061 q^{64} -1.00000 q^{65} +2.78119 q^{67} -38.2674 q^{68} -10.2167 q^{70} +0.799061 q^{71} -5.29557 q^{73} +25.0335 q^{74} -21.5146 q^{76} +5.10275 q^{77} -2.00693 q^{79} -11.3085 q^{80} -23.7831 q^{82} -1.80662 q^{83} +7.59518 q^{85} +16.2210 q^{86} +10.6809 q^{88} +3.93754 q^{89} +3.85102 q^{91} +40.5629 q^{92} +3.70203 q^{94} +4.27014 q^{95} +4.43863 q^{97} +20.7738 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.65299	1.87595	0.937975	−	0.346702i	$-0.112699\pi$
$2$	2.65299	1.87595	0.937975	+	0.346702i	$0.112699\pi$
$3$	0	0
$3$	0	0
$4$	5.03838	2.51919
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.85102	1.45555	0.727774	−	0.685817i	$-0.240555\pi$
$7$	3.85102	1.45555	0.727774	+	0.685817i	$0.240555\pi$
$8$	8.06081	2.84993
$9$	0	0
$10$	−2.65299	−0.838951
$11$	1.32504	0.399514	0.199757	−	0.979845i	$-0.435985\pi$
$11$	1.32504	0.399514	0.199757	+	0.979845i	$0.435985\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	10.2167	2.73054
$15$	0	0
$16$	11.3085	2.82713
$17$	−7.59518	−1.84210	−0.921051	−	0.389443i	$-0.872668\pi$
$17$	−7.59518	−1.84210	−0.921051	+	0.389443i	$0.872668\pi$
$18$	0	0
$19$	−4.27014	−0.979637	−0.489819	−	0.871824i	$-0.662937\pi$
$19$	−4.27014	−0.979637	−0.489819	+	0.871824i	$0.662937\pi$
$20$	−5.03838	−1.12662
$21$	0	0
$22$	3.51532	0.749469
$23$	8.05078	1.67870	0.839352	−	0.543589i	$-0.182935\pi$
$23$	8.05078	1.67870	0.839352	+	0.543589i	$0.182935\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.65299	0.520295
$27$	0	0
$28$	19.4029	3.66680
$29$	−6.70141	−1.24442	−0.622210	−	0.782850i	$-0.713765\pi$
$29$	−6.70141	−1.24442	−0.622210	+	0.782850i	$0.713765\pi$
$30$	0	0
$31$	−2.87472	−0.516315	−0.258158	−	0.966103i	$-0.583115\pi$
$31$	−2.87472	−0.516315	−0.258158	+	0.966103i	$0.583115\pi$
$32$	13.8798	2.45363
$33$	0	0
$34$	−20.1500	−3.45569
$35$	−3.85102	−0.650941
$36$	0	0
$37$	9.43592	1.55126	0.775628	−	0.631190i	$-0.217433\pi$
$37$	9.43592	1.55126	0.775628	+	0.631190i	$0.217433\pi$
$38$	−11.3287	−1.83775
$39$	0	0
$40$	−8.06081	−1.27453
$41$	−8.96461	−1.40004	−0.700019	−	0.714125i	$-0.746825\pi$
$41$	−8.96461	−1.40004	−0.700019	+	0.714125i	$0.746825\pi$
$42$	0	0
$43$	6.11422	0.932410	0.466205	−	0.884677i	$-0.345621\pi$
$43$	6.11422	0.932410	0.466205	+	0.884677i	$0.345621\pi$
$44$	6.67605	1.00645
$45$	0	0
$46$	21.3587	3.14916
$47$	1.39542	0.203542	0.101771	−	0.994808i	$-0.467549\pi$
$47$	1.39542	0.203542	0.101771	+	0.994808i	$0.467549\pi$
$48$	0	0
$49$	7.83033	1.11862
$50$	2.65299	0.375190
$51$	0	0
$52$	5.03838	0.698698
$53$	−12.1466	−1.66846	−0.834231	−	0.551415i	$-0.814088\pi$
$53$	−12.1466	−1.66846	−0.834231	+	0.551415i	$0.814088\pi$
$54$	0	0
$55$	−1.32504	−0.178668
$56$	31.0423	4.14820
$57$	0	0
$58$	−17.7788	−2.33447
$59$	11.5080	1.49822	0.749109	−	0.662447i	$-0.230482\pi$
$59$	11.5080	1.49822	0.749109	+	0.662447i	$0.230482\pi$
$60$	0	0
$61$	−9.09356	−1.16431	−0.582156	−	0.813077i	$-0.697791\pi$
$61$	−9.09356	−1.16431	−0.582156	+	0.813077i	$0.697791\pi$
$62$	−7.62662	−0.968582
$63$	0	0
$64$	14.2061	1.77576
$65$	−1.00000	−0.124035
$66$	0	0
$67$	2.78119	0.339776	0.169888	−	0.985463i	$-0.445659\pi$
$67$	2.78119	0.339776	0.169888	+	0.985463i	$0.445659\pi$
$68$	−38.2674	−4.64060
$69$	0	0
$70$	−10.2167	−1.22113
$71$	0.799061	0.0948311	0.0474155	−	0.998875i	$-0.484902\pi$
$71$	0.799061	0.0948311	0.0474155	+	0.998875i	$0.484902\pi$
$72$	0	0
$73$	−5.29557	−0.619800	−0.309900	−	0.950769i	$-0.600296\pi$
$73$	−5.29557	−0.619800	−0.309900	+	0.950769i	$0.600296\pi$
$74$	25.0335	2.91008
$75$	0	0
$76$	−21.5146	−2.46789
$77$	5.10275	0.581512
$78$	0	0
$79$	−2.00693	−0.225798	−0.112899	−	0.993606i	$-0.536014\pi$
$79$	−2.00693	−0.225798	−0.112899	+	0.993606i	$0.536014\pi$
$80$	−11.3085	−1.26433
$81$	0	0
$82$	−23.7831	−2.62640
$83$	−1.80662	−0.198303	−0.0991513	−	0.995072i	$-0.531613\pi$
$83$	−1.80662	−0.198303	−0.0991513	+	0.995072i	$0.531613\pi$
$84$	0	0
$85$	7.59518	0.823813
$86$	16.2210	1.74916
$87$	0	0
$88$	10.6809	1.13859
$89$	3.93754	0.417378	0.208689	−	0.977982i	$-0.433080\pi$
$89$	3.93754	0.417378	0.208689	+	0.977982i	$0.433080\pi$
$90$	0	0
$91$	3.85102	0.403696
$92$	40.5629	4.22897
$93$	0	0
$94$	3.70203	0.381836
$95$	4.27014	0.438107
$96$	0	0
$97$	4.43863	0.450675	0.225337	−	0.974281i	$-0.427652\pi$
$97$	4.43863	0.450675	0.225337	+	0.974281i	$0.427652\pi$
$98$	20.7738	2.09847
$99$	0	0
$100$	5.03838	0.503838
$101$	4.39604	0.437423	0.218711	−	0.975790i	$-0.429815\pi$
$101$	4.39604	0.437423	0.218711	+	0.975790i	$0.429815\pi$
$102$	0	0
$103$	−7.47218	−0.736256	−0.368128	−	0.929775i	$-0.620001\pi$
$103$	−7.47218	−0.736256	−0.368128	+	0.929775i	$0.620001\pi$
$104$	8.06081	0.790427
$105$	0	0
$106$	−32.2248	−3.12995
