LMFDB - Embedded newform 2275.2.a.y.1.1

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			1.1
Root			$-1.92480$ 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)

$f(q)$	$=$	\(q-1.92480 q^{2} +2.17582 q^{3} +1.70485 q^{4} -4.18801 q^{6} -1.00000 q^{7} +0.568104 q^{8} +1.73417 q^{9} +2.81053 q^{11} +3.70944 q^{12} +1.00000 q^{13} +1.92480 q^{14} -4.50319 q^{16} +1.19307 q^{17} -3.33794 q^{18} -6.29938 q^{19} -2.17582 q^{21} -5.40970 q^{22} -8.36458 q^{23} +1.23609 q^{24} -1.92480 q^{26} -2.75420 q^{27} -1.70485 q^{28} -8.83084 q^{29} -10.4001 q^{31} +7.53152 q^{32} +6.11519 q^{33} -2.29642 q^{34} +2.95651 q^{36} -3.26617 q^{37} +12.1250 q^{38} +2.17582 q^{39} +8.99882 q^{41} +4.18801 q^{42} +7.66037 q^{43} +4.79153 q^{44} +16.1001 q^{46} -0.713672 q^{47} -9.79810 q^{48} +1.00000 q^{49} +2.59590 q^{51} +1.70485 q^{52} -3.33974 q^{53} +5.30129 q^{54} -0.568104 q^{56} -13.7063 q^{57} +16.9976 q^{58} -4.96578 q^{59} -13.9960 q^{61} +20.0181 q^{62} -1.73417 q^{63} -5.49029 q^{64} -11.7705 q^{66} +15.0427 q^{67} +2.03401 q^{68} -18.1998 q^{69} -2.79774 q^{71} +0.985191 q^{72} +5.39152 q^{73} +6.28671 q^{74} -10.7395 q^{76} -2.81053 q^{77} -4.18801 q^{78} +3.24377 q^{79} -11.1952 q^{81} -17.3209 q^{82} +13.0623 q^{83} -3.70944 q^{84} -14.7447 q^{86} -19.2143 q^{87} +1.59667 q^{88} -5.05128 q^{89} -1.00000 q^{91} -14.2604 q^{92} -22.6287 q^{93} +1.37368 q^{94} +16.3872 q^{96} +5.50717 q^{97} -1.92480 q^{98} +4.87395 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.92480	−1.36104	−0.680519	−	0.732730i	$-0.738246\pi$
$2$	−1.92480	−1.36104	−0.680519	+	0.732730i	$0.738246\pi$
$3$	2.17582	1.25621	0.628104	−	0.778129i	$-0.283831\pi$
$3$	2.17582	1.25621	0.628104	+	0.778129i	$0.283831\pi$
$4$	1.70485	0.852425
$5$	0	0
$5$	0	0
$6$	−4.18801	−1.70975
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.568104	0.200855
$9$	1.73417	0.578058
$10$	0	0
$11$	2.81053	0.847406	0.423703	−	0.905801i	$-0.360730\pi$
$11$	2.81053	0.847406	0.423703	+	0.905801i	$0.360730\pi$
$12$	3.70944	1.07082
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.92480	0.514424
$15$	0	0
$16$	−4.50319	−1.12580
$17$	1.19307	0.289362	0.144681	−	0.989478i	$-0.453784\pi$
$17$	1.19307	0.289362	0.144681	+	0.989478i	$0.453784\pi$
$18$	−3.33794	−0.786759
$19$	−6.29938	−1.44518	−0.722588	−	0.691279i	$-0.757048\pi$
$19$	−6.29938	−1.44518	−0.722588	+	0.691279i	$0.757048\pi$
$20$	0	0
$21$	−2.17582	−0.474802
$22$	−5.40970	−1.15335
$23$	−8.36458	−1.74414	−0.872068	−	0.489385i	$-0.837221\pi$
$23$	−8.36458	−1.74414	−0.872068	+	0.489385i	$0.837221\pi$
$24$	1.23609	0.252316
$25$	0	0
$26$	−1.92480	−0.377484
$27$	−2.75420	−0.530047
$28$	−1.70485	−0.322186
$29$	−8.83084	−1.63985	−0.819923	−	0.572474i	$-0.805984\pi$
$29$	−8.83084	−1.63985	−0.819923	+	0.572474i	$0.805984\pi$
$30$	0	0
$31$	−10.4001	−1.86792	−0.933959	−	0.357381i	$-0.883670\pi$
$31$	−10.4001	−1.86792	−0.933959	+	0.357381i	$0.883670\pi$
$32$	7.53152	1.33140
$33$	6.11519	1.06452
$34$	−2.29642	−0.393833
$35$	0	0
$36$	2.95651	0.492751
$37$	−3.26617	−0.536955	−0.268477	−	0.963286i	$-0.586520\pi$
$37$	−3.26617	−0.536955	−0.268477	+	0.963286i	$0.586520\pi$
$38$	12.1250	1.96694
$39$	2.17582	0.348409
$40$	0	0
$41$	8.99882	1.40538	0.702690	−	0.711496i	$-0.251982\pi$
$41$	8.99882	1.40538	0.702690	+	0.711496i	$0.251982\pi$
$42$	4.18801	0.646224
$43$	7.66037	1.16820	0.584098	−	0.811683i	$-0.301448\pi$
$43$	7.66037	1.16820	0.584098	+	0.811683i	$0.301448\pi$
$44$	4.79153	0.722350
$45$	0	0
$46$	16.1001	2.37384
$47$	−0.713672	−0.104100	−0.0520499	−	0.998644i	$-0.516575\pi$
$47$	−0.713672	−0.104100	−0.0520499	+	0.998644i	$0.516575\pi$
$48$	−9.79810	−1.41423
$49$	1.00000	0.142857
$50$	0	0
$51$	2.59590	0.363499
$52$	1.70485	0.236420
$53$	−3.33974	−0.458749	−0.229374	−	0.973338i	$-0.573668\pi$
$53$	−3.33974	−0.458749	−0.229374	+	0.973338i	$0.573668\pi$
$54$	5.30129	0.721414
$55$	0	0
$56$	−0.568104	−0.0759161
$57$	−13.7063	−1.81544
$58$	16.9976	2.23189
$59$	−4.96578	−0.646489	−0.323245	−	0.946315i	$-0.604774\pi$
$59$	−4.96578	−0.646489	−0.323245	+	0.946315i	$0.604774\pi$
$60$	0	0
$61$	−13.9960	−1.79201	−0.896005	−	0.444044i	$-0.853543\pi$
$61$	−13.9960	−1.79201	−0.896005	+	0.444044i	$0.853543\pi$
$62$	20.0181	2.54231
$63$	−1.73417	−0.218485
$64$	−5.49029	−0.686286
$65$	0	0
$66$	−11.7705	−1.44885
$67$	15.0427	1.83776	0.918879	−	0.394540i	$-0.129096\pi$
$67$	15.0427	1.83776	0.918879	+	0.394540i	$0.129096\pi$
$68$	2.03401	0.246660
$69$	−18.1998	−2.19100
$70$	0	0
$71$	−2.79774	−0.332030	−0.166015	−	0.986123i	$-0.553090\pi$
$71$	−2.79774	−0.332030	−0.166015	+	0.986123i	$0.553090\pi$
$72$	0.985191	0.116106
$73$	5.39152	0.631029	0.315515	−	0.948921i	$-0.397823\pi$
$73$	5.39152	0.631029	0.315515	+	0.948921i	$0.397823\pi$
$74$	6.28671	0.730816
$75$	0	0
$76$	−10.7395	−1.23190
$77$	−2.81053	−0.320289
$78$	−4.18801	−0.474198
$79$	3.24377	0.364953	0.182477	−	0.983210i	$-0.441589\pi$
$79$	3.24377	0.364953	0.182477	+	0.983210i	$0.441589\pi$
$80$	0	0
$81$	−11.1952	−1.24391
$82$	−17.3209	−1.91278
$83$	13.0623	1.43377	0.716887	−	0.697190i	$-0.245566\pi$
$83$	13.0623	1.43377	0.716887	+	0.697190i	$0.245566\pi$
