LMFDB - Embedded newform 275.4.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,4,Mod(1,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("275.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	275.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:	$16.2255252516$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 24x^{3} + 31x^{2} + 108x - 84 $ x^5 - 2x^4 - 24x^3 + 31x^2 + 108x - 84
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.68861$ of defining polynomial
Character	$\chi$	$=$	275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.68861 q^{2} -5.92895 q^{3} -0.771363 q^{4} -15.9407 q^{6} +13.6511 q^{7} -23.5828 q^{8} +8.15246 q^{9} +11.0000 q^{11} +4.57337 q^{12} -5.37969 q^{13} +36.7026 q^{14} -57.2341 q^{16} +85.3345 q^{17} +21.9188 q^{18} +51.6873 q^{19} -80.9370 q^{21} +29.5747 q^{22} +163.144 q^{23} +139.821 q^{24} -14.4639 q^{26} +111.746 q^{27} -10.5300 q^{28} +16.5882 q^{29} +35.6828 q^{31} +34.7821 q^{32} -65.2185 q^{33} +229.431 q^{34} -6.28851 q^{36} +55.5336 q^{37} +138.967 q^{38} +31.8959 q^{39} +71.0253 q^{41} -217.608 q^{42} +296.317 q^{43} -8.48499 q^{44} +438.632 q^{46} -190.018 q^{47} +339.338 q^{48} -156.646 q^{49} -505.944 q^{51} +4.14969 q^{52} +623.984 q^{53} +300.442 q^{54} -321.932 q^{56} -306.451 q^{57} +44.5994 q^{58} -468.122 q^{59} -181.218 q^{61} +95.9373 q^{62} +111.290 q^{63} +551.388 q^{64} -175.347 q^{66} +682.370 q^{67} -65.8239 q^{68} -967.275 q^{69} -579.969 q^{71} -192.258 q^{72} +190.416 q^{73} +149.308 q^{74} -39.8696 q^{76} +150.163 q^{77} +85.7558 q^{78} -718.663 q^{79} -882.654 q^{81} +190.960 q^{82} -1503.91 q^{83} +62.4318 q^{84} +796.681 q^{86} -98.3509 q^{87} -259.411 q^{88} +756.827 q^{89} -73.4390 q^{91} -125.843 q^{92} -211.562 q^{93} -510.885 q^{94} -206.221 q^{96} +707.394 q^{97} -421.161 q^{98} +89.6771 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 12 q^{3} + 12 q^{4} + 8 q^{6} + 24 q^{7} + 27 q^{8} + 31 q^{9} + 55 q^{11} + 3 q^{12} + 111 q^{13} + 47 q^{14} - 56 q^{16} + 40 q^{17} + 217 q^{18} + 205 q^{19} - 94 q^{21} + 22 q^{22} + 287 q^{23}+ \cdots + 341 q^{99}+O(q^{100}) $ 5 * q + 2 * q^2 + 12 * q^3 + 12 * q^4 + 8 * q^6 + 24 * q^7 + 27 * q^8 + 31 * q^9 + 55 * q^11 + 3 * q^12 + 111 * q^13 + 47 * q^14 - 56 * q^16 + 40 * q^17 + 217 * q^18 + 205 * q^19 - 94 * q^21 + 22 * q^22 + 287 * q^23 + 273 * q^24 - 354 * q^26 + 270 * q^27 + 460 * q^28 + 251 * q^29 - 289 * q^31 + 248 * q^32 + 132 * q^33 + 522 * q^34 - 722 * q^36 + 224 * q^37 + 540 * q^38 + 538 * q^39 - 462 * q^41 + 175 * q^42 + 593 * q^43 + 132 * q^44 - 972 * q^46 + 766 * q^47 + 992 * q^48 + 75 * q^49 - 820 * q^51 + 696 * q^53 + 281 * q^54 - 527 * q^56 + 1170 * q^57 + 1461 * q^58 - 22 * q^59 + 720 * q^61 - 998 * q^62 - 168 * q^63 - 317 * q^64 + 88 * q^66 + 1230 * q^67 - 109 * q^68 - 514 * q^69 + 951 * q^71 - 1590 * q^72 + 666 * q^73 + 873 * q^74 + 290 * q^76 + 264 * q^77 + 56 * q^78 - 588 * q^79 + 1885 * q^81 - 1807 * q^82 - 867 * q^83 + 1321 * q^84 - 411 * q^86 + 766 * q^87 + 297 * q^88 - 51 * q^89 + 2172 * q^91 - 4137 * q^92 - 1916 * q^93 - 865 * q^94 + 603 * q^96 + 2849 * q^97 - 4104 * q^98 + 341 * 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 $

$2$	2.68861	0.950568	0.475284	−	0.879832i	$-0.342345\pi$
$2$	2.68861	0.950568	0.475284	+	0.879832i	$0.342345\pi$
$3$	−5.92895	−1.14103	−0.570514	−	0.821288i	$-0.693256\pi$
$3$	−5.92895	−1.14103	−0.570514	+	0.821288i	$0.693256\pi$
$4$	−0.771363	−0.0964203
$5$	0	0
$5$	0	0
$6$	−15.9407	−1.08462
$7$	13.6511	0.737092	0.368546	−	0.929609i	$-0.379856\pi$
$7$	13.6511	0.737092	0.368546	+	0.929609i	$0.379856\pi$
$8$	−23.5828	−1.04222
$9$	8.15246	0.301943
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	4.57337	0.110018
$13$	−5.37969	−0.114774	−0.0573869	−	0.998352i	$-0.518277\pi$
$13$	−5.37969	−0.114774	−0.0573869	+	0.998352i	$0.518277\pi$
$14$	36.7026	0.700657
$15$	0	0
$16$	−57.2341	−0.894283
$17$	85.3345	1.21745	0.608725	−	0.793381i	$-0.291681\pi$
$17$	85.3345	1.21745	0.608725	+	0.793381i	$0.291681\pi$
$18$	21.9188	0.287017
$19$	51.6873	0.624099	0.312049	−	0.950066i	$-0.398985\pi$
$19$	51.6873	0.624099	0.312049	+	0.950066i	$0.398985\pi$
$20$	0	0
$21$	−80.9370	−0.841043
$22$	29.5747	0.286607
$23$	163.144	1.47904	0.739520	−	0.673134i	$-0.235052\pi$
$23$	163.144	1.47904	0.739520	+	0.673134i	$0.235052\pi$
$24$	139.821	1.18920
$25$	0	0
$26$	−14.4639	−0.109100
$27$	111.746	0.796502
$28$	−10.5300	−0.0710707
$29$	16.5882	0.106219	0.0531097	−	0.998589i	$-0.483087\pi$
$29$	16.5882	0.106219	0.0531097	+	0.998589i	$0.483087\pi$
$30$	0	0
$31$	35.6828	0.206736	0.103368	−	0.994643i	$-0.467038\pi$
$31$	35.6828	0.206736	0.103368	+	0.994643i	$0.467038\pi$
$32$	34.7821	0.192146
$33$	−65.2185	−0.344033
$34$	229.431	1.15727
$35$	0	0
$36$	−6.28851	−0.0291135
$37$	55.5336	0.246748	0.123374	−	0.992360i	$-0.460629\pi$
$37$	55.5336	0.246748	0.123374	+	0.992360i	$0.460629\pi$
$38$	138.967	0.593248
$39$	31.8959	0.130960
$40$	0	0
$41$	71.0253	0.270544	0.135272	−	0.990809i	$-0.456809\pi$
$41$	71.0253	0.270544	0.135272	+	0.990809i	$0.456809\pi$
$42$	−217.608	−0.799468
$43$	296.317	1.05088	0.525440	−	0.850831i	$-0.323901\pi$
$43$	296.317	1.05088	0.525440	+	0.850831i	$0.323901\pi$
$44$	−8.48499	−0.0290718
$45$	0	0