$107$	−1.14145	−0.110348	−0.0551741	−	0.998477i	$-0.517571\pi$
$107$	−1.14145	−0.110348	−0.0551741	+	0.998477i	$0.517571\pi$
$108$	0	0
$109$	1.10458	0.105799	0.0528997	−	0.998600i	$-0.483154\pi$
$109$	1.10458	0.105799	0.0528997	+	0.998600i	$0.483154\pi$
$110$	−3.51532	−0.335173
$111$	0	0
$112$	43.5493	4.11502
$113$	−14.3725	−1.35205	−0.676025	−	0.736879i	$-0.736299\pi$
$113$	−14.3725	−1.35205	−0.676025	+	0.736879i	$0.736299\pi$
$114$	0	0
$115$	−8.05078	−0.750739
$116$	−33.7642	−3.13493
$117$	0	0
$118$	30.5307	2.81058
$119$	−29.2492	−2.68127
$120$	0	0
$121$	−9.24427	−0.840388
$122$	−24.1252	−2.18419
$123$	0	0
$124$	−14.4839	−1.30070
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	22.2927	1.97816	0.989078	−	0.147390i	$-0.0470872\pi$
$127$	22.2927	1.97816	0.989078	+	0.147390i	$0.0470872\pi$
$128$	9.92897	0.877605
$129$	0	0
$130$	−2.65299	−0.232683
$131$	14.8469	1.29718	0.648590	−	0.761138i	$-0.275359\pi$
$131$	14.8469	1.29718	0.648590	+	0.761138i	$0.275359\pi$
$132$	0	0
$133$	−16.4444	−1.42591
$134$	7.37848	0.637404
$135$	0	0
$136$	−61.2233	−5.24985
$137$	10.0396	0.857745	0.428872	−	0.903365i	$-0.358911\pi$
$137$	10.0396	0.857745	0.428872	+	0.903365i	$0.358911\pi$
$138$	0	0
$139$	11.4219	0.968790	0.484395	−	0.874850i	$-0.339040\pi$
$139$	11.4219	0.968790	0.484395	+	0.874850i	$0.339040\pi$
$140$	−19.4029	−1.63984
$141$	0	0
$142$	2.11990	0.177898
$143$	1.32504	0.110805
$144$	0	0
$145$	6.70141	0.556521
$146$	−14.0491	−1.16271
$147$	0	0
$148$	47.5418	3.90791
$149$	−17.2528	−1.41341	−0.706703	−	0.707510i	$-0.749818\pi$
$149$	−17.2528	−1.41341	−0.706703	+	0.707510i	$0.749818\pi$
$150$	0	0
$151$	−13.6881	−1.11392	−0.556961	−	0.830538i	$-0.688033\pi$
$151$	−13.6881	−1.11392	−0.556961	+	0.830538i	$0.688033\pi$
$152$	−34.4208	−2.79189
$153$	0	0
$154$	13.5376	1.09089
$155$	2.87472	0.230903
$156$	0	0
$157$	1.22987	0.0981540	0.0490770	−	0.998795i	$-0.484372\pi$
$157$	1.22987	0.0981540	0.0490770	+	0.998795i	$0.484372\pi$
$158$	−5.32439	−0.423585
$159$	0	0
$160$	−13.8798	−1.09730
$161$	31.0037	2.44343
$162$	0	0
$163$	−6.43863	−0.504313	−0.252156	−	0.967687i	$-0.581140\pi$
$163$	−6.43863	−0.504313	−0.252156	+	0.967687i	$0.581140\pi$
$164$	−45.1671	−3.52696
$165$	0	0
$166$	−4.79296	−0.372006
$167$	4.57739	0.354209	0.177105	−	0.984192i	$-0.443327\pi$
$167$	4.57739	0.354209	0.177105	+	0.984192i	$0.443327\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	20.1500	1.54543
$171$	0	0
$172$	30.8058	2.34892
$173$	−17.0645	−1.29739	−0.648693	−	0.761050i	$-0.724684\pi$
$173$	−17.0645	−1.29739	−0.648693	+	0.761050i	$0.724684\pi$
$174$	0	0
$175$	3.85102	0.291109
$176$	14.9842	1.12948
$177$	0	0
$178$	10.4463	0.782981
$179$	−15.4595	−1.15550	−0.577749	−	0.816214i	$-0.696069\pi$
$179$	−15.4595	−1.15550	−0.577749	+	0.816214i	$0.696069\pi$
$180$	0	0
$181$	0.758003	0.0563419	0.0281710	−	0.999603i	$-0.491032\pi$
$181$	0.758003	0.0563419	0.0281710	+	0.999603i	$0.491032\pi$
$182$	10.2167	0.757314
$183$	0	0
$184$	64.8958	4.78418
$185$	−9.43592	−0.693743
$186$	0	0
$187$	−10.0639	−0.735946
$188$	7.03064	0.512762
$189$	0	0
$190$	11.3287	0.821867
$191$	−23.8336	−1.72454	−0.862269	−	0.506450i	$-0.830958\pi$
$191$	−23.8336	−1.72454	−0.862269	+	0.506450i	$0.830958\pi$
$192$	0	0
$193$	−18.7512	−1.34974	−0.674871	−	0.737936i	$-0.735801\pi$
$193$	−18.7512	−1.34974	−0.674871	+	0.737936i	$0.735801\pi$
$194$	11.7757	0.845444
$195$	0	0
$196$	39.4522	2.81801
$197$	1.55671	0.110911	0.0554554	−	0.998461i	$-0.482339\pi$
$197$	1.55671	0.110911	0.0554554	+	0.998461i	$0.482339\pi$
$198$	0	0
$199$	9.40931	0.667009	0.333504	−	0.942749i	$-0.391769\pi$
$199$	9.40931	0.667009	0.333504	+	0.942749i	$0.391769\pi$
$200$	8.06081	0.569985
$201$	0	0
$202$	11.6627	0.820583
$203$	−25.8072	−1.81131
$204$	0	0
$205$	8.96461	0.626116
$206$	−19.8237	−1.38118
$207$	0	0
$208$	11.3085	0.784105
$209$	−5.65810	−0.391379
$210$	0	0
$211$	18.9184	1.30240	0.651200	−	0.758906i	$-0.274266\pi$
$211$	18.9184	1.30240	0.651200	+	0.758906i	$0.274266\pi$
$212$	−61.1991	−4.20317
$213$	0	0
$214$	−3.02826	−0.207008
$215$	−6.11422	−0.416986
$216$	0	0
$217$	−11.0706	−0.751522
$218$	2.93044	0.198474
$219$	0	0
$220$	−6.67605	−0.450099
$221$	−7.59518	−0.510907
$222$	0	0
$223$	1.31944	0.0883566	0.0441783	−	0.999024i	$-0.485933\pi$
$223$	1.31944	0.0883566	0.0441783	+	0.999024i	$0.485933\pi$
$224$	53.4514	3.57138
$225$	0	0
$226$	−38.1301	−2.53638
$227$	19.3036	1.28122	0.640612	−	0.767864i	$-0.278681\pi$
$227$	19.3036	1.28122	0.640612	+	0.767864i	$0.278681\pi$
$228$	0	0
$229$	−11.8565	−0.783502	−0.391751	−	0.920071i	$-0.628131\pi$
$229$	−11.8565	−0.783502	−0.391751	+	0.920071i	$0.628131\pi$
$230$	−21.3587	−1.40835