$84$	−3.70944	−0.404733
$85$	0	0
$86$	−14.7447	−1.58996
$87$	−19.2143	−2.05999
$88$	1.59667	0.170206
$89$	−5.05128	−0.535434	−0.267717	−	0.963498i	$-0.586269\pi$
$89$	−5.05128	−0.535434	−0.267717	+	0.963498i	$0.586269\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−14.2604	−1.48675
$93$	−22.6287	−2.34649
$94$	1.37368	0.141684
$95$	0	0
$96$	16.3872	1.67251
$97$	5.50717	0.559169	0.279584	−	0.960121i	$-0.409803\pi$
$97$	5.50717	0.559169	0.279584	+	0.960121i	$0.409803\pi$
$98$	−1.92480	−0.194434
$99$	4.87395	0.489850
$100$	0	0
$101$	−8.96883	−0.892432	−0.446216	−	0.894925i	$-0.647229\pi$
$101$	−8.96883	−0.892432	−0.446216	+	0.894925i	$0.647229\pi$
$102$	−4.99659	−0.494736
$103$	4.36408	0.430005	0.215003	−	0.976613i	$-0.431024\pi$
$103$	4.36408	0.430005	0.215003	+	0.976613i	$0.431024\pi$
$104$	0.568104	0.0557072
$105$	0	0
$106$	6.42833	0.624375
$107$	−12.0245	−1.16245	−0.581224	−	0.813743i	$-0.697426\pi$
$107$	−12.0245	−1.16245	−0.581224	+	0.813743i	$0.697426\pi$
$108$	−4.69550	−0.451825
$109$	−4.55332	−0.436129	−0.218064	−	0.975934i	$-0.569974\pi$
$109$	−4.55332	−0.436129	−0.218064	+	0.975934i	$0.569974\pi$
$110$	0	0
$111$	−7.10658	−0.674526
$112$	4.50319	0.425511
$113$	−9.21896	−0.867247	−0.433623	−	0.901094i	$-0.642765\pi$
$113$	−9.21896	−0.867247	−0.433623	+	0.901094i	$0.642765\pi$
$114$	26.3818	2.47089
$115$	0	0
$116$	−15.0553	−1.39785
$117$	1.73417	0.160324
$118$	9.55812	0.879896
$119$	−1.19307	−0.109369
$120$	0	0
$121$	−3.10093	−0.281903
$122$	26.9396	2.43899
$123$	19.5798	1.76545
$124$	−17.7306	−1.59226
$125$	0	0
$126$	3.33794	0.297367
$127$	−3.94474	−0.350039	−0.175019	−	0.984565i	$-0.555999\pi$
$127$	−3.94474	−0.350039	−0.175019	+	0.984565i	$0.555999\pi$
$128$	−4.49534	−0.397336
$129$	16.6676	1.46750
$130$	0	0
$131$	6.07419	0.530704	0.265352	−	0.964152i	$-0.414512\pi$
$131$	6.07419	0.530704	0.265352	+	0.964152i	$0.414512\pi$
$132$	10.4255	0.907422
$133$	6.29938	0.546225
$134$	−28.9542	−2.50126
$135$	0	0
$136$	0.677788	0.0581199
$137$	−2.14481	−0.183244	−0.0916219	−	0.995794i	$-0.529205\pi$
$137$	−2.14481	−0.183244	−0.0916219	+	0.995794i	$0.529205\pi$
$138$	35.0309	2.98203
$139$	−1.86383	−0.158088	−0.0790439	−	0.996871i	$-0.525187\pi$
$139$	−1.86383	−0.158088	−0.0790439	+	0.996871i	$0.525187\pi$
$140$	0	0
$141$	−1.55282	−0.130771
$142$	5.38508	0.451906
$143$	2.81053	0.235028
$144$	−7.80931	−0.650776
$145$	0	0
$146$	−10.3776	−0.858855
$147$	2.17582	0.179458
$148$	−5.56833	−0.457714
$149$	−12.0952	−0.990875	−0.495437	−	0.868644i	$-0.664992\pi$
$149$	−12.0952	−0.990875	−0.495437	+	0.868644i	$0.664992\pi$
$150$	0	0
$151$	11.7598	0.957002	0.478501	−	0.878087i	$-0.341180\pi$
$151$	11.7598	0.957002	0.478501	+	0.878087i	$0.341180\pi$
$152$	−3.57870	−0.290271
$153$	2.06899	0.167268
$154$	5.40970	0.435926
$155$	0	0
$156$	3.70944	0.296993
$157$	5.18116	0.413502	0.206751	−	0.978394i	$-0.433711\pi$
$157$	5.18116	0.413502	0.206751	+	0.978394i	$0.433711\pi$
$158$	−6.24361	−0.496715
$159$	−7.26666	−0.576284
$160$	0	0
$161$	8.36458	0.659221
$162$	21.5484	1.69300
$163$	−5.74455	−0.449948	−0.224974	−	0.974365i	$-0.572230\pi$
$163$	−5.74455	−0.449948	−0.224974	+	0.974365i	$0.572230\pi$
$164$	15.3416	1.19798
$165$	0	0
$166$	−25.1423	−1.95142
$167$	−3.93279	−0.304328	−0.152164	−	0.988355i	$-0.548624\pi$
$167$	−3.93279	−0.304328	−0.152164	+	0.988355i	$0.548624\pi$
$168$	−1.23609	−0.0953664
$169$	1.00000	0.0769231
$170$	0	0
$171$	−10.9242	−0.835396
$172$	13.0598	0.995799
$173$	−21.1392	−1.60718	−0.803592	−	0.595181i	$-0.797080\pi$
$173$	−21.1392	−1.60718	−0.803592	+	0.595181i	$0.797080\pi$
$174$	36.9836	2.80372
$175$	0	0
$176$	−12.6563	−0.954007
$177$	−10.8046	−0.812125
$178$	9.72269	0.728747
$179$	8.90147	0.665327	0.332664	−	0.943046i	$-0.392053\pi$
$179$	8.90147	0.665327	0.332664	+	0.943046i	$0.392053\pi$
$180$	0	0
$181$	8.19773	0.609333	0.304666	−	0.952459i	$-0.401455\pi$
$181$	8.19773	0.609333	0.304666	+	0.952459i	$0.401455\pi$
$182$	1.92480	0.142676
$183$	−30.4528	−2.25114
$184$	−4.75195	−0.350319
$185$	0	0
$186$	43.5558	3.19367
$187$	3.35316	0.245207
$188$	−1.21670	−0.0887373
$189$	2.75420	0.200339
$190$	0	0
$191$	17.7302	1.28291	0.641457	−	0.767159i	$-0.278330\pi$
$191$	17.7302	1.28291	0.641457	+	0.767159i	$0.278330\pi$
$192$	−11.9459	−0.862118
$193$	−9.34627	−0.672759	−0.336380	−	0.941726i	$-0.609203\pi$
$193$	−9.34627	−0.672759	−0.336380	+	0.941726i	$0.609203\pi$
$194$	−10.6002	−0.761050
$195$	0	0
$196$	1.70485	0.121775
$197$	16.2254	1.15601	0.578007	−	0.816032i	$-0.303830\pi$
$197$	16.2254	1.15601	0.578007	+	0.816032i	$0.303830\pi$
$198$	−9.38136	−0.666704
$199$	3.26666	0.231568	0.115784	−	0.993274i	$-0.463062\pi$
$199$	3.26666	0.231568	0.115784	+	0.993274i	$0.463062\pi$
$200$	0	0
$201$	32.7301	2.30861
$202$	17.2632	1.21463
$203$	8.83084	0.619803
$204$	4.42562	0.309856
$205$	0	0
$206$	−8.39997	−0.585253
$207$	−14.5056	−1.00821
$208$	−4.50319	−0.312240
$209$	−17.7046	−1.22465
$210$	0	0
$211$	−0.943434	−0.0649487	−0.0324744	−	0.999473i	$-0.510339\pi$
$211$	−0.943434	−0.0649487	−0.0324744	+	0.999473i	$0.510339\pi$
$212$	−5.69376	−0.391049