$46$	438.632	1.40593
$47$	−190.018	−0.589723	−0.294862	−	0.955540i	$-0.595274\pi$
$47$	−190.018	−0.589723	−0.294862	+	0.955540i	$0.595274\pi$
$48$	339.338	1.02040
$49$	−156.646	−0.456695
$50$	0	0
$51$	−505.944	−1.38914
$52$	4.14969	0.0110665
$53$	623.984	1.61719	0.808593	−	0.588369i	$-0.200230\pi$
$53$	623.984	1.61719	0.808593	+	0.588369i	$0.200230\pi$
$54$	300.442	0.757129
$55$	0	0
$56$	−321.932	−0.768214
$57$	−306.451	−0.712113
$58$	44.5994	0.100969
$59$	−468.122	−1.03295	−0.516477	−	0.856301i	$-0.672757\pi$
$59$	−468.122	−1.03295	−0.516477	+	0.856301i	$0.672757\pi$
$60$	0	0
$61$	−181.218	−0.380371	−0.190186	−	0.981748i	$-0.560909\pi$
$61$	−181.218	−0.380371	−0.190186	+	0.981748i	$0.560909\pi$
$62$	95.9373	0.196517
$63$	111.290	0.222560
$64$	551.388	1.07693
$65$	0	0
$66$	−175.347	−0.327026
$67$	682.370	1.24425	0.622125	−	0.782918i	$-0.286270\pi$
$67$	682.370	1.24425	0.622125	+	0.782918i	$0.286270\pi$
$68$	−65.8239	−0.117387
$69$	−967.275	−1.68763
$70$	0	0
$71$	−579.969	−0.969432	−0.484716	−	0.874672i	$-0.661077\pi$
$71$	−579.969	−0.969432	−0.484716	+	0.874672i	$0.661077\pi$
$72$	−192.258	−0.314692
$73$	190.416	0.305295	0.152648	−	0.988281i	$-0.451220\pi$
$73$	190.416	0.305295	0.152648	+	0.988281i	$0.451220\pi$
$74$	149.308	0.234551
$75$	0	0
$76$	−39.8696	−0.0601758
$77$	150.163	0.222242
$78$	85.7558	0.124486
$79$	−718.663	−1.02349	−0.511746	−	0.859137i	$-0.671001\pi$
$79$	−718.663	−1.02349	−0.511746	+	0.859137i	$0.671001\pi$
$80$	0	0
$81$	−882.654	−1.21077
$82$	190.960	0.257170
$83$	−1503.91	−1.98886	−0.994428	−	0.105415i	$-0.966383\pi$
$83$	−1503.91	−1.98886	−0.994428	+	0.105415i	$0.966383\pi$
$84$	62.4318	0.0810936
$85$	0	0
$86$	796.681	0.998933
$87$	−98.3509	−0.121199
$88$	−259.411	−0.314242
$89$	756.827	0.901387	0.450694	−	0.892679i	$-0.351177\pi$
$89$	756.827	0.901387	0.450694	+	0.892679i	$0.351177\pi$
$90$	0	0
$91$	−73.4390	−0.0845988
$92$	−125.843	−0.142610
$93$	−211.562	−0.235892
$94$	−510.885	−0.560572
$95$	0	0
$96$	−206.221	−0.219243
$97$	707.394	0.740464	0.370232	−	0.928939i	$-0.379278\pi$
$97$	707.394	0.740464	0.370232	+	0.928939i	$0.379278\pi$
$98$	−421.161	−0.434119
$99$	89.6771	0.0910393
$100$	0	0
$101$	2001.84	1.97219	0.986094	−	0.166190i	$-0.0531464\pi$
$101$	2001.84	1.97219	0.986094	+	0.166190i	$0.0531464\pi$
$102$	−1360.29	−1.32048
$103$	−1107.75	−1.05971	−0.529853	−	0.848089i	$-0.677753\pi$
$103$	−1107.75	−1.05971	−0.529853	+	0.848089i	$0.677753\pi$
$104$	126.868	0.119620
$105$	0	0
$106$	1677.65	1.53724
$107$	59.1852	0.0534734	0.0267367	−	0.999643i	$-0.491488\pi$
$107$	59.1852	0.0534734	0.0267367	+	0.999643i	$0.491488\pi$
