LMFDB - Embedded newform 2275.2.a.y.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2275,2,Mod(1,2275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2275 = 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$18.1659664598$
Analytic rank:	$1$
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} - 8x^{5} - 2x^{4} + 16x^{3} + 5x^{2} - 6x - 2 $ x^7 - 8x^5 - 2x^4 + 16x^3 + 5x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 455)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.700339$ of defining polynomial
Character	$\chi$	$=$	2275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.700339 q^{2} -1.90743 q^{3} -1.50953 q^{4} -1.33585 q^{6} -1.00000 q^{7} -2.45786 q^{8} +0.638282 q^{9} +1.45508 q^{11} +2.87931 q^{12} +1.00000 q^{13} -0.700339 q^{14} +1.29772 q^{16} +6.90822 q^{17} +0.447014 q^{18} -2.77210 q^{19} +1.90743 q^{21} +1.01905 q^{22} +5.28255 q^{23} +4.68819 q^{24} +0.700339 q^{26} +4.50481 q^{27} +1.50953 q^{28} +0.605210 q^{29} -4.55409 q^{31} +5.82456 q^{32} -2.77546 q^{33} +4.83810 q^{34} -0.963503 q^{36} -6.64638 q^{37} -1.94141 q^{38} -1.90743 q^{39} -8.10358 q^{41} +1.33585 q^{42} -2.01992 q^{43} -2.19648 q^{44} +3.69958 q^{46} +0.402167 q^{47} -2.47530 q^{48} +1.00000 q^{49} -13.1769 q^{51} -1.50953 q^{52} -6.04672 q^{53} +3.15489 q^{54} +2.45786 q^{56} +5.28757 q^{57} +0.423852 q^{58} +0.173303 q^{59} -12.8552 q^{61} -3.18941 q^{62} -0.638282 q^{63} +1.48373 q^{64} -1.94377 q^{66} -11.6242 q^{67} -10.4281 q^{68} -10.0761 q^{69} +16.3101 q^{71} -1.56881 q^{72} +1.79758 q^{73} -4.65472 q^{74} +4.18455 q^{76} -1.45508 q^{77} -1.33585 q^{78} -3.07858 q^{79} -10.5074 q^{81} -5.67525 q^{82} +15.0052 q^{83} -2.87931 q^{84} -1.41463 q^{86} -1.15439 q^{87} -3.57638 q^{88} -0.0343671 q^{89} -1.00000 q^{91} -7.97415 q^{92} +8.68660 q^{93} +0.281653 q^{94} -11.1099 q^{96} +2.49513 q^{97} +0.700339 q^{98} +0.928753 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{3} + 2 q^{4} - 8 q^{6} - 7 q^{7} + 6 q^{8} + q^{9} - 6 q^{11} - 3 q^{12} + 7 q^{13} - 4 q^{16} - 2 q^{17} + q^{18} - 10 q^{19} - 2 q^{21} - 18 q^{22} - 10 q^{23} - 5 q^{24} + 8 q^{27} - 2 q^{28}+ \cdots - 18 q^{99}+O(q^{100}) $ 7 * q + 2 * q^3 + 2 * q^4 - 8 * q^6 - 7 * q^7 + 6 * q^8 + q^9 - 6 * q^11 - 3 * q^12 + 7 * q^13 - 4 * q^16 - 2 * q^17 + q^18 - 10 * q^19 - 2 * q^21 - 18 * q^22 - 10 * q^23 - 5 * q^24 + 8 * q^27 - 2 * q^28 - 26 * q^29 - 6 * q^31 + 7 * q^32 - 4 * q^33 - q^34 - 6 * q^36 + 4 * q^37 + q^38 + 2 * q^39 - 28 * q^41 + 8 * q^42 - 2 * q^43 - 18 * q^46 + 20 * q^47 - 20 * q^48 + 7 * q^49 - 32 * q^51 + 2 * q^52 - 8 * q^53 + 11 * q^54 - 6 * q^56 - 14 * q^57 + 12 * q^58 - 28 * q^59 - 26 * q^61 + 22 * q^62 - q^63 - 8 * q^64 + 2 * q^66 - 2 * q^67 + 2 * q^68 - 22 * q^69 - 14 * q^71 - 4 * q^72 + 2 * q^73 - 12 * q^74 - 36 * q^76 + 6 * q^77 - 8 * q^78 + 14 * q^79 - 33 * q^81 - 19 * q^82 + 28 * q^83 + 3 * q^84 - 4 * q^86 - 22 * q^87 - 24 * q^88 - 30 * q^89 - 7 * q^91 - 42 * q^92 - 2 * q^93 + 14 * q^94 + 18 * q^97 - 18 * q^99

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 $

$2$	0.700339	0.495214	0.247607	−	0.968861i	$-0.420356\pi$
$2$	0.700339	0.495214	0.247607	+	0.968861i	$0.420356\pi$
$3$	−1.90743	−1.10125	−0.550627	−	0.834751i	$-0.685611\pi$
$3$	−1.90743	−1.10125	−0.550627	+	0.834751i	$0.685611\pi$
$4$	−1.50953	−0.754763
$5$	0	0
$5$	0	0
$6$	−1.33585	−0.545357
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.45786	−0.868984
$9$	0.638282	0.212761
$10$	0	0
$11$	1.45508	0.438724	0.219362	−	0.975644i	$-0.429602\pi$
$11$	1.45508	0.438724	0.219362	+	0.975644i	$0.429602\pi$
$12$	2.87931	0.831186
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.700339	−0.187173
$15$	0	0
$16$	1.29772	0.324429
$17$	6.90822	1.67549	0.837745	−	0.546062i	$-0.183874\pi$
$17$	6.90822	1.67549	0.837745	+	0.546062i	$0.183874\pi$
$18$	0.447014	0.105362
$19$	−2.77210	−0.635962	−0.317981	−	0.948097i	$-0.603005\pi$
$19$	−2.77210	−0.635962	−0.317981	+	0.948097i	$0.603005\pi$
$20$	0	0
$21$	1.90743	0.416235
$22$	1.01905	0.217262
$23$	5.28255	1.10149	0.550744	−	0.834674i	$-0.314344\pi$
$23$	5.28255	1.10149	0.550744	+	0.834674i	$0.314344\pi$
$24$	4.68819	0.956972
$25$	0	0
$26$	0.700339	0.137348
$27$	4.50481	0.866950
$28$	1.50953	0.285273
$29$	0.605210	0.112385	0.0561924	−	0.998420i	$-0.482104\pi$
$29$	0.605210	0.112385	0.0561924	+	0.998420i	$0.482104\pi$
$30$	0	0
$31$	−4.55409	−0.817939	−0.408969	−	0.912548i	$-0.634112\pi$
$31$	−4.55409	−0.817939	−0.408969	+	0.912548i	$0.634112\pi$
$32$	5.82456	1.02965
$33$	−2.77546	−0.483146
$34$	4.83810	0.829727
$35$	0	0
$36$	−0.963503	−0.160584
$37$	−6.64638	−1.09266	−0.546329	−	0.837571i	$-0.683975\pi$
$37$	−6.64638	−1.09266	−0.546329	+	0.837571i	$0.683975\pi$
$38$	−1.94141	−0.314938
$39$	−1.90743	−0.305433
$40$	0	0
$41$	−8.10358	−1.26557	−0.632783	−	0.774329i	$-0.718088\pi$
$41$	−8.10358	−1.26557	−0.632783	+	0.774329i	$0.718088\pi$
$42$	1.33585	0.206126
$43$	−2.01992	−0.308034	−0.154017	−	0.988068i	$-0.549221\pi$
$43$	−2.01992	−0.308034	−0.154017	+	0.988068i	$0.549221\pi$
$44$	−2.19648	−0.331132
$45$	0	0
$46$	3.69958	0.545473
$47$	0.402167	0.0586620	0.0293310	−	0.999570i	$-0.490662\pi$
$47$	0.402167	0.0586620	0.0293310	+	0.999570i	$0.490662\pi$
$48$	−2.47530	−0.357279
$49$	1.00000	0.142857
$50$	0	0
$51$	−13.1769	−1.84514
$52$	−1.50953	−0.209334
$53$	−6.04672	−0.830580	−0.415290	−	0.909689i	$-0.636320\pi$
$53$	−6.04672	−0.830580	−0.415290	+	0.909689i	$0.636320\pi$
$54$	3.15489	0.429326