$213$	−6.08736	−0.417099
$214$	23.1447	1.58214
$215$	0	0
$216$	−1.56467	−0.106463
$217$	10.4001	0.706006
$218$	8.76422	0.593588
$219$	11.7309	0.792704
$220$	0	0
$221$	1.19307	0.0802546
$222$	13.6787	0.918056
$223$	11.8394	0.792828	0.396414	−	0.918072i	$-0.370255\pi$
$223$	11.8394	0.792828	0.396414	+	0.918072i	$0.370255\pi$
$224$	−7.53152	−0.503221
$225$	0	0
$226$	17.7446	1.18036
$227$	−24.1996	−1.60618	−0.803091	−	0.595856i	$-0.796813\pi$
$227$	−24.1996	−1.60618	−0.803091	+	0.595856i	$0.796813\pi$
$228$	−23.3672	−1.54753
$229$	5.12821	0.338882	0.169441	−	0.985540i	$-0.445804\pi$
$229$	5.12821	0.338882	0.169441	+	0.985540i	$0.445804\pi$
$230$	0	0
$231$	−6.11519	−0.402350
$232$	−5.01684	−0.329371
$233$	−3.79626	−0.248701	−0.124351	−	0.992238i	$-0.539685\pi$
$233$	−3.79626	−0.248701	−0.124351	+	0.992238i	$0.539685\pi$
$234$	−3.33794	−0.218208
$235$	0	0
$236$	−8.46591	−0.551083
$237$	7.05786	0.458457
$238$	2.29642	0.148855
$239$	−23.0656	−1.49199	−0.745995	−	0.665952i	$-0.768026\pi$
$239$	−23.0656	−1.49199	−0.745995	+	0.665952i	$0.768026\pi$
$240$	0	0
$241$	17.5163	1.12833	0.564163	−	0.825664i	$-0.309199\pi$
$241$	17.5163	1.12833	0.564163	+	0.825664i	$0.309199\pi$
$242$	5.96867	0.383681
$243$	−16.0960	−1.03256
$244$	−23.8612	−1.52755
$245$	0	0
$246$	−37.6871	−2.40284
$247$	−6.29938	−0.400820
$248$	−5.90835	−0.375181
$249$	28.4212	1.80112
$250$	0	0
$251$	−11.1841	−0.705933	−0.352966	−	0.935636i	$-0.614827\pi$
$251$	−11.1841	−0.705933	−0.352966	+	0.935636i	$0.614827\pi$
$252$	−2.95651	−0.186242
$253$	−23.5089	−1.47799
$254$	7.59283	0.476416
$255$	0	0
$256$	19.6332	1.22708
$257$	26.2399	1.63680	0.818402	−	0.574647i	$-0.194861\pi$
$257$	26.2399	1.63680	0.818402	+	0.574647i	$0.194861\pi$
$258$	−32.0817	−1.99732
$259$	3.26617	0.202950
$260$	0	0
$261$	−15.3142	−0.947926
$262$	−11.6916	−0.722309
$263$	−8.21782	−0.506733	−0.253366	−	0.967370i	$-0.581538\pi$
$263$	−8.21782	−0.506733	−0.253366	+	0.967370i	$0.581538\pi$
$264$	3.47406	0.213814
$265$	0	0
$266$	−12.1250	−0.743434
$267$	−10.9907	−0.672617
$268$	25.6455	1.56655
$269$	−1.32400	−0.0807259	−0.0403629	−	0.999185i	$-0.512851\pi$
$269$	−1.32400	−0.0807259	−0.0403629	+	0.999185i	$0.512851\pi$
$270$	0	0
$271$	20.5385	1.24762	0.623812	−	0.781575i	$-0.285583\pi$
$271$	20.5385	1.24762	0.623812	+	0.781575i	$0.285583\pi$
$272$	−5.37262	−0.325763
$273$	−2.17582	−0.131686
$274$	4.12834	0.249402
$275$	0	0
$276$	−31.0279	−1.86766
$277$	−0.773982	−0.0465041	−0.0232520	−	0.999730i	$-0.507402\pi$
$277$	−0.773982	−0.0465041	−0.0232520	+	0.999730i	$0.507402\pi$
$278$	3.58749	0.215164
$279$	−18.0356	−1.07976
$280$	0	0
$281$	−24.7568	−1.47687	−0.738435	−	0.674325i	$-0.764435\pi$
$281$	−24.7568	−1.47687	−0.738435	+	0.674325i	$0.764435\pi$
$282$	2.98886	0.177984
$283$	−26.6656	−1.58511	−0.792553	−	0.609803i	$-0.791249\pi$
$283$	−26.6656	−1.58511	−0.792553	+	0.609803i	$0.791249\pi$
$284$	−4.76972	−0.283031
$285$	0	0
$286$	−5.40970	−0.319882
$287$	−8.99882	−0.531184
$288$	13.0610	0.769625
$289$	−15.5766	−0.916270
$290$	0	0
$291$	11.9826	0.702432
$292$	9.19173	0.537905
$293$	−1.73248	−0.101213	−0.0506063	−	0.998719i	$-0.516115\pi$
$293$	−1.73248	−0.101213	−0.0506063	+	0.998719i	$0.516115\pi$
$294$	−4.18801	−0.244250
$295$	0	0
$296$	−1.85552	−0.107850
$297$	−7.74076	−0.449165
$298$	23.2808	1.34862
$299$	−8.36458	−0.483736
$300$	0	0
$301$	−7.66037	−0.441537
$302$	−22.6353	−1.30252
$303$	−19.5145	−1.12108
$304$	28.3673	1.62697
$305$	0	0
$306$	−3.98239	−0.227658
$307$	−2.96635	−0.169298	−0.0846492	−	0.996411i	$-0.526977\pi$
$307$	−2.96635	−0.169298	−0.0846492	+	0.996411i	$0.526977\pi$
$308$	−4.79153	−0.273023
$309$	9.49542	0.540176
$310$	0	0
$311$	−16.6681	−0.945163	−0.472581	−	0.881287i	$-0.656678\pi$
$311$	−16.6681	−0.945163	−0.472581	+	0.881287i	$0.656678\pi$
$312$	1.23609	0.0699798
$313$	−6.90930	−0.390537	−0.195268	−	0.980750i	$-0.562558\pi$
$313$	−6.90930	−0.390537	−0.195268	+	0.980750i	$0.562558\pi$
$314$	−9.97270	−0.562792
$315$	0	0
$316$	5.53015	0.311095
$317$	0.521668	0.0292998	0.0146499	−	0.999893i	$-0.495337\pi$
$317$	0.521668	0.0292998	0.0146499	+	0.999893i	$0.495337\pi$
$318$	13.9869	0.784345
$319$	−24.8193	−1.38962
$320$	0	0
$321$	−26.1630	−1.46028
$322$	−16.1001	−0.897226
$323$	−7.51560	−0.418179
$324$	−19.0861	−1.06034
$325$	0	0
$326$	11.0571	0.612397
$327$	−9.90718	−0.547868
$328$	5.11227	0.282278
$329$	0.713672	0.0393460
$330$	0	0
$331$	−0.865626	−0.0475791	−0.0237895	−	0.999717i	$-0.507573\pi$
$331$	−0.865626	−0.0475791	−0.0237895	+	0.999717i	$0.507573\pi$
$332$	22.2693	1.22218
$333$	−5.66410	−0.310391
$334$	7.56983	0.414202
$335$	0	0
$336$	9.79810	0.534530
$337$	10.8695	0.592099	0.296049	−	0.955173i	$-0.404331\pi$
$337$	10.8695	0.592099	0.296049	+	0.955173i	$0.404331\pi$
$338$	−1.92480	−0.104695
$339$	−20.0588	−1.08944
$340$	0	0
$341$	−29.2298	−1.58288
$342$	21.0269	1.13701
$343$	−1.00000	−0.0539949
$344$	4.35189	0.234638
$345$	0	0
$346$	40.6887	2.18744
$347$	−35.5330	−1.90751	−0.953755	−	0.300586i	$-0.902818\pi$
$347$	−35.5330	−1.90751	−0.953755	+	0.300586i	$0.902818\pi$
$348$	−32.7575	−1.75598