$108$	−86.1968	−0.0767990
$109$	−466.505	−0.409937	−0.204968	−	0.978769i	$-0.565709\pi$
$109$	−466.505	−0.409937	−0.204968	+	0.978769i	$0.565709\pi$
$110$	0	0
$111$	−329.256	−0.281546
$112$	−781.311	−0.659169
$113$	82.6677	0.0688205	0.0344103	−	0.999408i	$-0.489045\pi$
$113$	82.6677	0.0688205	0.0344103	+	0.999408i	$0.489045\pi$
$114$	−823.929	−0.676912
$115$	0	0
$116$	−12.7956	−0.0102417
$117$	−43.8578	−0.0346551
$118$	−1258.60	−0.981893
$119$	1164.91	0.897374
$120$	0	0
$121$	121.000	0.0909091
$122$	−487.226	−0.361569
$123$	−421.106	−0.308698
$124$	−27.5244	−0.0199336
$125$	0	0
$126$	299.217	0.211558
$127$	2761.92	1.92977	0.964883	−	0.262679i	$-0.0846060\pi$
$127$	2761.92	1.92977	0.964883	+	0.262679i	$0.0846060\pi$
$128$	1204.21	0.831550
$129$	−1756.85	−1.19908
$130$	0	0
$131$	2472.38	1.64895	0.824475	−	0.565898i	$-0.191471\pi$
$131$	2472.38	1.64895	0.824475	+	0.565898i	$0.191471\pi$
$132$	50.3071	0.0331717
$133$	705.590	0.460018
$134$	1834.63	1.18274
$135$	0	0
$136$	−2012.43	−1.26885
$137$	−1282.29	−0.799663	−0.399831	−	0.916589i	$-0.630931\pi$
$137$	−1282.29	−0.799663	−0.399831	+	0.916589i	$0.630931\pi$
$138$	−2600.63	−1.60420
$139$	2128.82	1.29902	0.649510	−	0.760353i	$-0.274974\pi$
$139$	2128.82	1.29902	0.649510	+	0.760353i	$0.274974\pi$
$140$	0	0
$141$	1126.61	0.672891
$142$	−1559.31	−0.921511
$143$	−59.1766	−0.0346056
$144$	−466.599	−0.270023
$145$	0	0
$146$	511.956	0.290204
$147$	928.748	0.521101
$148$	−42.8365	−0.0237915
$149$	1955.28	1.07505	0.537526	−	0.843247i	$-0.319359\pi$
$149$	1955.28	1.07505	0.537526	+	0.843247i	$0.319359\pi$
$150$	0	0
$151$	1381.37	0.744465	0.372233	−	0.928139i	$-0.378592\pi$
$151$	1381.37	0.744465	0.372233	+	0.928139i	$0.378592\pi$
$152$	−1218.93	−0.650449
$153$	695.687	0.367601
$154$	403.729	0.211256
$155$	0	0
$156$	−24.6033	−0.0126272
$157$	1039.78	0.528556	0.264278	−	0.964447i	$-0.414866\pi$
$157$	1039.78	0.528556	0.264278	+	0.964447i	$0.414866\pi$
$158$	−1932.21	−0.972899
$159$	−3699.57	−1.84525
$160$	0	0
$161$	2227.11	1.09019
$162$	−2373.11	−1.15092
$163$	−3323.13	−1.59686	−0.798429	−	0.602088i	$-0.794335\pi$
$163$	−3323.13	−1.59686	−0.798429	+	0.602088i	$0.794335\pi$
$164$	−54.7863	−0.0260859
$165$	0	0
$166$	−4043.42	−1.89054
$167$	−94.5679	−0.0438196	−0.0219098	−	0.999760i	$-0.506975\pi$
$167$	−94.5679	−0.0438196	−0.0219098	+	0.999760i	$0.506975\pi$
$168$	1908.72	0.876553
$169$	−2168.06	−0.986827
$170$	0	0
$171$	421.379	0.188442
$172$	−228.568	−0.101326
$173$	2167.07	0.952368	0.476184	−	0.879346i	$-0.342020\pi$
$173$	2167.07	0.952368	0.476184	+	0.879346i	$0.342020\pi$
$174$	−264.427	−0.115208