$55$	0	0
$56$	2.45786	0.328445
$57$	5.28757	0.700356
$58$	0.423852	0.0556545
$59$	0.173303	0.0225622	0.0112811	−	0.999936i	$-0.496409\pi$
$59$	0.173303	0.0225622	0.0112811	+	0.999936i	$0.496409\pi$
$60$	0	0
$61$	−12.8552	−1.64593	−0.822967	−	0.568089i	$-0.807683\pi$
$61$	−12.8552	−1.64593	−0.822967	+	0.568089i	$0.807683\pi$
$62$	−3.18941	−0.405055
$63$	−0.638282	−0.0804160
$64$	1.48373	0.185466
$65$	0	0
$66$	−1.94377	−0.239261
$67$	−11.6242	−1.42012	−0.710059	−	0.704142i	$-0.751332\pi$
$67$	−11.6242	−1.42012	−0.710059	+	0.704142i	$0.751332\pi$
$68$	−10.4281	−1.26460
$69$	−10.0761	−1.21302
$70$	0	0
$71$	16.3101	1.93565	0.967824	−	0.251630i	$-0.0809665\pi$
$71$	16.3101	1.93565	0.967824	+	0.251630i	$0.0809665\pi$
$72$	−1.56881	−0.184886
$73$	1.79758	0.210391	0.105195	−	0.994452i	$-0.466453\pi$
$73$	1.79758	0.210391	0.105195	+	0.994452i	$0.466453\pi$
$74$	−4.65472	−0.541100
$75$	0	0
$76$	4.18455	0.480001
$77$	−1.45508	−0.165822
$78$	−1.33585	−0.151255
$79$	−3.07858	−0.346367	−0.173183	−	0.984890i	$-0.555405\pi$
$79$	−3.07858	−0.346367	−0.173183	+	0.984890i	$0.555405\pi$
$80$	0	0
$81$	−10.5074	−1.16749
$82$	−5.67525	−0.626727
$83$	15.0052	1.64704	0.823518	−	0.567290i	$-0.192008\pi$
$83$	15.0052	1.64704	0.823518	+	0.567290i	$0.192008\pi$
$84$	−2.87931	−0.314159
$85$	0	0
$86$	−1.41463	−0.152543
$87$	−1.15439	−0.123764
$88$	−3.57638	−0.381244
$89$	−0.0343671	−0.00364290	−0.00182145	−	0.999998i	$-0.500580\pi$
$89$	−0.0343671	−0.00364290	−0.00182145	+	0.999998i	$0.500580\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−7.97415	−0.831362
$93$	8.68660	0.900758
$94$	0.281653	0.0290503
$95$	0	0
$96$	−11.1099	−1.13390
$97$	2.49513	0.253342	0.126671	−	0.991945i	$-0.459571\pi$
$97$	2.49513	0.253342	0.126671	+	0.991945i	$0.459571\pi$
$98$	0.700339	0.0707449
$99$	0.928753	0.0933432
$100$	0	0
$101$	−16.1272	−1.60472	−0.802359	−	0.596842i	$-0.796422\pi$
$101$	−16.1272	−1.60472	−0.802359	+	0.596842i	$0.796422\pi$
$102$	−9.22832	−0.913740
$103$	16.1870	1.59495	0.797475	−	0.603352i	$-0.206169\pi$
$103$	16.1870	1.59495	0.797475	+	0.603352i	$0.206169\pi$
$104$	−2.45786	−0.241013
$105$	0	0
$106$	−4.23475	−0.411315
$107$	5.15383	0.498239	0.249120	−	0.968473i	$-0.419859\pi$
$107$	5.15383	0.498239	0.249120	+	0.968473i	$0.419859\pi$
$108$	−6.80012	−0.654342
$109$	−12.9954	−1.24473	−0.622366	−	0.782726i	$-0.713829\pi$
$109$	−12.9954	−1.24473	−0.622366	+	0.782726i	$0.713829\pi$
$110$	0	0
$111$	12.6775	1.20329
$112$	−1.29772	−0.122623
$113$	−11.7837	−1.10851	−0.554256	−	0.832346i	$-0.686997\pi$
$113$	−11.7837	−1.10851	−0.554256	+	0.832346i	$0.686997\pi$
$114$	3.70309	0.346827
$115$	0	0
$116$	−0.913580	−0.0848238
$117$	0.638282	0.0590092
$118$	0.121371	0.0111731
$119$	−6.90822	−0.633276
$120$	0	0
$121$	−8.88274	−0.807521
$122$	−9.00297	−0.815090
$123$	15.4570	1.39371
$124$	6.87451	0.617350
$125$	0	0
$126$	−0.447014	−0.0398232
$127$	−18.5773	−1.64847	−0.824235	−	0.566249i	$-0.808394\pi$
$127$	−18.5773	−1.64847	−0.824235	+	0.566249i	$0.808394\pi$
$128$	−10.6100	−0.937800
$129$	3.85285	0.339224
$130$	0	0
$131$	−2.85636	−0.249562	−0.124781	−	0.992184i	$-0.539823\pi$
$131$	−2.85636	−0.249562	−0.124781	+	0.992184i	$0.539823\pi$
$132$	4.18963	0.364661
$133$	2.77210	0.240371
$134$	−8.14086	−0.703263
$135$	0	0
$136$	−16.9794	−1.45597
$137$	6.15921	0.526216	0.263108	−	0.964766i	$-0.415252\pi$
$137$	6.15921	0.526216	0.263108	+	0.964766i	$0.415252\pi$
$138$	−7.05668	−0.600704
$139$	−9.07283	−0.769548	−0.384774	−	0.923011i	$-0.625721\pi$
$139$	−9.07283	−0.769548	−0.384774	+	0.923011i	$0.625721\pi$
$140$	0	0
$141$	−0.767104	−0.0646018
$142$	11.4226	0.958560
$143$	1.45508	0.121680
$144$	0.828310	0.0690259
$145$	0	0
$146$	1.25892	0.104189
$147$	−1.90743	−0.157322
$148$	10.0329	0.824697
$149$	−4.84627	−0.397022	−0.198511	−	0.980099i	$-0.563611\pi$
$149$	−4.84627	−0.397022	−0.198511	+	0.980099i	$0.563611\pi$
$150$	0	0
$151$	−16.5437	−1.34631	−0.673153	−	0.739504i	$-0.735060\pi$
$151$	−16.5437	−1.34631	−0.673153	+	0.739504i	$0.735060\pi$
$152$	6.81342	0.552641
$153$	4.40940	0.356478
$154$	−1.01905	−0.0821174
$155$	0	0
$156$	2.87931	0.230529
$157$	−3.22956	−0.257747	−0.128874	−	0.991661i	$-0.541136\pi$
$157$	−3.22956	−0.257747	−0.128874	+	0.991661i	$0.541136\pi$
$158$	−2.15605	−0.171526
$159$	11.5337	0.914680
$160$	0	0
$161$	−5.28255	−0.416324
$162$	−7.35877	−0.578160
$163$	−3.90918	−0.306191	−0.153095	−	0.988211i	$-0.548924\pi$
$163$	−3.90918	−0.306191	−0.153095	+	0.988211i	$0.548924\pi$
$164$	12.2326	0.955203
$165$	0	0
$166$	10.5087	0.815636
$167$	17.8513	1.38138	0.690689	−	0.723152i	$-0.257307\pi$
$167$	17.8513	1.38138	0.690689	+	0.723152i	$0.257307\pi$
$168$	−4.68819	−0.361701
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.76938	−0.135308
$172$	3.04912	0.232493
$173$	−9.36517	−0.712021	−0.356010	−	0.934482i	$-0.615863\pi$
$173$	−9.36517	−0.712021	−0.356010	+	0.934482i	$0.615863\pi$
$174$	−0.808468	−0.0612898
$175$	0	0
$176$	1.88829	0.142335
$177$	−0.330563	−0.0248467
$178$	−0.0240686	−0.00180402
$179$	−1.81219	−0.135449	−0.0677246	−	0.997704i	$-0.521574\pi$
$179$	−1.81219	−0.135449	−0.0677246	+	0.997704i	$0.521574\pi$
$180$	0	0
$181$	23.6654	1.75904	0.879518	−	0.475866i	$-0.157865\pi$
$181$	23.6654	1.75904	0.879518	+	0.475866i	$0.157865\pi$
$182$	−0.700339	−0.0519126
$183$	24.5203	1.81259
$184$	−12.9838	−0.957176
$185$	0	0
$186$	6.08356	0.446069
$187$	10.0520	0.735077
$188$	−0.607081	−0.0442759
$189$	−4.50481	−0.327676