$349$	−5.15115	−0.275735	−0.137867	−	0.990451i	$-0.544025\pi$
$349$	−5.15115	−0.275735	−0.137867	+	0.990451i	$0.544025\pi$
$350$	0	0
$351$	−2.75420	−0.147008
$352$	21.1675	1.12823
$353$	19.3223	1.02842	0.514210	−	0.857664i	$-0.328085\pi$
$353$	19.3223	1.02842	0.514210	+	0.857664i	$0.328085\pi$
$354$	20.7967	1.10533
$355$	0	0
$356$	−8.61167	−0.456418
$357$	−2.59590	−0.137390
$358$	−17.1335	−0.905536
$359$	27.3104	1.44139	0.720695	−	0.693252i	$-0.243823\pi$
$359$	27.3104	1.44139	0.720695	+	0.693252i	$0.243823\pi$
$360$	0	0
$361$	20.6822	1.08854
$362$	−15.7790	−0.829325
$363$	−6.74706	−0.354129
$364$	−1.70485	−0.0893584
$365$	0	0
$366$	58.6155	3.06388
$367$	−21.5189	−1.12328	−0.561639	−	0.827382i	$-0.689829\pi$
$367$	−21.5189	−1.12328	−0.561639	+	0.827382i	$0.689829\pi$
$368$	37.6673	1.96354
$369$	15.6055	0.812391
$370$	0	0
$371$	3.33974	0.173391
$372$	−38.5786	−2.00021
$373$	13.1467	0.680712	0.340356	−	0.940297i	$-0.389452\pi$
$373$	13.1467	0.680712	0.340356	+	0.940297i	$0.389452\pi$
$374$	−6.45415	−0.333736
$375$	0	0
$376$	−0.405440	−0.0209090
$377$	−8.83084	−0.454811
$378$	−5.30129	−0.272669
$379$	13.5057	0.693739	0.346869	−	0.937913i	$-0.387245\pi$
$379$	13.5057	0.693739	0.346869	+	0.937913i	$0.387245\pi$
$380$	0	0
$381$	−8.58302	−0.439722
$382$	−34.1271	−1.74610
$383$	6.29376	0.321596	0.160798	−	0.986987i	$-0.448593\pi$
$383$	6.29376	0.321596	0.160798	+	0.986987i	$0.448593\pi$
$384$	−9.78104	−0.499137
$385$	0	0
$386$	17.9897	0.915651
$387$	13.2844	0.675285
$388$	9.38890	0.476649
$389$	−1.06542	−0.0540187	−0.0270094	−	0.999635i	$-0.508598\pi$
$389$	−1.06542	−0.0540187	−0.0270094	+	0.999635i	$0.508598\pi$
$390$	0	0
$391$	−9.97954	−0.504687
$392$	0.568104	0.0286936
$393$	13.2163	0.666675
$394$	−31.2307	−1.57338
$395$	0	0
$396$	8.30935	0.417560
$397$	26.4337	1.32667	0.663336	−	0.748322i	$-0.269140\pi$
$397$	26.4337	1.32667	0.663336	+	0.748322i	$0.269140\pi$
$398$	−6.28767	−0.315173
$399$	13.7063	0.686173
$400$	0	0
$401$	−8.35061	−0.417010	−0.208505	−	0.978021i	$-0.566860\pi$
$401$	−8.35061	−0.417010	−0.208505	+	0.978021i	$0.566860\pi$
$402$	−62.9989	−3.14210
$403$	−10.4001	−0.518067
$404$	−15.2905	−0.760731
$405$	0	0
$406$	−16.9976	−0.843576
$407$	−9.17965	−0.455018
$408$	1.47474	0.0730106
$409$	−3.53087	−0.174590	−0.0872952	−	0.996182i	$-0.527822\pi$
$409$	−3.53087	−0.174590	−0.0872952	+	0.996182i	$0.527822\pi$
$410$	0	0
$411$	−4.66672	−0.230192
$412$	7.44009	0.366547
$413$	4.96578	0.244350
$414$	27.9204	1.37222
$415$	0	0
$416$	7.53152	0.369263
$417$	−4.05535	−0.198591
$418$	34.0778	1.66680
$419$	−10.8978	−0.532391	−0.266196	−	0.963919i	$-0.585767\pi$
$419$	−10.8978	−0.532391	−0.266196	+	0.963919i	$0.585767\pi$
$420$	0	0
$421$	9.30767	0.453628	0.226814	−	0.973938i	$-0.427169\pi$
$421$	9.30767	0.453628	0.226814	+	0.973938i	$0.427169\pi$
$422$	1.81592	0.0883977
$423$	−1.23763	−0.0601757
$424$	−1.89732	−0.0921421
$425$	0	0
$426$	11.7169	0.567688
$427$	13.9960	0.677316
$428$	−20.4999	−0.990900
$429$	6.11519	0.295244
$430$	0	0
$431$	23.1386	1.11455	0.557273	−	0.830329i	$-0.311848\pi$
$431$	23.1386	1.11455	0.557273	+	0.830329i	$0.311848\pi$
$432$	12.4027	0.596725
$433$	−9.82823	−0.472314	−0.236157	−	0.971715i	$-0.575888\pi$
$433$	−9.82823	−0.472314	−0.236157	+	0.971715i	$0.575888\pi$
$434$	−20.0181	−0.960902
$435$	0	0
$436$	−7.76272	−0.371767
$437$	52.6917	2.52058
$438$	−22.5797	−1.07890
$439$	27.8149	1.32753	0.663766	−	0.747940i	$-0.268957\pi$
$439$	27.8149	1.32753	0.663766	+	0.747940i	$0.268957\pi$
$440$	0	0
$441$	1.73417	0.0825797
$442$	−2.29642	−0.109230
$443$	34.8056	1.65367	0.826833	−	0.562448i	$-0.190140\pi$
$443$	34.8056	1.65367	0.826833	+	0.562448i	$0.190140\pi$
$444$	−12.1156	−0.574983
$445$	0	0
$446$	−22.7885	−1.07907
$447$	−26.3169	−1.24474
$448$	5.49029	0.259392
$449$	−28.3071	−1.33589	−0.667947	−	0.744209i	$-0.732827\pi$
$449$	−28.3071	−1.33589	−0.667947	+	0.744209i	$0.732827\pi$
$450$	0	0
$451$	25.2914	1.19093
$452$	−15.7169	−0.739263
$453$	25.5872	1.20219
$454$	46.5793	2.18608
$455$	0	0
$456$	−7.78660	−0.364641
$457$	−8.56304	−0.400562	−0.200281	−	0.979738i	$-0.564186\pi$
$457$	−8.56304	−0.400562	−0.200281	+	0.979738i	$0.564186\pi$
$458$	−9.87078	−0.461231
$459$	−3.28596	−0.153375
$460$	0	0
$461$	−25.8221	−1.20265	−0.601327	−	0.799003i	$-0.705361\pi$
$461$	−25.8221	−1.20265	−0.601327	+	0.799003i	$0.705361\pi$
$462$	11.7705	0.547614
$463$	36.7093	1.70602	0.853012	−	0.521891i	$-0.174773\pi$
$463$	36.7093	1.70602	0.853012	+	0.521891i	$0.174773\pi$
$464$	39.7669	1.84613
$465$	0	0
$466$	7.30704	0.338492
$467$	6.06854	0.280818	0.140409	−	0.990094i	$-0.455158\pi$
$467$	6.06854	0.280818	0.140409	+	0.990094i	$0.455158\pi$
$468$	2.95651	0.136665
$469$	−15.0427	−0.694607
$470$	0	0
$471$	11.2733	0.519444
$472$	−2.82108	−0.129851
$473$	21.5297	0.989936
$474$	−13.5850	−0.623978
$475$	0	0
$476$	−2.03401	−0.0932285
$477$	−5.79170	−0.265184
$478$	44.3966	2.03065
$479$	3.49589	0.159731	0.0798657	−	0.996806i	$-0.474551\pi$
$479$	3.49589	0.159731	0.0798657	+	0.996806i	$0.474551\pi$
$480$	0	0
$481$	−3.26617	−0.148924
$482$	−33.7154	−1.53569
$483$	18.1998	0.828119
$484$	−5.28663	−0.240301