$175$	0	0
$176$	−629.575	−0.269636
$177$	2775.47	1.17863
$178$	2034.81	0.856830
$179$	−947.375	−0.395587	−0.197794	−	0.980244i	$-0.563378\pi$
$179$	−947.375	−0.395587	−0.197794	+	0.980244i	$0.563378\pi$
$180$	0	0
$181$	1927.25	0.791443	0.395722	−	0.918371i	$-0.370495\pi$
$181$	1927.25	0.791443	0.395722	+	0.918371i	$0.370495\pi$
$182$	−197.449	−0.0804170
$183$	1074.44	0.434014
$184$	−3847.40	−1.54149
$185$	0	0
$186$	−568.808	−0.224231
$187$	938.680	0.367075
$188$	146.573	0.0568613
$189$	1525.46	0.587096
$190$	0	0
$191$	−3448.79	−1.30652	−0.653260	−	0.757133i	$-0.726599\pi$
$191$	−3448.79	−1.30652	−0.653260	+	0.757133i	$0.726599\pi$
$192$	−3269.15	−1.22881
$193$	−3292.91	−1.22813	−0.614064	−	0.789256i	$-0.710466\pi$
$193$	−3292.91	−1.22813	−0.614064	+	0.789256i	$0.710466\pi$
$194$	1901.91	0.703861
$195$	0	0
$196$	120.831	0.0440347
$197$	−2583.89	−0.934489	−0.467244	−	0.884128i	$-0.654753\pi$
$197$	−2583.89	−0.934489	−0.467244	+	0.884128i	$0.654753\pi$
$198$	241.107	0.0865390
$199$	−3701.37	−1.31851	−0.659254	−	0.751921i	$-0.729128\pi$
$199$	−3701.37	−1.31851	−0.659254	+	0.751921i	$0.729128\pi$
$200$	0	0
$201$	−4045.74	−1.41972
$202$	5382.18	1.87470
$203$	226.449	0.0782935
$204$	390.266	0.133942
$205$	0	0
$206$	−2978.31	−1.00732
$207$	1330.03	0.446586
$208$	307.902	0.102640
$209$	568.560	0.188173
$210$	0	0
$211$	−2345.86	−0.765382	−0.382691	−	0.923876i	$-0.625003\pi$
$211$	−2345.86	−0.765382	−0.382691	+	0.923876i	$0.625003\pi$
$212$	−481.318	−0.155930
$213$	3438.61	1.10615
$214$	159.126	0.0508301
$215$	0	0
$216$	−2635.29	−0.830132
$217$	487.111	0.152384
$218$	−1254.25	−0.389673
$219$	−1128.97	−0.348350
$220$	0	0
$221$	−459.073	−0.139731
$222$	−885.242	−0.267629
$223$	6393.37	1.91987	0.959937	−	0.280216i	$-0.0904061\pi$
$223$	6393.37	1.91987	0.959937	+	0.280216i	$0.0904061\pi$
$224$	474.815	0.141629
$225$	0	0
$226$	222.261	0.0654186
$227$	1060.41	0.310051	0.155026	−	0.987910i	$-0.450454\pi$
$227$	1060.41	0.310051	0.155026	+	0.987910i	$0.450454\pi$
$228$	236.385	0.0686622
$229$	6219.64	1.79478	0.897392	−	0.441234i	$-0.145459\pi$
$229$	6219.64	1.79478	0.897392	+	0.441234i	$0.145459\pi$
$230$	0	0
$231$	−890.307	−0.253584
$232$	−391.197	−0.110704
$233$	5816.62	1.63545	0.817724	−	0.575610i	$-0.195235\pi$
$233$	5816.62	1.63545	0.817724	+	0.575610i	$0.195235\pi$
$234$	−117.917	−0.0329421
$235$	0	0
$236$	361.092	0.0995977
$237$	4260.92	1.16783
$238$	3132.00	0.853015
$239$	−1682.36	−0.455326	−0.227663	−	0.973740i	$-0.573109\pi$
$239$	−1682.36	−0.455326	−0.227663	+	0.973740i	$0.573109\pi$
$240$	0	0
$241$	4905.91	1.31128	0.655638	−	0.755076i	$-0.272400\pi$