$231$	0	0
$232$	−54.0188	−3.54650
$233$	−11.3232	−0.741807	−0.370904	−	0.928671i	$-0.620952\pi$
$233$	−11.3232	−0.741807	−0.370904	+	0.928671i	$0.620952\pi$
$234$	0	0
$235$	−1.39542	−0.0910269
$236$	57.9818	3.77430
$237$	0	0
$238$	−77.5979	−5.02992
$239$	−5.95266	−0.385046	−0.192523	−	0.981292i	$-0.561667\pi$
$239$	−5.95266	−0.385046	−0.192523	+	0.981292i	$0.561667\pi$
$240$	0	0
$241$	−9.23267	−0.594728	−0.297364	−	0.954764i	$-0.596108\pi$
$241$	−9.23267	−0.594728	−0.297364	+	0.954764i	$0.596108\pi$
$242$	−24.5250	−1.57653
$243$	0	0
$244$	−45.8168	−2.93312
$245$	−7.83033	−0.500261
$246$	0	0
$247$	−4.27014	−0.271702
$248$	−23.1726	−1.47146
$249$	0	0
$250$	−2.65299	−0.167790
$251$	−11.5170	−0.726949	−0.363474	−	0.931604i	$-0.618410\pi$
$251$	−11.5170	−0.726949	−0.363474	+	0.931604i	$0.618410\pi$
$252$	0	0
$253$	10.6676	0.670666
$254$	59.1424	3.71092
$255$	0	0
$256$	−2.07063	−0.129415
$257$	−5.23633	−0.326633	−0.163317	−	0.986574i	$-0.552219\pi$
$257$	−5.23633	−0.326633	−0.163317	+	0.986574i	$0.552219\pi$
$258$	0	0
$259$	36.3379	2.25793
$260$	−5.03838	−0.312467
$261$	0	0
$262$	39.3887	2.43344
$263$	−2.88416	−0.177845	−0.0889224	−	0.996039i	$-0.528342\pi$
$263$	−2.88416	−0.177845	−0.0889224	+	0.996039i	$0.528342\pi$
$264$	0	0
$265$	12.1466	0.746159
$266$	−43.6268	−2.67493
$267$	0	0
$268$	14.0127	0.855961
$269$	−3.15367	−0.192283	−0.0961413	−	0.995368i	$-0.530650\pi$
$269$	−3.15367	−0.192283	−0.0961413	+	0.995368i	$0.530650\pi$
$270$	0	0
$271$	1.22558	0.0744486	0.0372243	−	0.999307i	$-0.488148\pi$
$271$	1.22558	0.0744486	0.0372243	+	0.999307i	$0.488148\pi$
$272$	−85.8902	−5.20786
$273$	0	0
$274$	26.6351	1.60909
$275$	1.32504	0.0799028
$276$	0	0
$277$	−12.8469	−0.771895	−0.385948	−	0.922521i	$-0.626125\pi$
$277$	−12.8469	−0.771895	−0.385948	+	0.922521i	$0.626125\pi$
$278$	30.3021	1.81740
$279$	0	0
$280$	−31.0423	−1.85513
$281$	−18.3551	−1.09497	−0.547486	−	0.836815i	$-0.684415\pi$
$281$	−18.3551	−1.09497	−0.547486	+	0.836815i	$0.684415\pi$
$282$	0	0
$283$	−9.89254	−0.588050	−0.294025	−	0.955798i	$-0.594995\pi$
$283$	−9.89254	−0.588050	−0.294025	+	0.955798i	$0.594995\pi$
$284$	4.02597	0.238897
$285$	0	0
$286$	3.51532	0.207865
$287$	−34.5229	−2.03782
$288$	0	0
$289$	40.6867	2.39334
$290$	17.7788	1.04401
$291$	0	0
$292$	−26.6811	−1.56139
$293$	24.9647	1.45845	0.729227	−	0.684272i	$-0.239880\pi$
$293$	24.9647	1.45845	0.729227	+	0.684272i	$0.239880\pi$
$294$	0	0
$295$	−11.5080	−0.670023
$296$	76.0612	4.42097
$297$	0	0
$298$	−45.7716	−2.65148
$299$	8.05078	0.465589
$300$	0	0
$301$	23.5460	1.35717
$302$	−36.3145	−2.08966
$303$	0	0
$304$	−48.2890	−2.76956
$305$	9.09356	0.520696
$306$	0	0
$307$	24.8558	1.41860	0.709298	−	0.704908i	$-0.249012\pi$
$307$	24.8558	1.41860	0.709298	+	0.704908i	$0.249012\pi$
$308$	25.7096	1.46494
$309$	0	0
$310$	7.62662	0.433163
$311$	16.7327	0.948823	0.474411	−	0.880303i	$-0.342661\pi$
$311$	16.7327	0.948823	0.474411	+	0.880303i	$0.342661\pi$
$312$	0	0
$313$	19.8037	1.11937	0.559687	−	0.828704i	$-0.310921\pi$
$313$	19.8037	1.11937	0.559687	+	0.828704i	$0.310921\pi$
$314$	3.26283	0.184132
$315$	0	0
$316$	−10.1117	−0.568828
$317$	4.85226	0.272530	0.136265	−	0.990672i	$-0.456490\pi$
$317$	4.85226	0.272530	0.136265	+	0.990672i	$0.456490\pi$
$318$	0	0
$319$	−8.87962	−0.497163
$320$	−14.2061	−0.794143
$321$	0	0
$322$	82.2526	4.58376
$323$	32.4325	1.80459
$324$	0	0
$325$	1.00000	0.0554700
$326$	−17.0817	−0.946066
$327$	0	0
$328$	−72.2620	−3.99000
$329$	5.37377	0.296266
$330$	0	0
$331$	33.4540	1.83880	0.919399	−	0.393327i	$-0.128676\pi$
$331$	33.4540	1.83880	0.919399	+	0.393327i	$0.128676\pi$
$332$	−9.10245	−0.499562
$333$	0	0
$334$	12.1438	0.664479
$335$	−2.78119	−0.151953
$336$	0	0
$337$	−6.15846	−0.335473	−0.167736	−	0.985832i	$-0.553646\pi$
$337$	−6.15846	−0.335473	−0.167736	+	0.985832i	$0.553646\pi$
$338$	2.65299	0.144304
$339$	0	0
$340$	38.2674	2.07534
$341$	−3.80912	−0.206275
$342$	0	0
$343$	3.19761	0.172655
$344$	49.2856	2.65730
$345$	0	0
$346$	−45.2719	−2.43383
$347$	20.8825	1.12103	0.560516	−	0.828144i	$-0.310603\pi$
$347$	20.8825	1.12103	0.560516	+	0.828144i	$0.310603\pi$
$348$	0	0
$349$	−3.86478	−0.206877	−0.103439	−	0.994636i	$-0.532984\pi$
$349$	−3.86478	−0.206877	−0.103439	+	0.994636i	$0.532984\pi$
$350$	10.2167	0.546107
$351$	0	0
$352$	18.3913	0.980260
$353$	9.41438	0.501077	0.250538	−	0.968107i	$-0.419392\pi$
$353$	9.41438	0.501077	0.250538	+	0.968107i	$0.419392\pi$
$354$	0	0
$355$	−0.799061	−0.0424097
$356$	19.8388	1.05146
$357$	0	0
$358$	−41.0140	−2.16766
$359$	12.4732	0.658308	0.329154	−	0.944276i	$-0.393236\pi$
$359$	12.4732	0.658308	0.329154	+	0.944276i	$0.393236\pi$
$360$	0	0
$361$	−0.765908	−0.0403110