$485$	0	0
$486$	30.9816	1.40535
$487$	16.5742	0.751047	0.375524	−	0.926813i	$-0.377463\pi$
$487$	16.5742	0.751047	0.375524	+	0.926813i	$0.377463\pi$
$488$	−7.95121	−0.359934
$489$	−12.4991	−0.565228
$490$	0	0
$491$	36.6167	1.65249	0.826245	−	0.563311i	$-0.190473\pi$
$491$	36.6167	1.65249	0.826245	+	0.563311i	$0.190473\pi$
$492$	33.3806	1.50491
$493$	−10.5358	−0.474509
$494$	12.1250	0.545531
$495$	0	0
$496$	46.8337	2.10289
$497$	2.79774	0.125496
$498$	−54.7050	−2.45139
$499$	18.3547	0.821669	0.410835	−	0.911710i	$-0.365237\pi$
$499$	18.3547	0.821669	0.410835	+	0.911710i	$0.365237\pi$
$500$	0	0
$501$	−8.55702	−0.382300
$502$	21.5271	0.960801
$503$	−18.2690	−0.814573	−0.407287	−	0.913300i	$-0.633525\pi$
$503$	−18.2690	−0.814573	−0.407287	+	0.913300i	$0.633525\pi$
$504$	−0.985191	−0.0438839
$505$	0	0
$506$	45.2499	2.01160
$507$	2.17582	0.0966314
$508$	−6.72519	−0.298382
$509$	14.6448	0.649118	0.324559	−	0.945865i	$-0.394784\pi$
$509$	14.6448	0.649118	0.324559	+	0.945865i	$0.394784\pi$
$510$	0	0
$511$	−5.39152	−0.238507
$512$	−28.7993	−1.27276
$513$	17.3498	0.766011
$514$	−50.5066	−2.22775
$515$	0	0
$516$	28.4157	1.25093
$517$	−2.00580	−0.0882148
$518$	−6.28671	−0.276222
$519$	−45.9950	−2.01896
$520$	0	0
$521$	−26.1781	−1.14688	−0.573441	−	0.819247i	$-0.694392\pi$
$521$	−26.1781	−1.14688	−0.573441	+	0.819247i	$0.694392\pi$
$522$	29.4768	1.29016
$523$	4.27496	0.186931	0.0934654	−	0.995623i	$-0.470206\pi$
$523$	4.27496	0.186931	0.0934654	+	0.995623i	$0.470206\pi$
$524$	10.3556	0.452386
$525$	0	0
$526$	15.8177	0.689683
$527$	−12.4081	−0.540504
$528$	−27.5378	−1.19843
$529$	46.9662	2.04201
$530$	0	0
$531$	−8.61152	−0.373708
$532$	10.7395	0.465616
$533$	8.99882	0.389782
$534$	21.1548	0.915457
$535$	0	0
$536$	8.54582	0.369123
$537$	19.3680	0.835789
$538$	2.54844	0.109871
$539$	2.81053	0.121058
$540$	0	0
$541$	−3.51523	−0.151132	−0.0755658	−	0.997141i	$-0.524076\pi$
$541$	−3.51523	−0.151132	−0.0755658	+	0.997141i	$0.524076\pi$
$542$	−39.5324	−1.69806
$543$	17.8368	0.765448
$544$	8.98563	0.385256
$545$	0	0
$546$	4.18801	0.179230
$547$	11.0351	0.471825	0.235913	−	0.971774i	$-0.424192\pi$
$547$	11.0351	0.471825	0.235913	+	0.971774i	$0.424192\pi$
$548$	−3.65659	−0.156202
$549$	−24.2716	−1.03589
$550$	0	0
$551$	55.6288	2.36987
$552$	−10.3394	−0.440073
$553$	−3.24377	−0.137939
$554$	1.48976	0.0632938
$555$	0	0
$556$	−3.17755	−0.134758
$557$	−12.4828	−0.528915	−0.264457	−	0.964397i	$-0.585193\pi$
$557$	−12.4828	−0.528915	−0.264457	+	0.964397i	$0.585193\pi$
$558$	34.7149	1.46960
$559$	7.66037	0.323999
$560$	0	0
$561$	7.29585	0.308031
$562$	47.6520	2.01008
$563$	−18.1095	−0.763226	−0.381613	−	0.924322i	$-0.624631\pi$
$563$	−18.1095	−0.763226	−0.381613	+	0.924322i	$0.624631\pi$
$564$	−2.64732	−0.111472
$565$	0	0
$566$	51.3260	2.15739
$567$	11.1952	0.470153
$568$	−1.58941	−0.0666900
$569$	−1.04701	−0.0438930	−0.0219465	−	0.999759i	$-0.506986\pi$
$569$	−1.04701	−0.0438930	−0.0219465	+	0.999759i	$0.506986\pi$
$570$	0	0
$571$	−13.7363	−0.574847	−0.287423	−	0.957804i	$-0.592799\pi$
$571$	−13.7363	−0.574847	−0.287423	+	0.957804i	$0.592799\pi$
$572$	4.79153	0.200344
$573$	38.5777	1.61161
$574$	17.3209	0.722961
$575$	0	0
$576$	−9.52111	−0.396713
$577$	4.52720	0.188470	0.0942348	−	0.995550i	$-0.469960\pi$
$577$	4.52720	0.188470	0.0942348	+	0.995550i	$0.469960\pi$
$578$	29.9818	1.24708
$579$	−20.3358	−0.845125
$580$	0	0
$581$	−13.0623	−0.541915
$582$	−23.0641	−0.956037
$583$	−9.38644	−0.388747
$584$	3.06294	0.126745
$585$	0	0
$586$	3.33468	0.137754
$587$	9.33888	0.385457	0.192729	−	0.981252i	$-0.438266\pi$
$587$	9.33888	0.385457	0.192729	+	0.981252i	$0.438266\pi$
$588$	3.70944	0.152975
$589$	65.5143	2.69947
$590$	0	0
$591$	35.3036	1.45219
$592$	14.7082	0.604502
$593$	−40.1137	−1.64727	−0.823636	−	0.567119i	$-0.808058\pi$
$593$	−40.1137	−1.64727	−0.823636	+	0.567119i	$0.808058\pi$
$594$	14.8994	0.611330
$595$	0	0
$596$	−20.6204	−0.844647
$597$	7.10766	0.290897
$598$	16.1001	0.658384
$599$	12.0784	0.493509	0.246755	−	0.969078i	$-0.420636\pi$
$599$	12.0784	0.493509	0.246755	+	0.969078i	$0.420636\pi$
$600$	0	0
$601$	−12.0640	−0.492101	−0.246051	−	0.969257i	$-0.579133\pi$
$601$	−12.0640	−0.492101	−0.246051	+	0.969257i	$0.579133\pi$
$602$	14.7447	0.600948
$603$	26.0867	1.06233
$604$	20.0488	0.815773
$605$	0	0
$606$	37.5615	1.52583
$607$	5.03644	0.204423	0.102211	−	0.994763i	$-0.467408\pi$
$607$	5.03644	0.204423	0.102211	+	0.994763i	$0.467408\pi$
$608$	−47.4439	−1.92410
$609$	19.2143	0.778602
$610$	0	0
$611$	−0.713672	−0.0288721
$612$	3.52732	0.142584
$613$	−3.17511	−0.128241	−0.0641207	−	0.997942i	$-0.520424\pi$
$613$	−3.17511	−0.128241	−0.0641207	+	0.997942i	$0.520424\pi$
$614$	5.70962	0.230422
$615$	0	0
$616$	−1.59667	−0.0643318
$617$	−22.8841	−0.921280	−0.460640	−	0.887587i	$-0.652380\pi$
$617$	−22.8841	−0.921280	−0.460640	+	0.887587i	$0.652380\pi$
$618$	−18.2768	−0.735200
$619$	23.3096	0.936891	0.468446	−	0.883492i	$-0.344814\pi$
$619$	23.3096	0.936891	0.468446	+	0.883492i	$0.344814\pi$
$620$	0	0
$621$	23.0378	0.924473
$622$	32.0828	1.28640
$623$	5.05128	0.202375
$624$	−9.79810	−0.392238