$241$	4905.91	1.31128	0.655638	+	0.755076i	$0.272400\pi$
$242$	325.322	0.0864153
$243$	2216.07	0.585024
$244$	139.785	0.0366755
$245$	0	0
$246$	−1132.19	−0.293438
$247$	−278.062	−0.0716301
$248$	−841.501	−0.215465
$249$	8916.58	2.26934
$250$	0	0
$251$	5120.12	1.28757	0.643784	−	0.765208i	$-0.277364\pi$
$251$	5120.12	1.28757	0.643784	+	0.765208i	$0.277364\pi$
$252$	−85.8453	−0.0214593
$253$	1794.59	0.445948
$254$	7425.72	1.83437
$255$	0	0
$256$	−1173.44	−0.286485
$257$	−6898.84	−1.67447	−0.837233	−	0.546846i	$-0.815828\pi$
$257$	−6898.84	−1.67447	−0.837233	+	0.546846i	$0.815828\pi$
$258$	−4723.48	−1.13981
$259$	758.097	0.181876
$260$	0	0
$261$	135.235	0.0320722
$262$	6647.26	1.56744
$263$	−4931.25	−1.15617	−0.578087	−	0.815975i	$-0.696201\pi$
$263$	−4931.25	−1.15617	−0.578087	+	0.815975i	$0.696201\pi$
$264$	1538.03	0.358558
$265$	0	0
$266$	1897.06	0.437279
$267$	−4487.19	−1.02851
$268$	−526.355	−0.119971
$269$	−5426.92	−1.23006	−0.615028	−	0.788505i	$-0.710855\pi$
$269$	−5426.92	−1.23006	−0.615028	+	0.788505i	$0.710855\pi$
$270$	0	0
$271$	1514.61	0.339507	0.169753	−	0.985487i	$-0.445703\pi$
$271$	1514.61	0.339507	0.169753	+	0.985487i	$0.445703\pi$
$272$	−4884.04	−1.08875
$273$	435.416	0.0965296
$274$	−3447.59	−0.760134
$275$	0	0
$276$	746.120	0.162721
$277$	2872.72	0.623123	0.311562	−	0.950226i	$-0.399148\pi$
$277$	2872.72	0.623123	0.311562	+	0.950226i	$0.399148\pi$
$278$	5723.56	1.23481
$279$	290.903	0.0624226
$280$	0	0
$281$	3442.02	0.730724	0.365362	−	0.930866i	$-0.380945\pi$
$281$	3442.02	0.730724	0.365362	+	0.930866i	$0.380945\pi$
$282$	3029.01	0.639628
$283$	332.109	0.0697592	0.0348796	−	0.999392i	$-0.488895\pi$
$283$	332.109	0.0697592	0.0348796	+	0.999392i	$0.488895\pi$
$284$	447.367	0.0934730
$285$	0	0
$286$	−159.103	−0.0328950
$287$	969.577	0.199416
$288$	283.559	0.0580170
$289$	2368.98	0.482186
$290$	0	0
$291$	−4194.11	−0.844890
$292$	−146.880	−0.0294367
$293$	2263.84	0.451381	0.225691	−	0.974199i	$-0.427536\pi$
$293$	2263.84	0.451381	0.225691	+	0.974199i	$0.427536\pi$
$294$	2497.04	0.495342
$295$	0	0
$296$	−1309.64	−0.257166
$297$	1229.21	0.240154
$298$	5256.99	1.02191
$299$	−877.666	−0.169755
$300$	0	0
$301$	4045.06	0.774596
$302$	3713.97	0.707665
$303$	−11868.8	−2.25032
$304$	−2958.27	−0.558121
$305$	0	0
$306$	1870.43	0.349430
$307$	1041.17	0.193560	0.0967800	−	0.995306i	$-0.469146\pi$
$307$	1041.17	0.193560	0.0967800	+	0.995306i	$0.469146\pi$
$308$	−115.830	−0.0214286
$309$	6567.79	1.20915
$310$	0	0
$311$	−6108.88	−1.11384	−0.556918	−	0.830567i	$-0.688016\pi$
$311$	−6108.88	−1.11384	−0.556918	+	0.830567i	$0.688016\pi$