$190$	0	0
$191$	−20.2026	−1.46181	−0.730906	−	0.682478i	$-0.760902\pi$
$191$	−20.2026	−1.46181	−0.730906	+	0.682478i	$0.760902\pi$
$192$	−2.83011	−0.204245
$193$	3.11688	0.224358	0.112179	−	0.993688i	$-0.464217\pi$
$193$	3.11688	0.224358	0.112179	+	0.993688i	$0.464217\pi$
$194$	1.74743	0.125458
$195$	0	0
$196$	−1.50953	−0.107823
$197$	−16.1602	−1.15137	−0.575683	−	0.817673i	$-0.695264\pi$
$197$	−16.1602	−1.15137	−0.575683	+	0.817673i	$0.695264\pi$
$198$	0.650442	0.0462249
$199$	−15.5337	−1.10115	−0.550577	−	0.834785i	$-0.685592\pi$
$199$	−15.5337	−1.10115	−0.550577	+	0.834785i	$0.685592\pi$
$200$	0	0
$201$	22.1723	1.56391
$202$	−11.2945	−0.794679
$203$	−0.605210	−0.0424774
$204$	19.8909	1.39264
$205$	0	0
$206$	11.3364	0.789842
$207$	3.37176	0.234354
$208$	1.29772	0.0899805
$209$	−4.03363	−0.279012
$210$	0	0
$211$	6.40692	0.441071	0.220535	−	0.975379i	$-0.429220\pi$
$211$	6.40692	0.441071	0.220535	+	0.975379i	$0.429220\pi$
$212$	9.12767	0.626891
$213$	−31.1103	−2.13164
$214$	3.60943	0.246735
$215$	0	0
$216$	−11.0722	−0.753366
$217$	4.55409	0.309152
$218$	−9.10118	−0.616409
$219$	−3.42876	−0.231694
$220$	0	0
$221$	6.90822	0.464697
$222$	8.87854	0.595888
$223$	0.151299	0.0101317	0.00506586	−	0.999987i	$-0.498387\pi$
$223$	0.151299	0.0101317	0.00506586	+	0.999987i	$0.498387\pi$
$224$	−5.82456	−0.389170
$225$	0	0
$226$	−8.25255	−0.548952
$227$	−3.87255	−0.257030	−0.128515	−	0.991708i	$-0.541021\pi$
$227$	−3.87255	−0.257030	−0.128515	+	0.991708i	$0.541021\pi$
$228$	−7.98173	−0.528603
$229$	18.7262	1.23746	0.618730	−	0.785604i	$-0.287647\pi$
$229$	18.7262	1.23746	0.618730	+	0.785604i	$0.287647\pi$
$230$	0	0
$231$	2.77546	0.182612
$232$	−1.48752	−0.0976605
$233$	−5.25944	−0.344557	−0.172279	−	0.985048i	$-0.555113\pi$
$233$	−5.25944	−0.344557	−0.172279	+	0.985048i	$0.555113\pi$
$234$	0.447014	0.0292222
$235$	0	0
$236$	−0.261606	−0.0170291
$237$	5.87216	0.381438
$238$	−4.83810	−0.313607
$239$	10.0888	0.652593	0.326296	−	0.945268i	$-0.394199\pi$
$239$	10.0888	0.652593	0.326296	+	0.945268i	$0.394199\pi$
$240$	0	0
$241$	−1.54899	−0.0997791	−0.0498896	−	0.998755i	$-0.515887\pi$
$241$	−1.54899	−0.0997791	−0.0498896	+	0.998755i	$0.515887\pi$
$242$	−6.22093	−0.399896
$243$	6.52777	0.418757
$244$	19.4052	1.24229
$245$	0	0
$246$	10.8251	0.690186
$247$	−2.77210	−0.176384
$248$	11.1933	0.710775
$249$	−28.6214	−1.81381
$250$	0	0
$251$	−5.18164	−0.327062	−0.163531	−	0.986538i	$-0.552288\pi$
$251$	−5.18164	−0.327062	−0.163531	+	0.986538i	$0.552288\pi$
$252$	0.963503	0.0606950
$253$	7.68655	0.483249
$254$	−13.0104	−0.816346
$255$	0	0
$256$	−10.3981	−0.649878
$257$	−7.51750	−0.468929	−0.234464	−	0.972125i	$-0.575334\pi$
$257$	−7.51750	−0.468929	−0.234464	+	0.972125i	$0.575334\pi$
$258$	2.69830	0.167989
$259$	6.64638	0.412986
$260$	0	0
$261$	0.386295	0.0239111
$262$	−2.00042	−0.123586
$263$	−3.07554	−0.189646	−0.0948230	−	0.995494i	$-0.530229\pi$
$263$	−3.07554	−0.189646	−0.0948230	+	0.995494i	$0.530229\pi$
$264$	6.82170	0.419846
$265$	0	0
$266$	1.94141	0.119035
$267$	0.0655528	0.00401176
$268$	17.5470	1.07185
$269$	12.4216	0.757358	0.378679	−	0.925528i	$-0.376378\pi$
$269$	12.4216	0.757358	0.378679	+	0.925528i	$0.376378\pi$
$270$	0	0
$271$	15.2486	0.926284	0.463142	−	0.886284i	$-0.346722\pi$
$271$	15.2486	0.926284	0.463142	+	0.886284i	$0.346722\pi$
$272$	8.96492	0.543578
$273$	1.90743	0.115443
$274$	4.31353	0.260590
$275$	0	0
$276$	15.2101	0.915541
$277$	−21.6012	−1.29789	−0.648944	−	0.760836i	$-0.724789\pi$
$277$	−21.6012	−1.29789	−0.648944	+	0.760836i	$0.724789\pi$
$278$	−6.35406	−0.381091
$279$	−2.90680	−0.174025
$280$	0	0
$281$	−14.0128	−0.835932	−0.417966	−	0.908463i	$-0.637257\pi$
$281$	−14.0128	−0.835932	−0.417966	+	0.908463i	$0.637257\pi$
$282$	−0.537233	−0.0319917
$283$	−29.5809	−1.75840	−0.879200	−	0.476453i	$-0.841922\pi$
$283$	−29.5809	−1.75840	−0.879200	+	0.476453i	$0.841922\pi$
$284$	−24.6204	−1.46095
$285$	0	0
$286$	1.01905	0.0602577
$287$	8.10358	0.478339
$288$	3.71771	0.219068
$289$	30.7235	1.80727
$290$	0	0
$291$	−4.75927	−0.278994
$292$	−2.71349	−0.158795
$293$	−13.2920	−0.776525	−0.388263	−	0.921549i	$-0.626925\pi$
$293$	−13.2920	−0.776525	−0.388263	+	0.921549i	$0.626925\pi$
$294$	−1.33585	−0.0779081
$295$	0	0
$296$	16.3358	0.949502
$297$	6.55486	0.380352
$298$	−3.39403	−0.196611
$299$	5.28255	0.305498
$300$	0	0
$301$	2.01992	0.116426
$302$	−11.5862	−0.666710
$303$	30.7615	1.76720
$304$	−3.59740	−0.206325
$305$	0	0
$306$	3.08807	0.176533
$307$	28.9606	1.65287	0.826434	−	0.563033i	$-0.190366\pi$
$307$	28.9606	1.65287	0.826434	+	0.563033i	$0.190366\pi$
$308$	2.19648	0.125156
$309$	−30.8755	−1.75645
$310$	0	0
$311$	−0.0184394	−0.00104560	−0.000522802	−	1.00000i	$-0.500166\pi$
$311$	−0.0184394	−0.00104560	−0.000522802		1.00000i	$0.500166\pi$
$312$	4.68819	0.265416
$313$	−15.8102	−0.893644	−0.446822	−	0.894623i	$-0.647444\pi$
$313$	−15.8102	−0.893644	−0.446822	+	0.894623i	$0.647444\pi$
$314$	−2.26179	−0.127640
$315$	0	0
$316$	4.64719	0.261425
$317$	30.6020	1.71878	0.859390	−	0.511321i	$-0.170844\pi$
$317$	30.6020	1.71878	0.859390	+	0.511321i	$0.170844\pi$
$318$	8.07748	0.452963
$319$	0.880631	0.0493058
$320$	0	0
$321$	−9.83056	−0.548688
$322$	−3.69958	−0.206169
$323$	−19.1503	−1.06555
$324$	15.8613	0.881181
$325$	0	0
$326$	−2.73775	−0.151630
$327$	24.7878	1.37077
$328$	19.9174	1.09976
$329$	−0.402167	−0.0221722
$330$	0	0
$331$	−13.6302	−0.749186	−0.374593	−	0.927189i	$-0.622218\pi$