$362$	2.01098	0.105695
$363$	0	0
$364$	19.4029	1.01699
$365$	5.29557	0.277183
$366$	0	0
$367$	19.4001	1.01268	0.506338	−	0.862335i	$-0.330999\pi$
$367$	19.4001	1.01268	0.506338	+	0.862335i	$0.330999\pi$
$368$	91.0424	4.74591
$369$	0	0
$370$	−25.0335	−1.30143
$371$	−46.7767	−2.42853
$372$	0	0
$373$	−32.3781	−1.67647	−0.838237	−	0.545305i	$-0.816414\pi$
$373$	−32.3781	−1.67647	−0.838237	+	0.545305i	$0.816414\pi$
$374$	−26.6995	−1.38060
$375$	0	0
$376$	11.2482	0.580081
$377$	−6.70141	−0.345140
$378$	0	0
$379$	33.9363	1.74319	0.871595	−	0.490227i	$-0.163086\pi$
$379$	33.9363	1.74319	0.871595	+	0.490227i	$0.163086\pi$
$380$	21.5146	1.10368
$381$	0	0
$382$	−63.2304	−3.23515
$383$	5.74309	0.293458	0.146729	−	0.989177i	$-0.453125\pi$
$383$	5.74309	0.293458	0.146729	+	0.989177i	$0.453125\pi$
$384$	0	0
$385$	−5.10275	−0.260060
$386$	−49.7468	−2.53205
$387$	0	0
$388$	22.3635	1.13534
$389$	13.4229	0.680567	0.340283	−	0.940323i	$-0.389477\pi$
$389$	13.4229	0.680567	0.340283	+	0.940323i	$0.389477\pi$
$390$	0	0
$391$	−61.1471	−3.09234
$392$	63.1188	3.18798
$393$	0	0
$394$	4.12993	0.208063
$395$	2.00693	0.100980
$396$	0	0
$397$	13.5886	0.681993	0.340997	−	0.940065i	$-0.389236\pi$
$397$	13.5886	0.681993	0.340997	+	0.940065i	$0.389236\pi$
$398$	24.9629	1.25127
$399$	0	0
$400$	11.3085	0.565426
$401$	29.0300	1.44969	0.724843	−	0.688914i	$-0.241912\pi$
$401$	29.0300	1.44969	0.724843	+	0.688914i	$0.241912\pi$
$402$	0	0
$403$	−2.87472	−0.143200
$404$	22.1489	1.10195
$405$	0	0
$406$	−68.4664	−3.39793
$407$	12.5030	0.619749
$408$	0	0
$409$	8.53930	0.422241	0.211121	−	0.977460i	$-0.432289\pi$
$409$	8.53930	0.422241	0.211121	+	0.977460i	$0.432289\pi$
$410$	23.7831	1.17456
$411$	0	0
$412$	−37.6477	−1.85477
$413$	44.3176	2.18073
$414$	0	0
$415$	1.80662	0.0886836
$416$	13.8798	0.680515
$417$	0	0
$418$	−15.0109	−0.734208
$419$	−8.86601	−0.433133	−0.216567	−	0.976268i	$-0.569486\pi$
$419$	−8.86601	−0.433133	−0.216567	+	0.976268i	$0.569486\pi$
$420$	0	0
$421$	37.6228	1.83362	0.916812	−	0.399319i	$-0.130753\pi$
$421$	37.6228	1.83362	0.916812	+	0.399319i	$0.130753\pi$
$422$	50.1905	2.44324
$423$	0	0
$424$	−97.9113	−4.75499
$425$	−7.59518	−0.368420
$426$	0	0
$427$	−35.0195	−1.69471
$428$	−5.75106	−0.277988
$429$	0	0
$430$	−16.2210	−0.782246
$431$	11.4148	0.549833	0.274917	−	0.961468i	$-0.411350\pi$
$431$	11.4148	0.549833	0.274917	+	0.961468i	$0.411350\pi$
$432$	0	0
$433$	14.9679	0.719314	0.359657	−	0.933085i	$-0.382894\pi$
$433$	14.9679	0.719314	0.359657	+	0.933085i	$0.382894\pi$
$434$	−29.3703	−1.40982
$435$	0	0
$436$	5.56528	0.266529
$437$	−34.3779	−1.64452
$438$	0	0
$439$	−35.4067	−1.68987	−0.844934	−	0.534870i	$-0.820361\pi$
$439$	−35.4067	−1.68987	−0.844934	+	0.534870i	$0.820361\pi$
$440$	−10.6809	−0.509191
$441$	0	0
$442$	−20.1500	−0.958436
$443$	−1.39930	−0.0664829	−0.0332415	−	0.999447i	$-0.510583\pi$
$443$	−1.39930	−0.0664829	−0.0332415	+	0.999447i	$0.510583\pi$
$444$	0	0
$445$	−3.93754	−0.186657
$446$	3.50048	0.165753
$447$	0	0
$448$	54.7078	2.58470
$449$	9.58346	0.452272	0.226136	−	0.974096i	$-0.427391\pi$
$449$	9.58346	0.452272	0.226136	+	0.974096i	$0.427391\pi$
$450$	0	0
$451$	−11.8785	−0.559335
$452$	−72.4140	−3.40607
$453$	0	0
$454$	51.2123	2.40351
$455$	−3.85102	−0.180538
$456$	0	0
$457$	−24.2752	−1.13555	−0.567773	−	0.823185i	$-0.692195\pi$
$457$	−24.2752	−1.13555	−0.567773	+	0.823185i	$0.692195\pi$
$458$	−31.4553	−1.46981
$459$	0	0
$460$	−40.5629	−1.89125
$461$	23.8974	1.11301	0.556506	−	0.830844i	$-0.312142\pi$
$461$	23.8974	1.11301	0.556506	+	0.830844i	$0.312142\pi$
$462$	0	0
$463$	8.52472	0.396178	0.198089	−	0.980184i	$-0.436527\pi$
$463$	8.52472	0.396178	0.198089	+	0.980184i	$0.436527\pi$
$464$	−75.7830	−3.51814
$465$	0	0
$466$	−30.0404	−1.39159
$467$	12.8035	0.592476	0.296238	−	0.955114i	$-0.404268\pi$
$467$	12.8035	0.592476	0.296238	+	0.955114i	$0.404268\pi$
$468$	0	0
$469$	10.7104	0.494561
$470$	−3.70203	−0.170762
$471$	0	0
$472$	92.7640	4.26981
$473$	8.10158	0.372511
$474$	0	0
$475$	−4.27014	−0.195927
$476$	−147.368	−6.75462
$477$	0	0
$478$	−15.7924	−0.722327
$479$	−34.7314	−1.58692	−0.793459	−	0.608624i	$-0.791722\pi$
$479$	−34.7314	−1.58692	−0.793459	+	0.608624i	$0.791722\pi$
$480$	0	0
$481$	9.43592	0.430241
$482$	−24.4942	−1.11568
$483$	0	0
$484$	−46.5762	−2.11710
$485$	−4.43863	−0.201548
$486$	0	0
$487$	−26.5335	−1.20235	−0.601174	−	0.799118i	$-0.705300\pi$
$487$	−26.5335	−1.20235	−0.601174	+	0.799118i	$0.705300\pi$
$488$	−73.3015	−3.31820
$489$	0	0
$490$	−20.7738	−0.938466
$491$	−3.48405	−0.157233	−0.0786164	−	0.996905i	$-0.525050\pi$
$491$	−3.48405	−0.157233	−0.0786164	+	0.996905i	$0.525050\pi$
$492$	0	0
$493$	50.8984	2.29235