$625$	0	0
$626$	13.2990	0.531536
$627$	−38.5219	−1.53842
$628$	8.83311	0.352479
$629$	−3.89677	−0.155374
$630$	0	0
$631$	−43.8156	−1.74427	−0.872135	−	0.489264i	$-0.837265\pi$
$631$	−43.8156	−1.74427	−0.872135	+	0.489264i	$0.837265\pi$
$632$	1.84280	0.0733027
$633$	−2.05274	−0.0815891
$634$	−1.00411	−0.0398781
$635$	0	0
$636$	−12.3886	−0.491239
$637$	1.00000	0.0396214
$638$	47.7722	1.89132
$639$	−4.85176	−0.191933
$640$	0	0
$641$	−7.85705	−0.310335	−0.155167	−	0.987888i	$-0.549592\pi$
$641$	−7.85705	−0.310335	−0.155167	+	0.987888i	$0.549592\pi$
$642$	50.3585	1.98749
$643$	−14.2987	−0.563884	−0.281942	−	0.959431i	$-0.590979\pi$
$643$	−14.2987	−0.563884	−0.281942	+	0.959431i	$0.590979\pi$
$644$	14.2604	0.561937
$645$	0	0
$646$	14.4660	0.569158
$647$	7.23126	0.284290	0.142145	−	0.989846i	$-0.454600\pi$
$647$	7.23126	0.284290	0.142145	+	0.989846i	$0.454600\pi$
$648$	−6.36002	−0.249845
$649$	−13.9565	−0.547839
$650$	0	0
$651$	22.6287	0.886891
$652$	−9.79360	−0.383547
$653$	−1.18168	−0.0462425	−0.0231213	−	0.999733i	$-0.507360\pi$
$653$	−1.18168	−0.0462425	−0.0231213	+	0.999733i	$0.507360\pi$
$654$	19.0693	0.745669
$655$	0	0
$656$	−40.5234	−1.58217
$657$	9.34983	0.364772
$658$	−1.37368	−0.0535514
$659$	−43.4613	−1.69301	−0.846507	−	0.532378i	$-0.821299\pi$
$659$	−43.4613	−1.69301	−0.846507	+	0.532378i	$0.821299\pi$
$660$	0	0
$661$	−44.4243	−1.72790	−0.863952	−	0.503574i	$-0.832018\pi$
$661$	−44.4243	−1.72790	−0.863952	+	0.503574i	$0.832018\pi$
$662$	1.66616	0.0647570
$663$	2.59590	0.100816
$664$	7.42074	0.287981
$665$	0	0
$666$	10.9023	0.422454
$667$	73.8663	2.86011
$668$	−6.70482	−0.259417
$669$	25.7605	0.995957
$670$	0	0
$671$	−39.3363	−1.51856
$672$	−16.3872	−0.632150
$673$	35.3771	1.36369	0.681844	−	0.731498i	$-0.261178\pi$
$673$	35.3771	1.36369	0.681844	+	0.731498i	$0.261178\pi$
$674$	−20.9216	−0.805869
$675$	0	0
$676$	1.70485	0.0655712
$677$	31.5601	1.21295	0.606477	−	0.795101i	$-0.292582\pi$
$677$	31.5601	1.21295	0.606477	+	0.795101i	$0.292582\pi$
$678$	38.6091	1.48277
$679$	−5.50717	−0.211346
$680$	0	0
$681$	−52.6538	−2.01770
$682$	56.2615	2.15437
$683$	11.6781	0.446850	0.223425	−	0.974721i	$-0.428276\pi$
$683$	11.6781	0.446850	0.223425	+	0.974721i	$0.428276\pi$
$684$	−18.6242	−0.712113
$685$	0	0
$686$	1.92480	0.0734892
$687$	11.1581	0.425706
$688$	−34.4961	−1.31515
$689$	−3.33974	−0.127234
$690$	0	0
$691$	31.7942	1.20951	0.604755	−	0.796412i	$-0.293271\pi$
$691$	31.7942	1.20951	0.604755	+	0.796412i	$0.293271\pi$
$692$	−36.0392	−1.37000
$693$	−4.87395	−0.185146
$694$	68.3938	2.59619
$695$	0	0
$696$	−10.9157	−0.413759
$697$	10.7362	0.406664
$698$	9.91492	0.375285
$699$	−8.25996	−0.312421
$700$	0	0
$701$	−17.0874	−0.645382	−0.322691	−	0.946504i	$-0.604587\pi$
$701$	−17.0874	−0.645382	−0.322691	+	0.946504i	$0.604587\pi$
$702$	5.30129	0.200084
$703$	20.5748	0.775994
$704$	−15.4306	−0.581563
$705$	0	0
$706$	−37.1915	−1.39972
$707$	8.96883	0.337308
$708$	−18.4203	−0.692275
$709$	−13.9333	−0.523278	−0.261639	−	0.965166i	$-0.584263\pi$
$709$	−13.9333	−0.523278	−0.261639	+	0.965166i	$0.584263\pi$
$710$	0	0
$711$	5.62527	0.210964
$712$	−2.86965	−0.107545
$713$	86.9927	3.25790
$714$	4.99659	0.186993
$715$	0	0
$716$	15.1757	0.567142
$717$	−50.1865	−1.87425
$718$	−52.5671	−1.96179
$719$	18.6983	0.697329	0.348665	−	0.937248i	$-0.386635\pi$
$719$	18.6983	0.697329	0.348665	+	0.937248i	$0.386635\pi$
$720$	0	0
$721$	−4.36408	−0.162527
$722$	−39.8090	−1.48154
$723$	38.1123	1.41741
$724$	13.9759	0.519410
$725$	0	0
$726$	12.9867	0.481983
$727$	8.35476	0.309861	0.154931	−	0.987925i	$-0.450485\pi$
$727$	8.35476	0.309861	0.154931	+	0.987925i	$0.450485\pi$
$728$	−0.568104	−0.0210553
$729$	−1.43645	−0.0532017
$730$	0	0
$731$	9.13937	0.338032
$732$	−51.9175	−1.91893
$733$	−29.9487	−1.10618	−0.553090	−	0.833121i	$-0.686552\pi$
$733$	−29.9487	−1.10618	−0.553090	+	0.833121i	$0.686552\pi$
$734$	41.4196	1.52883
$735$	0	0
$736$	−62.9980	−2.32214
$737$	42.2779	1.55733
$738$	−30.0375	−1.10570
$739$	−13.0244	−0.479110	−0.239555	−	0.970883i	$-0.577002\pi$
$739$	−13.0244	−0.479110	−0.239555	+	0.970883i	$0.577002\pi$
$740$	0	0
$741$	−13.7063	−0.503513
$742$	−6.42833	−0.235992
$743$	11.4887	0.421479	0.210740	−	0.977542i	$-0.432413\pi$
$743$	11.4887	0.421479	0.210740	+	0.977542i	$0.432413\pi$
$744$	−12.8555	−0.471305
$745$	0	0
$746$	−25.3048	−0.926475
$747$	22.6523	0.828804
$748$	5.71663	0.209021
$749$	12.0245	0.439364
$750$	0	0
$751$	−19.4695	−0.710451	−0.355226	−	0.934781i	$-0.615596\pi$
$751$	−19.4695	−0.710451	−0.355226	+	0.934781i	$0.615596\pi$
$752$	3.21380	0.117195
$753$	−24.3345	−0.886798
$754$	16.9976	0.619016
$755$	0	0
$756$	4.69550	0.170774
$757$	26.5564	0.965208	0.482604	−	0.875839i	$-0.339691\pi$
$757$	26.5564	0.965208	0.482604	+	0.875839i	$0.339691\pi$
$758$	−25.9957	−0.944205
$759$	−51.1510	−1.85666
$760$	0	0
$761$	23.6311	0.856625	0.428312	−	0.903631i	$-0.359108\pi$
$761$	23.6311	0.856625	0.428312	+	0.903631i	$0.359108\pi$
$762$	16.5206	0.598478
$763$	4.55332	0.164841
$764$	30.2274	1.09359
$765$	0	0
$766$	−12.1142	−0.437704
$767$	−4.96578	−0.179304
$768$	42.7182	1.54146