$331$	−13.6302	−0.749186	−0.374593	+	0.927189i	$0.622218\pi$
$332$	−22.6507	−1.24312
$333$	−4.24227	−0.232475
$334$	12.5020	0.684079
$335$	0	0
$336$	2.47530	0.135039
$337$	−15.2982	−0.833347	−0.416673	−	0.909056i	$-0.636804\pi$
$337$	−15.2982	−0.833347	−0.416673	+	0.909056i	$0.636804\pi$
$338$	0.700339	0.0380934
$339$	22.4765	1.22075
$340$	0	0
$341$	−6.62658	−0.358849
$342$	−1.23917	−0.0670064
$343$	−1.00000	−0.0539949
$344$	4.96467	0.267677
$345$	0	0
$346$	−6.55879	−0.352603
$347$	9.47086	0.508422	0.254211	−	0.967149i	$-0.418184\pi$
$347$	9.47086	0.508422	0.254211	+	0.967149i	$0.418184\pi$
$348$	1.74259	0.0934126
$349$	14.8537	0.795100	0.397550	−	0.917580i	$-0.369860\pi$
$349$	14.8537	0.795100	0.397550	+	0.917580i	$0.369860\pi$
$350$	0	0
$351$	4.50481	0.240449
$352$	8.47521	0.451730
$353$	30.2040	1.60760	0.803798	−	0.594903i	$-0.202809\pi$
$353$	30.2040	1.60760	0.803798	+	0.594903i	$0.202809\pi$
$354$	−0.231506	−0.0123044
$355$	0	0
$356$	0.0518780	0.00274953
$357$	13.1769	0.697397
$358$	−1.26915	−0.0670764
$359$	−26.0944	−1.37721	−0.688605	−	0.725136i	$-0.741777\pi$
$359$	−26.0944	−1.37721	−0.688605	+	0.725136i	$0.741777\pi$
$360$	0	0
$361$	−11.3155	−0.595552
$362$	16.5738	0.871100
$363$	16.9432	0.889286
$364$	1.50953	0.0791206
$365$	0	0
$366$	17.1725	0.897622
$367$	−3.39089	−0.177003	−0.0885015	−	0.996076i	$-0.528208\pi$
$367$	−3.39089	−0.177003	−0.0885015	+	0.996076i	$0.528208\pi$
$368$	6.85526	0.357355
$369$	−5.17237	−0.269263
$370$	0	0
$371$	6.04672	0.313930
$372$	−13.1126	−0.679859
$373$	−29.9799	−1.55230	−0.776150	−	0.630549i	$-0.782830\pi$
$373$	−29.9799	−1.55230	−0.776150	+	0.630549i	$0.782830\pi$
$374$	7.03983	0.364021
$375$	0	0
$376$	−0.988468	−0.0509763
$377$	0.605210	0.0311699
$378$	−3.15489	−0.162270
$379$	7.48208	0.384329	0.192164	−	0.981363i	$-0.438449\pi$
$379$	7.48208	0.384329	0.192164	+	0.981363i	$0.438449\pi$
$380$	0	0
$381$	35.4349	1.81538
$382$	−14.1487	−0.723910
$383$	31.1435	1.59136	0.795678	−	0.605720i	$-0.207115\pi$
$383$	31.1435	1.59136	0.795678	+	0.605720i	$0.207115\pi$
$384$	20.2378	1.03276
$385$	0	0
$386$	2.18287	0.111105
$387$	−1.28928	−0.0655376
$388$	−3.76646	−0.191213
$389$	−18.5258	−0.939296	−0.469648	−	0.882854i	$-0.655619\pi$
$389$	−18.5258	−0.939296	−0.469648	+	0.882854i	$0.655619\pi$
$390$	0	0
$391$	36.4930	1.84553
$392$	−2.45786	−0.124141
$393$	5.44831	0.274831
$394$	−11.3176	−0.570173
$395$	0	0
$396$	−1.40198	−0.0704520
$397$	−20.0511	−1.00634	−0.503168	−	0.864189i	$-0.667832\pi$
$397$	−20.0511	−1.00634	−0.503168	+	0.864189i	$0.667832\pi$
$398$	−10.8788	−0.545307
$399$	−5.28757	−0.264710
$400$	0	0
$401$	14.4962	0.723904	0.361952	−	0.932197i	$-0.382110\pi$
$401$	14.4962	0.723904	0.361952	+	0.932197i	$0.382110\pi$
$402$	15.5281	0.774471
$403$	−4.55409	−0.226855
$404$	24.3444	1.21118
$405$	0	0
$406$	−0.423852	−0.0210354
$407$	−9.67103	−0.479375
$408$	32.3870	1.60340
$409$	−21.8195	−1.07891	−0.539453	−	0.842015i	$-0.681369\pi$
$409$	−21.8195	−1.07891	−0.539453	+	0.842015i	$0.681369\pi$
$410$	0	0
$411$	−11.7482	−0.579498
$412$	−24.4346	−1.20381
$413$	−0.173303	−0.00852769
$414$	2.36138	0.116055
$415$	0	0
$416$	5.82456	0.285572
$417$	17.3058	0.847468
$418$	−2.82491	−0.138171
$419$	36.1878	1.76789	0.883946	−	0.467589i	$-0.154877\pi$
$419$	36.1878	1.76789	0.883946	+	0.467589i	$0.154877\pi$
$420$	0	0
$421$	−9.45278	−0.460700	−0.230350	−	0.973108i	$-0.573987\pi$
$421$	−9.45278	−0.460700	−0.230350	+	0.973108i	$0.573987\pi$
$422$	4.48702	0.218425
$423$	0.256696	0.0124810
$424$	14.8620	0.721761
$425$	0	0
$426$	−21.7877	−1.05562
$427$	12.8552	0.622105
$428$	−7.77983	−0.376052
$429$	−2.77546	−0.134001
$430$	0	0
$431$	−8.32317	−0.400913	−0.200456	−	0.979703i	$-0.564243\pi$
$431$	−8.32317	−0.400913	−0.200456	+	0.979703i	$0.564243\pi$
$432$	5.84597	0.281264
$433$	−0.496464	−0.0238585	−0.0119293	−	0.999929i	$-0.503797\pi$
$433$	−0.496464	−0.0238585	−0.0119293	+	0.999929i	$0.503797\pi$
$434$	3.18941	0.153096
$435$	0	0
$436$	19.6169	0.939478
$437$	−14.6437	−0.700505
$438$	−2.40129	−0.114738
$439$	−36.3672	−1.73571	−0.867857	−	0.496815i	$-0.834503\pi$
$439$	−36.3672	−1.73571	−0.867857	+	0.496815i	$0.834503\pi$
$440$	0	0
$441$	0.638282	0.0303944
$442$	4.83810	0.230125
$443$	14.4235	0.685283	0.342641	−	0.939466i	$-0.388678\pi$
$443$	14.4235	0.685283	0.342641	+	0.939466i	$0.388678\pi$
$444$	−19.1370	−0.908201
$445$	0	0
$446$	0.105961	0.00501738
$447$	9.24391	0.437222
$448$	−1.48373	−0.0700996
$449$	−20.2268	−0.954562	−0.477281	−	0.878751i	$-0.658378\pi$
$449$	−20.2268	−0.954562	−0.477281	+	0.878751i	$0.658378\pi$
$450$	0	0
$451$	−11.7914	−0.555234
$452$	17.7877	0.836664
$453$	31.5559	1.48262
$454$	−2.71209	−0.127285
$455$	0	0
$456$	−12.9961	−0.608598
$457$	32.7158	1.53038	0.765190	−	0.643804i	$-0.222645\pi$
$457$	32.7158	1.53038	0.765190	+	0.643804i	$0.222645\pi$
$458$	13.1147	0.612808
$459$	31.1202	1.45257
$460$	0	0
$461$	30.7380	1.43161	0.715805	−	0.698301i	$-0.246060\pi$
$461$	30.7380	1.43161	0.715805	+	0.698301i	$0.246060\pi$
$462$	1.94377	0.0904322
$463$	−5.85096	−0.271917	−0.135959	−	0.990715i	$-0.543411\pi$
$463$	−5.85096	−0.271917	−0.135959	+	0.990715i	$0.543411\pi$
$464$	0.785392	0.0364609
$465$	0	0
$466$	−3.68339	−0.170630
$467$	−4.18312	−0.193572	−0.0967860	−	0.995305i	$-0.530856\pi$
$467$	−4.18312	−0.193572	−0.0967860	+	0.995305i	$0.530856\pi$
$468$	−0.963503	−0.0445380
$469$	11.6242	0.536754
$470$	0	0
$471$	6.16016	0.283845
$472$	−0.425955	−0.0196061
$473$	−2.93914	−0.135142