$494$	−11.3287	−0.509700
$495$	0	0
$496$	−32.5089	−1.45969
$497$	3.07720	0.138031
$498$	0	0
$499$	−16.4371	−0.735826	−0.367913	−	0.929860i	$-0.619928\pi$
$499$	−16.4371	−0.735826	−0.367913	+	0.929860i	$0.619928\pi$
$500$	−5.03838	−0.225323
$501$	0	0
$502$	−30.5546	−1.36372
$503$	4.86206	0.216788	0.108394	−	0.994108i	$-0.465429\pi$
$503$	4.86206	0.216788	0.108394	+	0.994108i	$0.465429\pi$
$504$	0	0
$505$	−4.39604	−0.195621
$506$	28.3011	1.25814
$507$	0	0
$508$	112.319	4.98335
$509$	4.97385	0.220462	0.110231	−	0.993906i	$-0.464841\pi$
$509$	4.97385	0.220462	0.110231	+	0.993906i	$0.464841\pi$
$510$	0	0
$511$	−20.3933	−0.902148
$512$	−25.3513	−1.12038
$513$	0	0
$514$	−13.8920	−0.612748
$515$	7.47218	0.329264
$516$	0	0
$517$	1.84898	0.0813181
$518$	96.4043	4.23576
$519$	0	0
$520$	−8.06081	−0.353490
$521$	36.7399	1.60960	0.804802	−	0.593543i	$-0.202271\pi$
$521$	36.7399	1.60960	0.804802	+	0.593543i	$0.202271\pi$
$522$	0	0
$523$	30.7121	1.34295	0.671474	−	0.741028i	$-0.265662\pi$
$523$	30.7121	1.34295	0.671474	+	0.741028i	$0.265662\pi$
$524$	74.8043	3.26784
$525$	0	0
$526$	−7.65165	−0.333628
$527$	21.8340	0.951105
$528$	0	0
$529$	41.8150	1.81804
$530$	32.2248	1.39976
$531$	0	0
$532$	−82.8530	−3.59213
$533$	−8.96461	−0.388300
$534$	0	0
$535$	1.14145	0.0493492
$536$	22.4186	0.968338
$537$	0	0
$538$	−8.36667	−0.360713
$539$	10.3755	0.446904
$540$	0	0
$541$	29.4655	1.26682	0.633411	−	0.773815i	$-0.281654\pi$
$541$	29.4655	1.26682	0.633411	+	0.773815i	$0.281654\pi$
$542$	3.25146	0.139662
$543$	0	0
$544$	−105.420	−4.51984
$545$	−1.10458	−0.0473149
$546$	0	0
$547$	19.9611	0.853476	0.426738	−	0.904375i	$-0.359663\pi$
$547$	19.9611	0.853476	0.426738	+	0.904375i	$0.359663\pi$
$548$	50.5836	2.16082
$549$	0	0
$550$	3.51532	0.149894
$551$	28.6159	1.21908
$552$	0	0
$553$	−7.72874	−0.328659
$554$	−34.0827	−1.44804
$555$	0	0
$556$	57.5477	2.44057
$557$	11.3108	0.479253	0.239626	−	0.970865i	$-0.422975\pi$
$557$	11.3108	0.479253	0.239626	+	0.970865i	$0.422975\pi$
$558$	0	0
$559$	6.11422	0.258604
$560$	−43.5493	−1.84029
$561$	0	0
$562$	−48.6959	−2.05411
$563$	−19.6530	−0.828274	−0.414137	−	0.910214i	$-0.635917\pi$
$563$	−19.6530	−0.828274	−0.414137	+	0.910214i	$0.635917\pi$
$564$	0	0
$565$	14.3725	0.604655
$566$	−26.2448	−1.10315
$567$	0	0
$568$	6.44108	0.270262
$569$	29.0448	1.21762	0.608811	−	0.793316i	$-0.291647\pi$
$569$	29.0448	1.21762	0.608811	+	0.793316i	$0.291647\pi$
$570$	0	0
$571$	−13.1644	−0.550912	−0.275456	−	0.961314i	$-0.588829\pi$
$571$	−13.1644	−0.550912	−0.275456	+	0.961314i	$0.588829\pi$
$572$	6.67605	0.279140
$573$	0	0
$574$	−91.5890	−3.82285
$575$	8.05078	0.335741
$576$	0	0
$577$	−16.0246	−0.667111	−0.333555	−	0.942730i	$-0.608248\pi$
$577$	−16.0246	−0.667111	−0.333555	+	0.942730i	$0.608248\pi$
$578$	107.942	4.48978
$579$	0	0
$580$	33.7642	1.40198
$581$	−6.95733	−0.288639
$582$	0	0
$583$	−16.0947	−0.666574
$584$	−42.6866	−1.76638
$585$	0	0
$586$	66.2313	2.73599
$587$	22.1311	0.913446	0.456723	−	0.889609i	$-0.349023\pi$
$587$	22.1311	0.913446	0.456723	+	0.889609i	$0.349023\pi$
$588$	0	0
$589$	12.2755	0.505802
$590$	−30.5307	−1.25693
$591$	0	0
$592$	106.706	4.38560
$593$	−5.63777	−0.231515	−0.115758	−	0.993277i	$-0.536930\pi$
$593$	−5.63777	−0.231515	−0.115758	+	0.993277i	$0.536930\pi$
$594$	0	0
$595$	29.2492	1.19910
$596$	−86.9263	−3.56064
$597$	0	0
$598$	21.3587	0.873421
$599$	42.6287	1.74176	0.870880	−	0.491496i	$-0.163550\pi$
$599$	42.6287	1.74176	0.870880	+	0.491496i	$0.163550\pi$
$600$	0	0
$601$	−15.6121	−0.636833	−0.318416	−	0.947951i	$-0.603151\pi$
$601$	−15.6121	−0.636833	−0.318416	+	0.947951i	$0.603151\pi$
$602$	62.4673	2.54598
$603$	0	0
$604$	−68.9659	−2.80618
$605$	9.24427	0.375833
$606$	0	0
$607$	40.6411	1.64957	0.824785	−	0.565447i	$-0.191296\pi$
$607$	40.6411	1.64957	0.824785	+	0.565447i	$0.191296\pi$
$608$	−59.2688	−2.40367
$609$	0	0
$610$	24.1252	0.976800
$611$	1.39542	0.0564525
$612$	0	0
$613$	−29.3709	−1.18628	−0.593141	−	0.805099i	$-0.702112\pi$
$613$	−29.3709	−1.18628	−0.593141	+	0.805099i	$0.702112\pi$
$614$	65.9424	2.66122
$615$	0	0
$616$	41.1323	1.65727
$617$	−36.4426	−1.46712	−0.733562	−	0.679622i	$-0.762144\pi$
$617$	−36.4426	−1.46712	−0.733562	+	0.679622i	$0.762144\pi$
$618$	0	0
$619$	30.0014	1.20586	0.602928	−	0.797795i	$-0.294001\pi$
$619$	30.0014	1.20586	0.602928	+	0.797795i	$0.294001\pi$
$620$	14.4839	0.581689
$621$	0	0
$622$	44.3917	1.77994
$623$	15.1635	0.607514
$624$	0	0
$625$	1.00000	0.0400000
$626$	52.5392	2.09989
$627$	0	0
$628$	6.19654	0.247269
$629$	−71.6675	−2.85757
$630$	0	0
$631$	22.6021	0.899777	0.449888	−	0.893085i	$-0.351464\pi$
$631$	22.6021	0.899777	0.449888	+	0.893085i	$0.351464\pi$