$312$	−752.195	−0.136489
$313$	2245.58	0.405519	0.202759	−	0.979229i	$-0.435009\pi$
$313$	2245.58	0.405519	0.202759	+	0.979229i	$0.435009\pi$
$314$	2795.56	0.502428
$315$	0	0
$316$	554.350	0.0986854
$317$	−6799.45	−1.20472	−0.602359	−	0.798226i	$-0.705772\pi$
$317$	−6799.45	−1.20472	−0.602359	+	0.798226i	$0.705772\pi$
$318$	−9946.72	−1.75404
$319$	182.471	0.0320263
$320$	0	0
$321$	−350.906	−0.0610146
$322$	5987.83	1.03630
$323$	4410.71	0.759809
$324$	680.846	0.116743
$325$	0	0
$326$	−8934.62	−1.51792
$327$	2765.89	0.467749
$328$	−1674.98	−0.281967
$329$	−2593.97	−0.434681
$330$	0	0
$331$	−1415.47	−0.235048	−0.117524	−	0.993070i	$-0.537496\pi$
$331$	−1415.47	−0.235048	−0.117524	+	0.993070i	$0.537496\pi$
$332$	1160.06	0.191766
$333$	452.736	0.0745038
$334$	−254.256	−0.0416536
$335$	0	0
$336$	4632.35	0.752130
$337$	7875.33	1.27299	0.636493	−	0.771282i	$-0.280384\pi$
$337$	7875.33	1.27299	0.636493	+	0.771282i	$0.280384\pi$
$338$	−5829.07	−0.938046
$339$	−490.133	−0.0785261
$340$	0	0
$341$	392.511	0.0623333
$342$	1132.92	0.179127
$343$	−6820.74	−1.07372
$344$	−6987.97	−1.09525
$345$	0	0
$346$	5826.42	0.905290
$347$	−8168.93	−1.26378	−0.631890	−	0.775058i	$-0.717720\pi$
$347$	−8168.93	−1.26378	−0.631890	+	0.775058i	$0.717720\pi$
$348$	75.8642	0.0116861
$349$	−7172.19	−1.10005	−0.550026	−	0.835147i	$-0.685382\pi$
$349$	−7172.19	−1.10005	−0.550026	+	0.835147i	$0.685382\pi$
$350$	0	0
$351$	−601.160	−0.0914175
$352$	382.603	0.0579340
$353$	9429.78	1.42180	0.710901	−	0.703292i	$-0.248287\pi$
$353$	9429.78	1.42180	0.710901	+	0.703292i	$0.248287\pi$
$354$	7462.17	1.12037
$355$	0	0
$356$	−583.788	−0.0869121
$357$	−6906.72	−1.02393
$358$	−2547.12	−0.376033
$359$	−3466.68	−0.509651	−0.254825	−	0.966987i	$-0.582018\pi$
$359$	−3466.68	−0.509651	−0.254825	+	0.966987i	$0.582018\pi$
$360$	0	0
$361$	−4187.43	−0.610501
$362$	5181.62	0.752321
$363$	−717.403	−0.103730
$364$	56.6481	0.00815705
$365$	0	0
$366$	2888.74	0.412560
$367$	−6298.17	−0.895808	−0.447904	−	0.894082i	$-0.647829\pi$
$367$	−6298.17	−0.895808	−0.447904	+	0.894082i	$0.647829\pi$
$368$	−9337.42	−1.32268
$369$	579.031	0.0816888
$370$	0	0
$371$	8518.10	1.19202
$372$	163.191	0.0227448
$373$	4618.79	0.641158	0.320579	−	0.947222i	$-0.396122\pi$
$373$	4618.79	0.641158	0.320579	+	0.947222i	$0.396122\pi$
$374$	2523.75	0.348930
$375$	0	0
$376$	4481.16	0.614623
$377$	−89.2397	−0.0121912
$378$	4101.38	0.558074
$379$	−14287.9	−1.93646	−0.968231	−	0.250058i	$-0.919550\pi$
$379$	−14287.9	−1.93646	−0.968231	+	0.250058i	$0.919550\pi$
$380$	0	0
$381$	−16375.3	−2.20192
$382$	−9272.45	−1.24194