$474$	4.11250	0.188894
$475$	0	0
$476$	10.4281	0.477973
$477$	−3.85951	−0.176715
$478$	7.06561	0.323173
$479$	−26.1706	−1.19577	−0.597883	−	0.801584i	$-0.703991\pi$
$479$	−26.1706	−1.19577	−0.597883	+	0.801584i	$0.703991\pi$
$480$	0	0
$481$	−6.64638	−0.303049
$482$	−1.08482	−0.0494121
$483$	10.0761	0.458478
$484$	13.4087	0.609487
$485$	0	0
$486$	4.57165	0.207374
$487$	−9.66726	−0.438065	−0.219033	−	0.975718i	$-0.570290\pi$
$487$	−9.66726	−0.438065	−0.219033	+	0.975718i	$0.570290\pi$
$488$	31.5961	1.43029
$489$	7.45648	0.337194
$490$	0	0
$491$	31.9408	1.44147	0.720733	−	0.693213i	$-0.243805\pi$
$491$	31.9408	1.44147	0.720733	+	0.693213i	$0.243805\pi$
$492$	−23.3327	−1.05192
$493$	4.18093	0.188299
$494$	−1.94141	−0.0873480
$495$	0	0
$496$	−5.90992	−0.265363
$497$	−16.3101	−0.731606
$498$	−20.0447	−0.898222
$499$	13.7690	0.616384	0.308192	−	0.951324i	$-0.400276\pi$
$499$	13.7690	0.616384	0.308192	+	0.951324i	$0.400276\pi$
$500$	0	0
$501$	−34.0502	−1.52125
$502$	−3.62890	−0.161966
$503$	−20.5862	−0.917895	−0.458947	−	0.888464i	$-0.651773\pi$
$503$	−20.5862	−0.917895	−0.458947	+	0.888464i	$0.651773\pi$
$504$	1.56881	0.0698802
$505$	0	0
$506$	5.38319	0.239312
$507$	−1.90743	−0.0847119
$508$	28.0429	1.24420
$509$	−31.0623	−1.37681	−0.688406	−	0.725326i	$-0.741689\pi$
$509$	−31.0623	−1.37681	−0.688406	+	0.725326i	$0.741689\pi$
$510$	0	0
$511$	−1.79758	−0.0795203
$512$	13.9378	0.615971
$513$	−12.4878	−0.551348
$514$	−5.26480	−0.232220
$515$	0	0
$516$	−5.81597	−0.256034
$517$	0.585185	0.0257364
$518$	4.65472	0.204517
$519$	17.8634	0.784116
$520$	0	0
$521$	29.1519	1.27717	0.638583	−	0.769553i	$-0.279521\pi$
$521$	29.1519	1.27717	0.638583	+	0.769553i	$0.279521\pi$
$522$	0.270537	0.0118411
$523$	−28.5267	−1.24738	−0.623692	−	0.781670i	$-0.714368\pi$
$523$	−28.5267	−1.24738	−0.623692	+	0.781670i	$0.714368\pi$
$524$	4.31175	0.188360
$525$	0	0
$526$	−2.15392	−0.0939154
$527$	−31.4607	−1.37045
$528$	−3.60177	−0.156747
$529$	4.90537	0.213277
$530$	0	0
$531$	0.110616	0.00480034
$532$	−4.18455	−0.181423
$533$	−8.10358	−0.351005
$534$	0.0459092	0.00198668
$535$	0	0
$536$	28.5705	1.23406
$537$	3.45662	0.149164
$538$	8.69933	0.375055
$539$	1.45508	0.0626748
$540$	0	0
$541$	7.95620	0.342064	0.171032	−	0.985266i	$-0.445290\pi$
$541$	7.95620	0.342064	0.171032	+	0.985266i	$0.445290\pi$
$542$	10.6792	0.458709
$543$	−45.1401	−1.93715
$544$	40.2373	1.72516
$545$	0	0
$546$	1.33585	0.0571689
$547$	−4.62795	−0.197877	−0.0989385	−	0.995094i	$-0.531545\pi$
$547$	−4.62795	−0.197877	−0.0989385	+	0.995094i	$0.531545\pi$
$548$	−9.29748	−0.397169
$549$	−8.20522	−0.350190
$550$	0	0
$551$	−1.67770	−0.0714725
$552$	24.7656	1.05409
$553$	3.07858	0.130914
$554$	−15.1281	−0.642733
$555$	0	0
$556$	13.6957	0.580826
$557$	−0.354869	−0.0150363	−0.00751814	−	0.999972i	$-0.502393\pi$
$557$	−0.354869	−0.0150363	−0.00751814	+	0.999972i	$0.502393\pi$
$558$	−2.03574	−0.0861798
$559$	−2.01992	−0.0854334
$560$	0	0
$561$	−19.1735	−0.809507
$562$	−9.81369	−0.413966
$563$	−29.4323	−1.24042	−0.620212	−	0.784434i	$-0.712953\pi$
$563$	−29.4323	−1.24042	−0.620212	+	0.784434i	$0.712953\pi$
$564$	1.15796	0.0487590
$565$	0	0
$566$	−20.7166	−0.870785
$567$	10.5074	0.441271
$568$	−40.0878	−1.68205
$569$	−29.4581	−1.23495	−0.617473	−	0.786592i	$-0.711844\pi$
$569$	−29.4581	−1.23495	−0.617473	+	0.786592i	$0.711844\pi$
$570$	0	0
$571$	17.7897	0.744475	0.372237	−	0.928138i	$-0.378591\pi$
$571$	17.7897	0.744475	0.372237	+	0.928138i	$0.378591\pi$
$572$	−2.19648	−0.0918396
$573$	38.5351	1.60983
$574$	5.67525	0.236880
$575$	0	0
$576$	0.947038	0.0394599
$577$	−21.0232	−0.875206	−0.437603	−	0.899168i	$-0.644172\pi$
$577$	−21.0232	−0.875206	−0.437603	+	0.899168i	$0.644172\pi$
$578$	21.5169	0.894984
$579$	−5.94523	−0.247075
$580$	0	0
$581$	−15.0052	−0.622521
$582$	−3.33310	−0.138162
$583$	−8.79847	−0.364395
$584$	−4.41820	−0.182826
$585$	0	0
$586$	−9.30889	−0.384547
$587$	−32.8396	−1.35544	−0.677718	−	0.735322i	$-0.737031\pi$
$587$	−32.8396	−1.35544	−0.677718	+	0.735322i	$0.737031\pi$
$588$	2.87931	0.118741
$589$	12.6244	0.520178
$590$	0	0
$591$	30.8244	1.26795
$592$	−8.62512	−0.354490
$593$	21.6461	0.888900	0.444450	−	0.895804i	$-0.353399\pi$
$593$	21.6461	0.888900	0.444450	+	0.895804i	$0.353399\pi$
$594$	4.59063	0.188356
$595$	0	0
$596$	7.31557	0.299657
$597$	29.6294	1.21265
$598$	3.69958	0.151287
$599$	17.5728	0.718007	0.359004	−	0.933336i	$-0.383117\pi$
$599$	17.5728	0.718007	0.359004	+	0.933336i	$0.383117\pi$
$600$	0	0
$601$	33.0025	1.34620	0.673100	−	0.739551i	$-0.264962\pi$
$601$	33.0025	1.34620	0.673100	+	0.739551i	$0.264962\pi$
$602$	1.41463	0.0576559
$603$	−7.41950	−0.302145
$604$	24.9731	1.01614
$605$	0	0
$606$	21.5435	0.875144
$607$	13.5183	0.548691	0.274346	−	0.961631i	$-0.411539\pi$
$607$	13.5183	0.548691	0.274346	+	0.961631i	$0.411539\pi$
$608$	−16.1462	−0.654816
$609$	1.15439	0.0467784
$610$	0	0
$611$	0.402167	0.0162699
$612$	−6.65609	−0.269057
$613$	−45.0024	−1.81763	−0.908814	−	0.417201i	$-0.863011\pi$
$613$	−45.0024	−1.81763	−0.908814	+	0.417201i	$0.863011\pi$
$614$	20.2822	0.818524
$615$	0	0
$616$	3.57638	0.144097
$617$	−17.5660	−0.707179	−0.353589	−	0.935401i	$-0.615039\pi$
$617$	−17.5660	−0.707179	−0.353589	+	0.935401i	$0.615039\pi$
$618$	−21.6233	−0.869817
$619$	−1.00286	−0.0403085	−0.0201543	−	0.999797i	$-0.506416\pi$
$619$	−1.00286	−0.0403085	−0.0201543	+	0.999797i	$0.506416\pi$
$620$	0	0
$621$	23.7969	0.954936
$622$	−0.0129138	−0.000517798 0