$769$	−22.0904	−0.796600	−0.398300	−	0.917255i	$-0.630400\pi$
$769$	−22.0904	−0.796600	−0.398300	+	0.917255i	$0.630400\pi$
$770$	0	0
$771$	57.0933	2.05616
$772$	−15.9340	−0.573477
$773$	−17.4021	−0.625910	−0.312955	−	0.949768i	$-0.601319\pi$
$773$	−17.4021	−0.625910	−0.312955	+	0.949768i	$0.601319\pi$
$774$	−25.5698	−0.919089
$775$	0	0
$776$	3.12865	0.112312
$777$	7.10658	0.254947
$778$	2.05071	0.0735215
$779$	−56.6870	−2.03102
$780$	0	0
$781$	−7.86312	−0.281364
$782$	19.2086	0.686898
$783$	24.3219	0.869195
$784$	−4.50319	−0.160828
$785$	0	0
$786$	−25.4387	−0.907370
$787$	−36.4732	−1.30013	−0.650065	−	0.759878i	$-0.725258\pi$
$787$	−36.4732	−1.30013	−0.650065	+	0.759878i	$0.725258\pi$
$788$	27.6619	0.985416
$789$	−17.8805	−0.636562
$790$	0	0
$791$	9.21896	0.327788
$792$	2.76891	0.0983889
$793$	−13.9960	−0.497014
$794$	−50.8796	−1.80565
$795$	0	0
$796$	5.56917	0.197394
$797$	39.2694	1.39099	0.695496	−	0.718530i	$-0.255185\pi$
$797$	39.2694	1.39099	0.695496	+	0.718530i	$0.255185\pi$
$798$	−26.3818	−0.933907
$799$	−0.851461	−0.0301225
$800$	0	0
$801$	−8.75980	−0.309512
$802$	16.0733	0.567566
$803$	15.1530	0.534738
$804$	55.8000	1.96791
$805$	0	0
$806$	20.0181	0.705109
$807$	−2.88079	−0.101409
$808$	−5.09523	−0.179249
$809$	−42.7378	−1.50258	−0.751291	−	0.659971i	$-0.770568\pi$
$809$	−42.7378	−1.50258	−0.751291	+	0.659971i	$0.770568\pi$
$810$	0	0
$811$	−21.1727	−0.743474	−0.371737	−	0.928338i	$-0.621238\pi$
$811$	−21.1727	−0.743474	−0.371737	+	0.928338i	$0.621238\pi$
$812$	15.0553	0.528336
$813$	44.6879	1.56727
$814$	17.6690	0.619298
$815$	0	0
$816$	−11.6898	−0.409226
$817$	−48.2556	−1.68825
$818$	6.79622	0.237624
$819$	−1.73417	−0.0605970
$820$	0	0
$821$	23.0210	0.803439	0.401720	−	0.915763i	$-0.368413\pi$
$821$	23.0210	0.803439	0.401720	+	0.915763i	$0.368413\pi$
$822$	8.98250	0.313301
$823$	−5.62443	−0.196055	−0.0980276	−	0.995184i	$-0.531253\pi$
$823$	−5.62443	−0.196055	−0.0980276	+	0.995184i	$0.531253\pi$
$824$	2.47925	0.0863687
$825$	0	0
$826$	−9.55812	−0.332570
$827$	35.6613	1.24007	0.620033	−	0.784576i	$-0.287119\pi$
$827$	35.6613	1.24007	0.620033	+	0.784576i	$0.287119\pi$
$828$	−24.7300	−0.859425
$829$	2.03485	0.0706732	0.0353366	−	0.999375i	$-0.488750\pi$
$829$	2.03485	0.0706732	0.0353366	+	0.999375i	$0.488750\pi$
$830$	0	0
$831$	−1.68404	−0.0584188
$832$	−5.49029	−0.190341
$833$	1.19307	0.0413374
$834$	7.80572	0.270290
$835$	0	0
$836$	−30.1837	−1.04392
$837$	28.6441	0.990083
$838$	20.9760	0.724605
$839$	37.9304	1.30950	0.654751	−	0.755845i	$-0.272774\pi$
$839$	37.9304	1.30950	0.654751	+	0.755845i	$0.272774\pi$
$840$	0	0
$841$	48.9837	1.68909
$842$	−17.9154	−0.617405
$843$	−53.8663	−1.85526
$844$	−1.60841	−0.0553639
$845$	0	0
$846$	2.38219	0.0819015
$847$	3.10093	0.106549
$848$	15.0395	0.516458
$849$	−58.0195	−1.99122
$850$	0	0
$851$	27.3201	0.936522
$852$	−10.3780	−0.355546
$853$	−11.6155	−0.397706	−0.198853	−	0.980029i	$-0.563722\pi$
$853$	−11.6155	−0.397706	−0.198853	+	0.980029i	$0.563722\pi$
$854$	−26.9396	−0.921853
$855$	0	0
$856$	−6.83114	−0.233484
$857$	15.1140	0.516283	0.258142	−	0.966107i	$-0.416890\pi$
$857$	15.1140	0.516283	0.258142	+	0.966107i	$0.416890\pi$
$858$	−11.7705	−0.401839
$859$	−50.2271	−1.71373	−0.856864	−	0.515543i	$-0.827590\pi$
$859$	−50.2271	−1.71373	−0.856864	+	0.515543i	$0.827590\pi$
$860$	0	0
$861$	−19.5798	−0.667277
$862$	−44.5371	−1.51694
$863$	1.49473	0.0508812	0.0254406	−	0.999676i	$-0.491901\pi$
$863$	1.49473	0.0508812	0.0254406	+	0.999676i	$0.491901\pi$
$864$	−20.7433	−0.705703
$865$	0	0
$866$	18.9174	0.642838
$867$	−33.8918	−1.15103
$868$	17.7306	0.601817
$869$	9.11672	0.309264
$870$	0	0
$871$	15.0427	0.509702
$872$	−2.58676	−0.0875986
$873$	9.55040	0.323232
$874$	−101.421	−3.43061
$875$	0	0
$876$	19.9995	0.675721
$877$	−37.9939	−1.28296	−0.641481	−	0.767139i	$-0.721680\pi$
$877$	−37.9939	−1.28296	−0.641481	+	0.767139i	$0.721680\pi$
$878$	−53.5381	−1.80682
$879$	−3.76956	−0.127144
$880$	0	0
$881$	19.9207	0.671146	0.335573	−	0.942014i	$-0.391070\pi$
$881$	19.9207	0.671146	0.335573	+	0.942014i	$0.391070\pi$
$882$	−3.33794	−0.112394
$883$	−27.4136	−0.922541	−0.461270	−	0.887260i	$-0.652606\pi$
$883$	−27.4136	−0.922541	−0.461270	+	0.887260i	$0.652606\pi$
$884$	2.03401	0.0684110
$885$	0	0
$886$	−66.9938	−2.25070
$887$	−40.2604	−1.35181	−0.675906	−	0.736988i	$-0.736247\pi$
$887$	−40.2604	−1.35181	−0.675906	+	0.736988i	$0.736247\pi$
$888$	−4.03728	−0.135482
$889$	3.94474	0.132302
$890$	0	0
$891$	−31.4643	−1.05409
$892$	20.1845	0.675826
$893$	4.49569	0.150443
$894$	50.6547	1.69415
$895$	0	0
$896$	4.49534	0.150179
$897$	−18.1998	−0.607673
$898$	54.4855	1.81820
$899$	91.8418	3.06310
$900$	0	0
$901$	−3.98455	−0.132745
$902$	−48.6809	−1.62090
$903$	−16.6676	−0.554662
$904$	−5.23733	−0.174191
$905$	0	0
$906$	−49.2503	−1.63623
$907$	49.1439	1.63180	0.815899	−	0.578195i	$-0.196243\pi$
$907$	49.1439	1.63180	0.815899	+	0.578195i	$0.196243\pi$
$908$	−41.2567	−1.36915
$909$	−15.5535	−0.515878
$910$	0	0
$911$	−6.31787	−0.209320	−0.104660	−	0.994508i	$-0.533375\pi$
$911$	−6.31787	−0.209320	−0.104660	+	0.994508i	$0.533375\pi$
$912$	61.7220	2.04382