$632$	−16.1775	−0.643507
$633$	0	0
$634$	12.8730	0.511253
$635$	−22.2927	−0.884659
$636$	0	0
$637$	7.83033	0.310249
$638$	−23.5576	−0.932654
$639$	0	0
$640$	−9.92897	−0.392477
$641$	12.9242	0.510474	0.255237	−	0.966879i	$-0.417846\pi$
$641$	12.9242	0.510474	0.255237	+	0.966879i	$0.417846\pi$
$642$	0	0
$643$	25.3973	1.00157	0.500786	−	0.865571i	$-0.333044\pi$
$643$	25.3973	1.00157	0.500786	+	0.865571i	$0.333044\pi$
$644$	156.208	6.15547
$645$	0	0
$646$	86.0432	3.38532
$647$	8.93472	0.351260	0.175630	−	0.984456i	$-0.443804\pi$
$647$	8.93472	0.351260	0.175630	+	0.984456i	$0.443804\pi$
$648$	0	0
$649$	15.2486	0.598559
$650$	2.65299	0.104059
$651$	0	0
$652$	−32.4403	−1.27046
$653$	−9.25365	−0.362123	−0.181062	−	0.983472i	$-0.557953\pi$
$653$	−9.25365	−0.362123	−0.181062	+	0.983472i	$0.557953\pi$
$654$	0	0
$655$	−14.8469	−0.580116
$656$	−101.376	−3.95809
$657$	0	0
$658$	14.2566	0.555780
$659$	−2.09713	−0.0816927	−0.0408463	−	0.999165i	$-0.513005\pi$
$659$	−2.09713	−0.0816927	−0.0408463	+	0.999165i	$0.513005\pi$
$660$	0	0
$661$	6.50965	0.253196	0.126598	−	0.991954i	$-0.459594\pi$
$661$	6.50965	0.253196	0.126598	+	0.991954i	$0.459594\pi$
$662$	88.7532	3.44949
$663$	0	0
$664$	−14.5628	−0.565148
$665$	16.4444	0.637686
$666$	0	0
$667$	−53.9515	−2.08901
$668$	23.0627	0.892321
$669$	0	0
$670$	−7.37848	−0.285056
$671$	−12.0493	−0.465159
$672$	0	0
$673$	0.460190	0.0177390	0.00886952	−	0.999961i	$-0.497177\pi$
$673$	0.460190	0.0177390	0.00886952	+	0.999961i	$0.497177\pi$
$674$	−16.3384	−0.629330
$675$	0	0
$676$	5.03838	0.193784
$677$	−42.9656	−1.65130	−0.825651	−	0.564182i	$-0.809192\pi$
$677$	−42.9656	−1.65130	−0.825651	+	0.564182i	$0.809192\pi$
$678$	0	0
$679$	17.0932	0.655979
$680$	61.2233	2.34781
$681$	0	0
$682$	−10.1056	−0.386962
$683$	−43.0121	−1.64581	−0.822906	−	0.568177i	$-0.807649\pi$
$683$	−43.0121	−1.64581	−0.822906	+	0.568177i	$0.807649\pi$
$684$	0	0
$685$	−10.0396	−0.383595
$686$	8.48324	0.323892
$687$	0	0
$688$	69.1428	2.63604
$689$	−12.1466	−0.462748
$690$	0	0
$691$	−24.3151	−0.924991	−0.462495	−	0.886622i	$-0.653046\pi$
$691$	−24.3151	−0.924991	−0.462495	+	0.886622i	$0.653046\pi$
$692$	−85.9772	−3.26836
$693$	0	0
$694$	55.4012	2.10300
$695$	−11.4219	−0.433256
$696$	0	0
$697$	68.0878	2.57901
$698$	−10.2532	−0.388091
$699$	0	0
$700$	19.4029	0.733360
$701$	31.0188	1.17156	0.585782	−	0.810469i	$-0.300788\pi$
$701$	31.0188	1.17156	0.585782	+	0.810469i	$0.300788\pi$
$702$	0	0
$703$	−40.2927	−1.51967
$704$	18.8236	0.709441
$705$	0	0
$706$	24.9763	0.939995
$707$	16.9292	0.636689
$708$	0	0
$709$	−25.1090	−0.942987	−0.471493	−	0.881870i	$-0.656285\pi$
$709$	−25.1090	−0.942987	−0.471493	+	0.881870i	$0.656285\pi$
$710$	−2.11990	−0.0795586
$711$	0	0
$712$	31.7397	1.18950
$713$	−23.1438	−0.866740
$714$	0	0
$715$	−1.32504	−0.0495536
$716$	−77.8909	−2.91092
$717$	0	0
$718$	33.0912	1.23495
$719$	39.6764	1.47968	0.739840	−	0.672783i	$-0.234901\pi$
$719$	39.6764	1.47968	0.739840	+	0.672783i	$0.234901\pi$
$720$	0	0
$721$	−28.7755	−1.07165
$722$	−2.03195	−0.0756214
$723$	0	0
$724$	3.81911	0.141936
$725$	−6.70141	−0.248884
$726$	0	0
$727$	−50.8470	−1.88581	−0.942906	−	0.333058i	$-0.891919\pi$
$727$	−50.8470	−1.88581	−0.942906	+	0.333058i	$0.891919\pi$
$728$	31.0423	1.15050
$729$	0	0
$730$	14.0491	0.519981
$731$	−46.4386	−1.71759
$732$	0	0
$733$	18.8468	0.696123	0.348062	−	0.937472i	$-0.386840\pi$
$733$	18.8468	0.696123	0.348062	+	0.937472i	$0.386840\pi$
$734$	51.4684	1.89973
$735$	0	0
$736$	111.743	4.11892
$737$	3.68518	0.135745
$738$	0	0
$739$	−5.02215	−0.184743	−0.0923714	−	0.995725i	$-0.529445\pi$
$739$	−5.02215	−0.184743	−0.0923714	+	0.995725i	$0.529445\pi$
$740$	−47.5418	−1.74767
$741$	0	0
$742$	−124.098	−4.55579
$743$	−4.15923	−0.152588	−0.0762938	−	0.997085i	$-0.524309\pi$
$743$	−4.15923	−0.152588	−0.0762938	+	0.997085i	$0.524309\pi$
$744$	0	0
$745$	17.2528	0.632094
$746$	−85.8990	−3.14498
$747$	0	0
$748$	−50.7058	−1.85399
$749$	−4.39575	−0.160617
$750$	0	0
$751$	−39.5384	−1.44278	−0.721389	−	0.692530i	$-0.756496\pi$
$751$	−39.5384	−1.44278	−0.721389	+	0.692530i	$0.756496\pi$
$752$	15.7801	0.575441
$753$	0	0
$754$	−17.7788	−0.647466
$755$	13.6881	0.498161
$756$	0	0
$757$	44.5966	1.62089	0.810446	−	0.585813i	$-0.199225\pi$
$757$	44.5966	1.62089	0.810446	+	0.585813i	$0.199225\pi$
$758$	90.0328	3.27014
$759$	0	0
$760$	34.4208	1.24857
$761$	22.0325	0.798677	0.399338	−	0.916804i	$-0.369240\pi$
$761$	22.0325	0.798677	0.399338	+	0.916804i	$0.369240\pi$
$762$	0	0
$763$	4.25375	0.153996
$764$	−120.083	−4.34444
$765$	0	0
$766$	15.2364	0.550513
$767$	11.5080	0.415531
$768$	0	0
$769$	−23.1103	−0.833380	−0.416690	−	0.909049i	$-0.636810\pi$