$623$	0.0343671	0.00137689
$624$	−2.47530	−0.0990914
$625$	0	0
$626$	−11.0725	−0.442545
$627$	7.69385	0.307263
$628$	4.87511	0.194538
$629$	−45.9146	−1.83074
$630$	0	0
$631$	15.7979	0.628905	0.314453	−	0.949273i	$-0.398179\pi$
$631$	15.7979	0.628905	0.314453	+	0.949273i	$0.398179\pi$
$632$	7.56670	0.300987
$633$	−12.2207	−0.485731
$634$	21.4318	0.851164
$635$	0	0
$636$	−17.4104	−0.690366
$637$	1.00000	0.0396214
$638$	0.616740	0.0244170
$639$	10.4104	0.411830
$640$	0	0
$641$	14.0303	0.554163	0.277081	−	0.960846i	$-0.410633\pi$
$641$	14.0303	0.554163	0.277081	+	0.960846i	$0.410633\pi$
$642$	−6.88472	−0.271718
$643$	−25.0198	−0.986686	−0.493343	−	0.869835i	$-0.664225\pi$
$643$	−25.0198	−0.986686	−0.493343	+	0.869835i	$0.664225\pi$
$644$	7.97415	0.314225
$645$	0	0
$646$	−13.4117	−0.527675
$647$	−16.1437	−0.634674	−0.317337	−	0.948313i	$-0.602789\pi$
$647$	−16.1437	−0.634674	−0.317337	+	0.948313i	$0.602789\pi$
$648$	25.8258	1.01453
$649$	0.252170	0.00989856
$650$	0	0
$651$	−8.68660	−0.340455
$652$	5.90101	0.231101
$653$	−25.7662	−1.00831	−0.504155	−	0.863613i	$-0.668196\pi$
$653$	−25.7662	−1.00831	−0.504155	+	0.863613i	$0.668196\pi$
$654$	17.3598	0.678823
$655$	0	0
$656$	−10.5162	−0.410587
$657$	1.14736	0.0447629
$658$	−0.281653	−0.0109800
$659$	−36.4389	−1.41946	−0.709728	−	0.704476i	$-0.751182\pi$
$659$	−36.4389	−1.41946	−0.709728	+	0.704476i	$0.751182\pi$
$660$	0	0
$661$	−3.88101	−0.150954	−0.0754770	−	0.997148i	$-0.524048\pi$
$661$	−3.88101	−0.150954	−0.0754770	+	0.997148i	$0.524048\pi$
$662$	−9.54579	−0.371008
$663$	−13.1769	−0.511750
$664$	−36.8807	−1.43125
$665$	0	0
$666$	−2.97102	−0.115125
$667$	3.19706	0.123790
$668$	−26.9471	−1.04261
$669$	−0.288592	−0.0111576
$670$	0	0
$671$	−18.7053	−0.722111
$672$	11.1099	0.428575
$673$	−22.3339	−0.860907	−0.430454	−	0.902613i	$-0.641646\pi$
$673$	−22.3339	−0.860907	−0.430454	+	0.902613i	$0.641646\pi$
$674$	−10.7139	−0.412685
$675$	0	0
$676$	−1.50953	−0.0580587
$677$	38.8547	1.49331	0.746654	−	0.665212i	$-0.231659\pi$
$677$	38.8547	1.49331	0.746654	+	0.665212i	$0.231659\pi$
$678$	15.7411	0.604535
$679$	−2.49513	−0.0957541
$680$	0	0
$681$	7.38660	0.283055
$682$	−4.64085	−0.177707
$683$	−23.7238	−0.907764	−0.453882	−	0.891062i	$-0.649961\pi$
$683$	−23.7238	−0.907764	−0.453882	+	0.891062i	$0.649961\pi$
$684$	2.67092	0.102125
$685$	0	0
$686$	−0.700339	−0.0267391
$687$	−35.7188	−1.36276
$688$	−2.62128	−0.0999354
$689$	−6.04672	−0.230362
$690$	0	0
$691$	−8.26899	−0.314567	−0.157284	−	0.987553i	$-0.550274\pi$
$691$	−8.26899	−0.314567	−0.157284	+	0.987553i	$0.550274\pi$
$692$	14.1370	0.537407
$693$	−0.928753	−0.0352804
$694$	6.63281	0.251778
$695$	0	0
$696$	2.83734	0.107549
$697$	−55.9813	−2.12044
$698$	10.4026	0.393745
$699$	10.0320	0.379445
$700$	0	0
$701$	−45.9011	−1.73366	−0.866830	−	0.498603i	$-0.833846\pi$
$701$	−45.9011	−1.73366	−0.866830	+	0.498603i	$0.833846\pi$
$702$	3.15489	0.119074
$703$	18.4244	0.694889
$704$	2.15895	0.0813684
$705$	0	0
$706$	21.1530	0.796104
$707$	16.1272	0.606526
$708$	0.498994	0.0187533
$709$	−22.1240	−0.830883	−0.415441	−	0.909620i	$-0.636373\pi$
$709$	−22.1240	−0.830883	−0.415441	+	0.909620i	$0.636373\pi$
$710$	0	0
$711$	−1.96500	−0.0736933
$712$	0.0844694	0.00316562
$713$	−24.0572	−0.900950
$714$	9.22832	0.345361
$715$	0	0
$716$	2.73554	0.102232
$717$	−19.2437	−0.718670
$718$	−18.2749	−0.682015
$719$	−32.5266	−1.21304	−0.606519	−	0.795069i	$-0.707435\pi$
$719$	−32.5266	−1.21304	−0.606519	+	0.795069i	$0.707435\pi$
$720$	0	0
$721$	−16.1870	−0.602834
$722$	−7.92467	−0.294926
$723$	2.95458	0.109882
$724$	−35.7235	−1.32765
$725$	0	0
$726$	11.8660	0.440387
$727$	4.35534	0.161531	0.0807653	−	0.996733i	$-0.474264\pi$
$727$	4.35534	0.161531	0.0807653	+	0.996733i	$0.474264\pi$
$728$	2.45786	0.0910943
$729$	19.0711	0.706336
$730$	0	0
$731$	−13.9540	−0.516108
$732$	−37.0140	−1.36808
$733$	−23.0953	−0.853044	−0.426522	−	0.904477i	$-0.640261\pi$
$733$	−23.0953	−0.853044	−0.426522	+	0.904477i	$0.640261\pi$
$734$	−2.37477	−0.0876544
$735$	0	0
$736$	30.7685	1.13414
$737$	−16.9141	−0.623040
$738$	−3.62241	−0.133343
$739$	6.08226	0.223739	0.111870	−	0.993723i	$-0.464316\pi$
$739$	6.08226	0.223739	0.111870	+	0.993723i	$0.464316\pi$
$740$	0	0
$741$	5.28757	0.194244
$742$	4.23475	0.155463
$743$	32.5498	1.19414	0.597069	−	0.802190i	$-0.296332\pi$
$743$	32.5498	1.19414	0.597069	+	0.802190i	$0.296332\pi$
$744$	−21.3504	−0.782744
$745$	0	0
$746$	−20.9961	−0.768721
$747$	9.57756	0.350425
$748$	−15.1738	−0.554809
$749$	−5.15383	−0.188317
$750$	0	0
$751$	35.7706	1.30529	0.652643	−	0.757666i	$-0.273660\pi$
$751$	35.7706	1.30529	0.652643	+	0.757666i	$0.273660\pi$
$752$	0.521899	0.0190317
$753$	9.88361	0.360179
$754$	0.423852	0.0154358
$755$	0	0
$756$	6.80012	0.247318
$757$	−3.25950	−0.118469	−0.0592343	−	0.998244i	$-0.518866\pi$
$757$	−3.25950	−0.118469	−0.0592343	+	0.998244i	$0.518866\pi$
$758$	5.23999	0.190325
$759$	−14.6615	−0.532180
$760$	0	0
$761$	1.73166	0.0627726	0.0313863	−	0.999507i	$-0.490008\pi$
$761$	1.73166	0.0627726	0.0313863	+	0.999507i	$0.490008\pi$
$762$	24.8164	0.899004
$763$	12.9954	0.470465
$764$	30.4964	1.10332
$765$	0	0
$766$	21.8110	0.788063
$767$	0.173303	0.00625762
$768$	19.8335	0.715681
$769$	36.5635	1.31851	0.659256	−	0.751919i	$-0.270871\pi$
$769$	36.5635	1.31851	0.659256	+	0.751919i	$0.270871\pi$
$770$	0	0
$771$	14.3391	0.516410
$772$	−4.70501	−0.169337
$773$	25.8773	0.930741	0.465370	−	0.885116i	$-0.345921\pi$