$913$	36.7119	1.21499
$914$	16.4821	0.545180
$915$	0	0
$916$	8.74284	0.288871
$917$	−6.07419	−0.200587
$918$	6.32481	0.208750
$919$	−51.5417	−1.70020	−0.850102	−	0.526618i	$-0.823460\pi$
$919$	−51.5417	−1.70020	−0.850102	+	0.526618i	$0.823460\pi$
$920$	0	0
$921$	−6.45422	−0.212674
$922$	49.7023	1.63686
$923$	−2.79774	−0.0920886
$924$	−10.4255	−0.342973
$925$	0	0
$926$	−70.6579	−2.32196
$927$	7.56807	0.248568
$928$	−66.5096	−2.18329
$929$	−23.5177	−0.771591	−0.385795	−	0.922584i	$-0.626073\pi$
$929$	−23.5177	−0.771591	−0.385795	+	0.922584i	$0.626073\pi$
$930$	0	0
$931$	−6.29938	−0.206454
$932$	−6.47206	−0.211999
$933$	−36.2668	−1.18732
$934$	−11.6807	−0.382205
$935$	0	0
$936$	0.985191	0.0322020
$937$	6.57820	0.214900	0.107450	−	0.994210i	$-0.465731\pi$
$937$	6.57820	0.214900	0.107450	+	0.994210i	$0.465731\pi$
$938$	28.9542	0.945387
$939$	−15.0334	−0.490595
$940$	0	0
$941$	−4.22190	−0.137630	−0.0688150	−	0.997629i	$-0.521922\pi$
$941$	−4.22190	−0.137630	−0.0688150	+	0.997629i	$0.521922\pi$
$942$	−21.6988	−0.706984
$943$	−75.2714	−2.45117
$944$	22.3618	0.727815
$945$	0	0
$946$	−41.4403	−1.34734
$947$	33.6690	1.09409	0.547047	−	0.837102i	$-0.315752\pi$
$947$	33.6690	1.09409	0.547047	+	0.837102i	$0.315752\pi$
$948$	12.0326	0.390800
$949$	5.39152	0.175016
$950$	0	0
$951$	1.13505	0.0368066
$952$	−0.677788	−0.0219672
$953$	−9.73077	−0.315211	−0.157605	−	0.987502i	$-0.550377\pi$
$953$	−9.73077	−0.315211	−0.157605	+	0.987502i	$0.550377\pi$
$954$	11.1478	0.360925
$955$	0	0
$956$	−39.3234	−1.27181
$957$	−54.0023	−1.74565
$958$	−6.72889	−0.217401
$959$	2.14481	0.0692597
$960$	0	0
$961$	77.1625	2.48911
$962$	6.28671	0.202692
$963$	−20.8525	−0.671963
$964$	29.8627	0.961813
$965$	0	0
$966$	−35.0309	−1.12710
$967$	6.14864	0.197727	0.0988634	−	0.995101i	$-0.468479\pi$
$967$	6.14864	0.197727	0.0988634	+	0.995101i	$0.468479\pi$
$968$	−1.76165	−0.0566217
$969$	−16.3526	−0.525320
$970$	0	0
$971$	36.9294	1.18512	0.592560	−	0.805526i	$-0.298117\pi$
$971$	36.9294	1.18512	0.592560	+	0.805526i	$0.298117\pi$
$972$	−27.4413	−0.880179
$973$	1.86383	0.0597516
$974$	−31.9019	−1.02220
$975$	0	0
$976$	63.0268	2.01744
$977$	−43.8003	−1.40130	−0.700648	−	0.713507i	$-0.747106\pi$
$977$	−43.8003	−1.40130	−0.700648	+	0.713507i	$0.747106\pi$
$978$	24.0582	0.769298
$979$	−14.1968	−0.453730
$980$	0	0
$981$	−7.89624	−0.252108
$982$	−70.4798	−2.24910
$983$	−32.4877	−1.03620	−0.518098	−	0.855321i	$-0.673360\pi$
$983$	−32.4877	−1.03620	−0.518098	+	0.855321i	$0.673360\pi$
$984$	11.1234	0.354599
$985$	0	0
$986$	20.2793	0.645825
$987$	1.55282	0.0494268
$988$	−10.7395	−0.341669
$989$	−64.0758	−2.03749
$990$	0	0
$991$	−13.7920	−0.438118	−0.219059	−	0.975712i	$-0.570299\pi$
$991$	−13.7920	−0.438118	−0.219059	+	0.975712i	$0.570299\pi$
$992$	−78.3287	−2.48694
$993$	−1.88344	−0.0597692
$994$	−5.38508	−0.170804
$995$	0	0
$996$	48.4538	1.53532
$997$	57.5296	1.82198	0.910991	−	0.412426i	$-0.135319\pi$
$997$	57.5296	1.82198	0.910991	+	0.412426i	$0.135319\pi$
$998$	−35.3291	−1.11832
$999$	8.99569	0.284611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2275.2.a.y.1.1		7
5.2	odd	4		455.2.c.b.274.2	✓	14
5.3	odd	4		455.2.c.b.274.13	yes	14
5.4	even	2		2275.2.a.w.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
455.2.c.b.274.2	✓	14	5.2	odd	4
455.2.c.b.274.13	yes	14	5.3	odd	4
2275.2.a.w.1.7		7	5.4	even	2
2275.2.a.y.1.1		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)

\(f(q)\)	\(=\)	\(q-1.92480 q^{2} +2.17582 q^{3} +1.70485 q^{4} -4.18801 q^{6} -1.00000 q^{7} +0.568104 q^{8} +1.73417 q^{9} +2.81053 q^{11} +3.70944 q^{12} +1.00000 q^{13} +1.92480 q^{14} -4.50319 q^{16} +1.19307 q^{17} -3.33794 q^{18} -6.29938 q^{19} -2.17582 q^{21} -5.40970 q^{22} -8.36458 q^{23} +1.23609 q^{24} -1.92480 q^{26} -2.75420 q^{27} -1.70485 q^{28} -8.83084 q^{29} -10.4001 q^{31} +7.53152 q^{32} +6.11519 q^{33} -2.29642 q^{34} +2.95651 q^{36} -3.26617 q^{37} +12.1250 q^{38} +2.17582 q^{39} +8.99882 q^{41} +4.18801 q^{42} +7.66037 q^{43} +4.79153 q^{44} +16.1001 q^{46} -0.713672 q^{47} -9.79810 q^{48} +1.00000 q^{49} +2.59590 q^{51} +1.70485 q^{52} -3.33974 q^{53} +5.30129 q^{54} -0.568104 q^{56} -13.7063 q^{57} +16.9976 q^{58} -4.96578 q^{59} -13.9960 q^{61} +20.0181 q^{62} -1.73417 q^{63} -5.49029 q^{64} -11.7705 q^{66} +15.0427 q^{67} +2.03401 q^{68} -18.1998 q^{69} -2.79774 q^{71} +0.985191 q^{72} +5.39152 q^{73} +6.28671 q^{74} -10.7395 q^{76} -2.81053 q^{77} -4.18801 q^{78} +3.24377 q^{79} -11.1952 q^{81} -17.3209 q^{82} +13.0623 q^{83} -3.70944 q^{84} -14.7447 q^{86} -19.2143 q^{87} +1.59667 q^{88} -5.05128 q^{89} -1.00000 q^{91} -14.2604 q^{92} -22.6287 q^{93} +1.37368 q^{94} +16.3872 q^{96} +5.50717 q^{97} -1.92480 q^{98} +4.87395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{3} + 2 q^{4} - 8 q^{6} - 7 q^{7} + 6 q^{8} + q^{9} - 6 q^{11} - 3 q^{12} + 7 q^{13} - 4 q^{16} - 2 q^{17} + q^{18} - 10 q^{19} - 2 q^{21} - 18 q^{22} - 10 q^{23} - 5 q^{24} + 8 q^{27} - 2 q^{28}+ \cdots - 18 q^{99}+O(q^{100}) \) 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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