$769$	−23.1103	−0.833380	−0.416690	+	0.909049i	$0.636810\pi$
$770$	−13.5376	−0.487860
$771$	0	0
$772$	−94.4757	−3.40025
$773$	−6.02117	−0.216566	−0.108283	−	0.994120i	$-0.534535\pi$
$773$	−6.02117	−0.216566	−0.108283	+	0.994120i	$0.534535\pi$
$774$	0	0
$775$	−2.87472	−0.103263
$776$	35.7790	1.28439
$777$	0	0
$778$	35.6108	1.27671
$779$	38.2801	1.37153
$780$	0	0
$781$	1.05879	0.0378864
$782$	−162.223	−5.80108
$783$	0	0
$784$	88.5494	3.16248
$785$	−1.22987	−0.0438958
$786$	0	0
$787$	1.17518	0.0418907	0.0209454	−	0.999781i	$-0.493332\pi$
$787$	1.17518	0.0418907	0.0209454	+	0.999781i	$0.493332\pi$
$788$	7.84328	0.279405
$789$	0	0
$790$	5.32439	0.189433
$791$	−55.3487	−1.96797
$792$	0	0
$793$	−9.09356	−0.322922
$794$	36.0505	1.27939
$795$	0	0
$796$	47.4077	1.68032
$797$	−5.83585	−0.206717	−0.103358	−	0.994644i	$-0.532959\pi$
$797$	−5.83585	−0.206717	−0.103358	+	0.994644i	$0.532959\pi$
$798$	0	0
$799$	−10.5984	−0.374946
$800$	13.8798	0.490726
$801$	0	0
$802$	77.0163	2.71954
$803$	−7.01684	−0.247619
$804$	0	0
$805$	−31.0037	−1.09274
$806$	−7.62662	−0.268636
$807$	0	0
$808$	35.4357	1.24662
$809$	−4.80880	−0.169068	−0.0845342	−	0.996421i	$-0.526940\pi$
$809$	−4.80880	−0.169068	−0.0845342	+	0.996421i	$0.526940\pi$
$810$	0	0
$811$	−28.8106	−1.01168	−0.505838	−	0.862629i	$-0.668817\pi$
$811$	−28.8106	−1.01168	−0.505838	+	0.862629i	$0.668817\pi$
$812$	−130.027	−4.56304
$813$	0	0
$814$	33.1703	1.16262
$815$	6.43863	0.225535
$816$	0	0
$817$	−26.1086	−0.913424
$818$	22.6547	0.792103
$819$	0	0
$820$	45.1671	1.57730
$821$	19.9631	0.696716	0.348358	−	0.937362i	$-0.386739\pi$
$821$	19.9631	0.696716	0.348358	+	0.937362i	$0.386739\pi$
$822$	0	0
$823$	−22.1074	−0.770617	−0.385309	−	0.922788i	$-0.625905\pi$
$823$	−22.1074	−0.770617	−0.385309	+	0.922788i	$0.625905\pi$
$824$	−60.2318	−2.09827
$825$	0	0
$826$	117.574	4.09094
$827$	30.0983	1.04662	0.523310	−	0.852142i	$-0.324697\pi$
$827$	30.0983	1.04662	0.523310	+	0.852142i	$0.324697\pi$
$828$	0	0
$829$	34.0028	1.18097	0.590483	−	0.807050i	$-0.298937\pi$
$829$	34.0028	1.18097	0.590483	+	0.807050i	$0.298937\pi$
$830$	4.79296	0.166366
$831$	0	0
$832$	14.2061	0.492507
$833$	−59.4727	−2.06061
$834$	0	0
$835$	−4.57739	−0.158407
$836$	−28.5077	−0.985958
$837$	0	0
$838$	−23.5215	−0.812536
$839$	2.89138	0.0998214	0.0499107	−	0.998754i	$-0.484106\pi$
$839$	2.89138	0.0998214	0.0499107	+	0.998754i	$0.484106\pi$
$840$	0	0
$841$	15.9088	0.548581
$842$	99.8132	3.43979
$843$	0	0
$844$	95.3184	3.28099
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−35.5998	−1.22323
$848$	−137.360	−4.71696
$849$	0	0
$850$	−20.1500	−0.691138
$851$	75.9665	2.60410
$852$	0	0
$853$	−48.3381	−1.65507	−0.827533	−	0.561418i	$-0.810256\pi$
$853$	−48.3381	−1.65507	−0.827533	+	0.561418i	$0.810256\pi$
$854$	−92.9064	−3.17919
$855$	0	0
$856$	−9.20102	−0.314484
$857$	−15.4173	−0.526644	−0.263322	−	0.964708i	$-0.584818\pi$
$857$	−15.4173	−0.526644	−0.263322	+	0.964708i	$0.584818\pi$
$858$	0	0
$859$	−19.0736	−0.650783	−0.325392	−	0.945579i	$-0.605496\pi$
$859$	−19.0736	−0.650783	−0.325392	+	0.945579i	$0.605496\pi$
$860$	−30.8058	−1.05047
$861$	0	0
$862$	30.2835	1.03146
$863$	3.62902	0.123533	0.0617667	−	0.998091i	$-0.480327\pi$
$863$	3.62902	0.123533	0.0617667	+	0.998091i	$0.480327\pi$
$864$	0	0
$865$	17.0645	0.580209
$866$	39.7099	1.34940
$867$	0	0
$868$	−55.7779	−1.89323
$869$	−2.65927	−0.0902094
$870$	0	0
$871$	2.78119	0.0942370
$872$	8.90379	0.301520
$873$	0	0
$874$	−91.2045	−3.08504
$875$	−3.85102	−0.130188
$876$	0	0
$877$	21.2327	0.716977	0.358488	−	0.933534i	$-0.383292\pi$
$877$	21.2327	0.716977	0.358488	+	0.933534i	$0.383292\pi$
$878$	−93.9337	−3.17011
$879$	0	0
$880$	−14.9842	−0.505118
$881$	−2.50694	−0.0844609	−0.0422304	−	0.999108i	$-0.513446\pi$
$881$	−2.50694	−0.0844609	−0.0422304	+	0.999108i	$0.513446\pi$
$882$	0	0
$883$	−32.6963	−1.10032	−0.550159	−	0.835060i	$-0.685433\pi$
$883$	−32.6963	−1.10032	−0.550159	+	0.835060i	$0.685433\pi$
$884$	−38.2674	−1.28707
$885$	0	0
$886$	−3.71235	−0.124719
$887$	3.06828	0.103023	0.0515113	−	0.998672i	$-0.483596\pi$
$887$	3.06828	0.103023	0.0515113	+	0.998672i	$0.483596\pi$
$888$	0	0
$889$	85.8495	2.87930
$890$	−10.4463	−0.350160
$891$	0	0
$892$	6.64787	0.222587
$893$	−5.95862	−0.199398
$894$	0	0
$895$	15.4595	0.516755
$896$	38.2366	1.27740
$897$	0	0
$898$	25.4249	0.848439
$899$	19.2647	0.642513
$900$	0	0
$901$	92.2555	3.07348
$902$	−31.5135	−1.04928
$903$	0	0
$904$	−115.854	−3.85324
$905$	−0.758003	−0.0251969
$906$	0	0
$907$	−1.47548	−0.0489926	−0.0244963	−	0.999700i	$-0.507798\pi$
$907$	−1.47548	−0.0489926	−0.0244963	+	0.999700i	$0.507798\pi$
$908$	97.2589	3.22765
$909$	0	0
$910$	−10.2167	−0.338681