$773$	25.8773	0.930741	0.465370	+	0.885116i	$0.345921\pi$
$774$	−0.902931	−0.0324552
$775$	0	0
$776$	−6.13266	−0.220150
$777$	−12.6775	−0.454802
$778$	−12.9744	−0.465153
$779$	22.4639	0.804853
$780$	0	0
$781$	23.7325	0.849214
$782$	25.5575	0.913934
$783$	2.72636	0.0974320
$784$	1.29772	0.0463471
$785$	0	0
$786$	3.81566	0.136100
$787$	7.46362	0.266049	0.133025	−	0.991113i	$-0.457531\pi$
$787$	7.46362	0.266049	0.133025	+	0.991113i	$0.457531\pi$
$788$	24.3942	0.869007
$789$	5.86637	0.208848
$790$	0	0
$791$	11.7837	0.418978
$792$	−2.28274	−0.0811137
$793$	−12.8552	−0.456500
$794$	−14.0426	−0.498352
$795$	0	0
$796$	23.4485	0.831109
$797$	31.6976	1.12278	0.561392	−	0.827550i	$-0.310266\pi$
$797$	31.6976	1.12278	0.561392	+	0.827550i	$0.310266\pi$
$798$	−3.70309	−0.131088
$799$	2.77826	0.0982876
$800$	0	0
$801$	−0.0219359	−0.000775067 0
$802$	10.1522	0.358488
$803$	2.61563	0.0923035
$804$	−33.4696	−1.18038
$805$	0	0
$806$	−3.18941	−0.112342
$807$	−23.6933	−0.834044
$808$	39.6384	1.39447
$809$	0.451010	0.0158567	0.00792834	−	0.999969i	$-0.497476\pi$
$809$	0.451010	0.0158567	0.00792834	+	0.999969i	$0.497476\pi$
$810$	0	0
$811$	39.6438	1.39208	0.696040	−	0.718003i	$-0.254944\pi$
$811$	39.6438	1.39208	0.696040	+	0.718003i	$0.254944\pi$
$812$	0.913580	0.0320604
$813$	−29.0855	−1.02007
$814$	−6.77300	−0.237393
$815$	0	0
$816$	−17.0999	−0.598618
$817$	5.59940	0.195898
$818$	−15.2811	−0.534290
$819$	−0.638282	−0.0223034
$820$	0	0
$821$	−55.4660	−1.93578	−0.967889	−	0.251380i	$-0.919116\pi$
$821$	−55.4660	−1.93578	−0.967889	+	0.251380i	$0.919116\pi$
$822$	−8.22775	−0.286976
$823$	13.8217	0.481795	0.240898	−	0.970551i	$-0.422558\pi$
$823$	13.8217	0.481795	0.240898	+	0.970551i	$0.422558\pi$
$824$	−39.7853	−1.38599
$825$	0	0
$826$	−0.121371	−0.00422304
$827$	15.7203	0.546648	0.273324	−	0.961922i	$-0.411877\pi$
$827$	15.7203	0.546648	0.273324	+	0.961922i	$0.411877\pi$
$828$	−5.08976	−0.176881
$829$	12.1723	0.422761	0.211381	−	0.977404i	$-0.432204\pi$
$829$	12.1723	0.422761	0.211381	+	0.977404i	$0.432204\pi$
$830$	0	0
$831$	41.2027	1.42930
$832$	1.48373	0.0514390
$833$	6.90822	0.239356
$834$	12.1199	0.419678
$835$	0	0
$836$	6.08886	0.210588
$837$	−20.5153	−0.709112
$838$	25.3438	0.875486
$839$	52.2198	1.80283	0.901414	−	0.432958i	$-0.142530\pi$
$839$	52.2198	1.80283	0.901414	+	0.432958i	$0.142530\pi$
$840$	0	0
$841$	−28.6337	−0.987370
$842$	−6.62015	−0.228145
$843$	26.7284	0.920574
$844$	−9.67141	−0.332904
$845$	0	0
$846$	0.179774	0.00618076
$847$	8.88274	0.305214
$848$	−7.84693	−0.269465
$849$	56.4234	1.93644
$850$	0	0
$851$	−35.1098	−1.20355
$852$	46.9617	1.60888
$853$	20.6391	0.706670	0.353335	−	0.935497i	$-0.385048\pi$
$853$	20.6391	0.706670	0.353335	+	0.935497i	$0.385048\pi$
$854$	9.00297	0.308075
$855$	0	0
$856$	−12.6674	−0.432962
$857$	−50.1996	−1.71479	−0.857393	−	0.514663i	$-0.827917\pi$
$857$	−50.1996	−1.71479	−0.857393	+	0.514663i	$0.827917\pi$
$858$	−1.94377	−0.0663591
$859$	−50.5385	−1.72435	−0.862176	−	0.506609i	$-0.830899\pi$
$859$	−50.5385	−1.72435	−0.862176	+	0.506609i	$0.830899\pi$
$860$	0	0
$861$	−15.4570	−0.526773
$862$	−5.82904	−0.198538
$863$	26.2768	0.894474	0.447237	−	0.894416i	$-0.352408\pi$
$863$	26.2768	0.894474	0.447237	+	0.894416i	$0.352408\pi$
$864$	26.2385	0.892652
$865$	0	0
$866$	−0.347693	−0.0118151
$867$	−58.6029	−1.99026
$868$	−6.87451	−0.233336
$869$	−4.47958	−0.151959
$870$	0	0
$871$	−11.6242	−0.393870
$872$	31.9408	1.08165
$873$	1.59259	0.0539012
$874$	−10.2556	−0.346900
$875$	0	0
$876$	5.17580	0.174874
$877$	33.1857	1.12060	0.560301	−	0.828289i	$-0.310686\pi$
$877$	33.1857	1.12060	0.560301	+	0.828289i	$0.310686\pi$
$878$	−25.4694	−0.859550
$879$	25.3535	0.855152
$880$	0	0
$881$	−46.7167	−1.57393	−0.786963	−	0.617000i	$-0.788348\pi$
$881$	−46.7167	−1.57393	−0.786963	+	0.617000i	$0.788348\pi$
$882$	0.447014	0.0150517
$883$	−4.10616	−0.138183	−0.0690917	−	0.997610i	$-0.522010\pi$
$883$	−4.10616	−0.138183	−0.0690917	+	0.997610i	$0.522010\pi$
$884$	−10.4281	−0.350736
$885$	0	0
$886$	10.1014	0.339362
$887$	10.5908	0.355605	0.177802	−	0.984066i	$-0.443101\pi$
$887$	10.5908	0.355605	0.177802	+	0.984066i	$0.443101\pi$
$888$	−31.1595	−1.04564
$889$	18.5773	0.623063
$890$	0	0
$891$	−15.2892	−0.512207
$892$	−0.228390	−0.00764705
$893$	−1.11484	−0.0373068
$894$	6.47387	0.216519
$895$	0	0
$896$	10.6100	0.354455
$897$	−10.0761	−0.336431
$898$	−14.1656	−0.472713
$899$	−2.75618	−0.0919238
$900$	0	0
$901$	−41.7721	−1.39163
$902$	−8.25796	−0.274960
$903$	−3.85285	−0.128215
$904$	28.9625	0.963280
$905$	0	0
$906$	22.0998	0.734217
$907$	58.6402	1.94712	0.973558	−	0.228440i	$-0.0733626\pi$
$907$	58.6402	1.94712	0.973558	+	0.228440i	$0.0733626\pi$
$908$	5.84571	0.193997
$909$	−10.2937	−0.341421
$910$	0	0
$911$	−21.7459	−0.720475	−0.360238	−	0.932861i	$-0.617304\pi$
$911$	−21.7459	−0.720475	−0.360238	+	0.932861i	$0.617304\pi$
$912$	6.86178	0.227216
$913$	21.8338	0.722594
$914$	22.9122	0.757867
$915$	0	0
$916$	−28.2676	−0.933989
$917$	2.85636	0.0943254
$918$	21.7947	0.719332
$919$	12.4445	0.410507	0.205254	−	0.978709i	$-0.434198\pi$
$919$	12.4445	0.410507	0.205254	+	0.978709i	$0.434198\pi$
$920$	0	0
$921$	−55.2403	−1.82023
$922$	21.5270	0.708954
$923$	16.3101	0.536852
$924$	−4.18963	−0.137829
$925$	0	0
$926$	−4.09766	−0.134657
$927$	10.3319	0.339343
$928$	3.52508	0.115716
$929$	5.71473	0.187494	0.0937471	−	0.995596i	$-0.470115\pi$
$929$	5.71473	0.187494	0.0937471	+	0.995596i	$0.470115\pi$