$911$	22.6325	0.749847	0.374924	−	0.927056i	$-0.377669\pi$
$911$	22.6325	0.749847	0.374924	+	0.927056i	$0.377669\pi$
$912$	0	0
$913$	−2.39384	−0.0792247
$914$	−64.4019	−2.13023
$915$	0	0
$916$	−59.7378	−1.97379
$917$	57.1756	1.88811
$918$	0	0
$919$	9.97443	0.329026	0.164513	−	0.986375i	$-0.447395\pi$
$919$	9.97443	0.329026	0.164513	+	0.986375i	$0.447395\pi$
$920$	−64.8958	−2.13955
$921$	0	0
$922$	63.3996	2.08796
$923$	0.799061	0.0263014
$924$	0	0
$925$	9.43592	0.310251
$926$	22.6160	0.743209
$927$	0	0
$928$	−93.0144	−3.05335
$929$	7.69857	0.252582	0.126291	−	0.991993i	$-0.459693\pi$
$929$	7.69857	0.252582	0.126291	+	0.991993i	$0.459693\pi$
$930$	0	0
$931$	−33.4366	−1.09584
$932$	−57.0506	−1.86875
$933$	0	0
$934$	33.9677	1.11146
$935$	10.0639	0.329125
$936$	0	0
$937$	−29.4106	−0.960804	−0.480402	−	0.877048i	$-0.659509\pi$
$937$	−29.4106	−0.960804	−0.480402	+	0.877048i	$0.659509\pi$
$938$	28.4147	0.927771
$939$	0	0
$940$	−7.03064	−0.229314
$941$	4.77162	0.155550	0.0777752	−	0.996971i	$-0.475218\pi$
$941$	4.77162	0.155550	0.0777752	+	0.996971i	$0.475218\pi$
$942$	0	0
$943$	−72.1721	−2.35025
$944$	130.139	4.23566
$945$	0	0
$946$	21.4934	0.698812
$947$	−12.2828	−0.399138	−0.199569	−	0.979884i	$-0.563954\pi$
$947$	−12.2828	−0.399138	−0.199569	+	0.979884i	$0.563954\pi$
$948$	0	0
$949$	−5.29557	−0.171902
$950$	−11.3287	−0.367550
$951$	0	0
$952$	−235.772	−7.64141
$953$	30.4055	0.984932	0.492466	−	0.870332i	$-0.336096\pi$
$953$	30.4055	0.984932	0.492466	+	0.870332i	$0.336096\pi$
$954$	0	0
$955$	23.8336	0.771237
$956$	−29.9918	−0.970004
$957$	0	0
$958$	−92.1422	−2.97698
$959$	38.6628	1.24849
$960$	0	0
$961$	−22.7360	−0.733418
$962$	25.0335	0.807111
$963$	0	0
$964$	−46.5177	−1.49823
$965$	18.7512	0.603623
$966$	0	0
$967$	−18.9287	−0.608706	−0.304353	−	0.952559i	$-0.598440\pi$
$967$	−18.9287	−0.608706	−0.304353	+	0.952559i	$0.598440\pi$
$968$	−74.5163	−2.39504
$969$	0	0
$970$	−11.7757	−0.378094
$971$	6.35781	0.204032	0.102016	−	0.994783i	$-0.467471\pi$
$971$	6.35781	0.204032	0.102016	+	0.994783i	$0.467471\pi$
$972$	0	0
$973$	43.9858	1.41012
$974$	−70.3933	−2.25555
$975$	0	0
$976$	−102.835	−3.29166
$977$	−32.8818	−1.05198	−0.525990	−	0.850491i	$-0.676305\pi$
$977$	−32.8818	−1.05198	−0.525990	+	0.850491i	$0.676305\pi$
$978$	0	0
$979$	5.21739	0.166749
$980$	−39.4522	−1.26025
$981$	0	0
$982$	−9.24316	−0.294961
$983$	−41.6121	−1.32722	−0.663610	−	0.748078i	$-0.730977\pi$
$983$	−41.6121	−1.32722	−0.663610	+	0.748078i	$0.730977\pi$
$984$	0	0
$985$	−1.55671	−0.0496008
$986$	135.033	4.30033
$987$	0	0
$988$	−21.5146	−0.684470
$989$	49.2242	1.56524
$990$	0	0
$991$	−47.6296	−1.51300	−0.756501	−	0.653992i	$-0.773093\pi$
$991$	−47.6296	−1.51300	−0.756501	+	0.653992i	$0.773093\pi$
$992$	−39.9007	−1.26685
$993$	0	0
$994$	8.16378	0.258940
$995$	−9.40931	−0.298295
$996$	0	0
$997$	−28.2366	−0.894263	−0.447132	−	0.894468i	$-0.647554\pi$
$997$	−28.2366	−0.894263	−0.447132	+	0.894468i	$0.647554\pi$
$998$	−43.6075	−1.38037
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.u.1.7	✓	7
3.2	odd	2		1755.2.a.v.1.1	yes	7
5.4	even	2		8775.2.a.bx.1.1		7
15.14	odd	2		8775.2.a.bw.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.u.1.7	✓	7	1.1	even	1	trivial
1755.2.a.v.1.1	yes	7	3.2	odd	2
8775.2.a.bw.1.7		7	15.14	odd	2
8775.2.a.bx.1.1		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)

\(f(q)\)	\(=\)	\(q+2.65299 q^{2} +5.03838 q^{4} -1.00000 q^{5} +3.85102 q^{7} +8.06081 q^{8} +O(q^{10})\)	\(q+2.65299 q^{2} +5.03838 q^{4} -1.00000 q^{5} +3.85102 q^{7} +8.06081 q^{8} -2.65299 q^{10} +1.32504 q^{11} +1.00000 q^{13} +10.2167 q^{14} +11.3085 q^{16} -7.59518 q^{17} -4.27014 q^{19} -5.03838 q^{20} +3.51532 q^{22} +8.05078 q^{23} +1.00000 q^{25} +2.65299 q^{26} +19.4029 q^{28} -6.70141 q^{29} -2.87472 q^{31} +13.8798 q^{32} -20.1500 q^{34} -3.85102 q^{35} +9.43592 q^{37} -11.3287 q^{38} -8.06081 q^{40} -8.96461 q^{41} +6.11422 q^{43} +6.67605 q^{44} +21.3587 q^{46} +1.39542 q^{47} +7.83033 q^{49} +2.65299 q^{50} +5.03838 q^{52} -12.1466 q^{53} -1.32504 q^{55} +31.0423 q^{56} -17.7788 q^{58} +11.5080 q^{59} -9.09356 q^{61} -7.62662 q^{62} +14.2061 q^{64} -1.00000 q^{65} +2.78119 q^{67} -38.2674 q^{68} -10.2167 q^{70} +0.799061 q^{71} -5.29557 q^{73} +25.0335 q^{74} -21.5146 q^{76} +5.10275 q^{77} -2.00693 q^{79} -11.3085 q^{80} -23.7831 q^{82} -1.80662 q^{83} +7.59518 q^{85} +16.2210 q^{86} +10.6809 q^{88} +3.93754 q^{89} +3.85102 q^{91} +40.5629 q^{92} +3.70203 q^{94} +4.27014 q^{95} +4.43863 q^{97} +20.7738 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 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

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