$930$	0	0
$931$	−2.77210	−0.0908518
$932$	7.93926	0.260059
$933$	0.0351719	0.00115147
$934$	−2.92960	−0.0958596
$935$	0	0
$936$	−1.56881	−0.0512781
$937$	−3.06267	−0.100053	−0.0500265	−	0.998748i	$-0.515931\pi$
$937$	−3.06267	−0.100053	−0.0500265	+	0.998748i	$0.515931\pi$
$938$	8.14086	0.265808
$939$	30.1568	0.984129
$940$	0	0
$941$	−29.6894	−0.967846	−0.483923	−	0.875111i	$-0.660788\pi$
$941$	−29.6894	−0.967846	−0.483923	+	0.875111i	$0.660788\pi$
$942$	4.31420	0.140564
$943$	−42.8076	−1.39401
$944$	0.224899	0.00731983
$945$	0	0
$946$	−2.05840	−0.0669243
$947$	−58.8922	−1.91374	−0.956870	−	0.290518i	$-0.906173\pi$
$947$	−58.8922	−1.91374	−0.956870	+	0.290518i	$0.906173\pi$
$948$	−8.86418	−0.287895
$949$	1.79758	0.0583520
$950$	0	0
$951$	−58.3711	−1.89281
$952$	16.9794	0.550306
$953$	−11.0841	−0.359048	−0.179524	−	0.983754i	$-0.557456\pi$
$953$	−11.0841	−0.359048	−0.179524	+	0.983754i	$0.557456\pi$
$954$	−2.70297	−0.0875118
$955$	0	0
$956$	−15.2294	−0.492553
$957$	−1.67974	−0.0542983
$958$	−18.3283	−0.592160
$959$	−6.15921	−0.198891
$960$	0	0
$961$	−10.2603	−0.330976
$962$	−4.65472	−0.150074
$963$	3.28960	0.106006
$964$	2.33824	0.0753096
$965$	0	0
$966$	7.05668	0.227045
$967$	−21.3482	−0.686513	−0.343256	−	0.939242i	$-0.611530\pi$
$967$	−21.3482	−0.686513	−0.343256	+	0.939242i	$0.611530\pi$
$968$	21.8325	0.701723
$969$	36.5277	1.17344
$970$	0	0
$971$	39.0975	1.25470	0.627349	−	0.778738i	$-0.284140\pi$
$971$	39.0975	1.25470	0.627349	+	0.778738i	$0.284140\pi$
$972$	−9.85384	−0.316062
$973$	9.07283	0.290862
$974$	−6.77036	−0.216936
$975$	0	0
$976$	−16.6824	−0.533989
$977$	31.6188	1.01158	0.505788	−	0.862658i	$-0.331202\pi$
$977$	31.6188	1.01158	0.505788	+	0.862658i	$0.331202\pi$
$978$	5.22206	0.166983
$979$	−0.0500069	−0.00159823
$980$	0	0
$981$	−8.29473	−0.264830
$982$	22.3694	0.713835
$983$	15.4039	0.491307	0.245653	−	0.969358i	$-0.420997\pi$
$983$	15.4039	0.491307	0.245653	+	0.969358i	$0.420997\pi$
$984$	−37.9911	−1.21111
$985$	0	0
$986$	2.92807	0.0932486
$987$	0.767104	0.0244172
$988$	4.18455	0.133128
$989$	−10.6703	−0.339296
$990$	0	0
$991$	53.9601	1.71410	0.857049	−	0.515235i	$-0.172295\pi$
$991$	53.9601	1.71410	0.857049	+	0.515235i	$0.172295\pi$
$992$	−26.5256	−0.842187
$993$	25.9987	0.825044
$994$	−11.4226	−0.362302
$995$	0	0
$996$	43.2047	1.36899
$997$	−20.7146	−0.656039	−0.328020	−	0.944671i	$-0.606381\pi$
$997$	−20.7146	−0.656039	−0.328020	+	0.944671i	$0.606381\pi$
$998$	9.64295	0.305242
$999$	−29.9406	−0.947280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2275.2.a.y.1.5		7
5.2	odd	4		455.2.c.b.274.10	yes	14
5.3	odd	4		455.2.c.b.274.5	✓	14
5.4	even	2		2275.2.a.w.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
455.2.c.b.274.5	✓	14	5.3	odd	4
455.2.c.b.274.10	yes	14	5.2	odd	4
2275.2.a.w.1.3		7	5.4	even	2
2275.2.a.y.1.5		7	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2275 = 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2275.a (trivial)

\(f(q)\)	\(=\)	\(q+0.700339 q^{2} -1.90743 q^{3} -1.50953 q^{4} -1.33585 q^{6} -1.00000 q^{7} -2.45786 q^{8} +0.638282 q^{9} +1.45508 q^{11} +2.87931 q^{12} +1.00000 q^{13} -0.700339 q^{14} +1.29772 q^{16} +6.90822 q^{17} +0.447014 q^{18} -2.77210 q^{19} +1.90743 q^{21} +1.01905 q^{22} +5.28255 q^{23} +4.68819 q^{24} +0.700339 q^{26} +4.50481 q^{27} +1.50953 q^{28} +0.605210 q^{29} -4.55409 q^{31} +5.82456 q^{32} -2.77546 q^{33} +4.83810 q^{34} -0.963503 q^{36} -6.64638 q^{37} -1.94141 q^{38} -1.90743 q^{39} -8.10358 q^{41} +1.33585 q^{42} -2.01992 q^{43} -2.19648 q^{44} +3.69958 q^{46} +0.402167 q^{47} -2.47530 q^{48} +1.00000 q^{49} -13.1769 q^{51} -1.50953 q^{52} -6.04672 q^{53} +3.15489 q^{54} +2.45786 q^{56} +5.28757 q^{57} +0.423852 q^{58} +0.173303 q^{59} -12.8552 q^{61} -3.18941 q^{62} -0.638282 q^{63} +1.48373 q^{64} -1.94377 q^{66} -11.6242 q^{67} -10.4281 q^{68} -10.0761 q^{69} +16.3101 q^{71} -1.56881 q^{72} +1.79758 q^{73} -4.65472 q^{74} +4.18455 q^{76} -1.45508 q^{77} -1.33585 q^{78} -3.07858 q^{79} -10.5074 q^{81} -5.67525 q^{82} +15.0052 q^{83} -2.87931 q^{84} -1.41463 q^{86} -1.15439 q^{87} -3.57638 q^{88} -0.0343671 q^{89} -1.00000 q^{91} -7.97415 q^{92} +8.68660 q^{93} +0.281653 q^{94} -11.1099 q^{96} +2.49513 q^{97} +0.700339 q^{98} +0.928753 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{3} + 2 q^{4} - 8 q^{6} - 7 q^{7} + 6 q^{8} + q^{9} - 6 q^{11} - 3 q^{12} + 7 q^{13} - 4 q^{16} - 2 q^{17} + q^{18} - 10 q^{19} - 2 q^{21} - 18 q^{22} - 10 q^{23} - 5 q^{24} + 8 q^{27} - 2 q^{28}+ \cdots - 18 q^{99}+O(q^{100}) \) 7 * q + 2 * q^3 + 2 * q^4 - 8 * q^6 - 7 * q^7 + 6 * q^8 + q^9 - 6 * q^11 - 3 * q^12 + 7 * q^13 - 4 * q^16 - 2 * q^17 + q^18 - 10 * q^19 - 2 * q^21 - 18 * q^22 - 10 * q^23 - 5 * q^24 + 8 * q^27 - 2 * q^28 - 26 * q^29 - 6 * q^31 + 7 * q^32 - 4 * q^33 - q^34 - 6 * q^36 + 4 * q^37 + q^38 + 2 * q^39 - 28 * q^41 + 8 * q^42 - 2 * q^43 - 18 * q^46 + 20 * q^47 - 20 * q^48 + 7 * q^49 - 32 * q^51 + 2 * q^52 - 8 * q^53 + 11 * q^54 - 6 * q^56 - 14 * q^57 + 12 * q^58 - 28 * q^59 - 26 * q^61 + 22 * q^62 - q^63 - 8 * q^64 + 2 * q^66 - 2 * q^67 + 2 * q^68 - 22 * q^69 - 14 * q^71 - 4 * q^72 + 2 * q^73 - 12 * q^74 - 36 * q^76 + 6 * q^77 - 8 * q^78 + 14 * q^79 - 33 * q^81 - 19 * q^82 + 28 * q^83 + 3 * q^84 - 4 * q^86 - 22 * q^87 - 24 * q^88 - 30 * q^89 - 7 * q^91 - 42 * q^92 - 2 * q^93 + 14 * q^94 + 18 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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