$383$	−2577.57	−0.343885	−0.171942	−	0.985107i	$-0.555004\pi$
$383$	−2577.57	−0.343885	−0.171942	+	0.985107i	$0.555004\pi$
$384$	−7139.72	−0.948821
$385$	0	0
$386$	−8853.36	−1.16742
$387$	2415.71	0.317306
$388$	−545.658	−0.0713958
$389$	−3174.47	−0.413758	−0.206879	−	0.978367i	$-0.566331\pi$
$389$	−3174.47	−0.413758	−0.206879	+	0.978367i	$0.566331\pi$
$390$	0	0
$391$	13921.8	1.80066
$392$	3694.16	0.475977
$393$	−14658.6	−1.88150
$394$	−6947.07	−0.888295
$395$	0	0
$396$	−69.1736	−0.00877804
$397$	4445.10	0.561947	0.280974	−	0.959716i	$-0.409343\pi$
$397$	4445.10	0.561947	0.280974	+	0.959716i	$0.409343\pi$
$398$	−9951.54	−1.25333
$399$	−4183.41	−0.524893
$400$	0	0
$401$	1203.58	0.149885	0.0749424	−	0.997188i	$-0.476123\pi$
$401$	1203.58	0.149885	0.0749424	+	0.997188i	$0.476123\pi$
$402$	−10877.4	−1.34954
$403$	−191.963	−0.0237279
$404$	−1544.15	−0.190159
$405$	0	0
$406$	608.832	0.0744233
$407$	610.870	0.0743973
$408$	11931.6	1.44780
$409$	−12734.1	−1.53951	−0.769757	−	0.638337i	$-0.779623\pi$
$409$	−12734.1	−1.53951	−0.769757	+	0.638337i	$0.779623\pi$
$410$	0	0
$411$	7602.66	0.912437
$412$	854.476	0.102177
$413$	−6390.40	−0.761382
$414$	3575.93	0.424511
$415$	0	0
$416$	−187.117	−0.0220533
$417$	−12621.6	−1.48222
$418$	1528.64	0.178871
$419$	9678.32	1.12844	0.564220	−	0.825624i	$-0.309177\pi$
$419$	9678.32	1.12844	0.564220	+	0.825624i	$0.309177\pi$
$420$	0	0
$421$	1273.39	0.147414	0.0737069	−	0.997280i	$-0.476517\pi$
$421$	1273.39	0.147414	0.0737069	+	0.997280i	$0.476517\pi$
$422$	−6307.10	−0.727547
$423$	−1549.12	−0.178063
$424$	−14715.3	−1.68547
$425$	0	0
$426$	9245.09	1.05147
$427$	−2473.84	−0.280369
$428$	−45.6533	−0.00515592
$429$	350.855	0.0394859
$430$	0	0
$431$	−14802.4	−1.65431	−0.827153	−	0.561977i	$-0.810041\pi$
$431$	−14802.4	−1.65431	−0.827153	+	0.561977i	$0.810041\pi$
$432$	−6395.69	−0.712298
$433$	9511.58	1.05565	0.527826	−	0.849352i	$-0.323007\pi$
$433$	9511.58	1.05565	0.527826	+	0.849352i	$0.323007\pi$
$434$	1309.65	0.144851
$435$	0	0
$436$	359.845	0.0395262
$437$	8432.48	0.923067
$438$	−3035.36	−0.331131
$439$	3527.72	0.383529	0.191764	−	0.981441i	$-0.438579\pi$
$439$	3527.72	0.383529	0.191764	+	0.981441i	$0.438579\pi$
$440$	0	0
$441$	−1277.05	−0.137896
$442$	−1234.27	−0.132824
$443$	−15306.6	−1.64162	−0.820809	−	0.571203i	$-0.806477\pi$
$443$	−15306.6	−1.64162	−0.820809	+	0.571203i	$0.806477\pi$
$444$	253.976	0.0271468
$445$	0	0
$446$	17189.3	1.82497
$447$	−11592.8	−1.22666
$448$	7527.08	0.793797
$449$	−1861.35	−0.195641	−0.0978204	−	0.995204i	$-0.531187\pi$
$449$	−1861.35	−0.195641	−0.0978204	+	0.995204i	$0.531187\pi$
$450$	0	0
$451$	781.278	0.0815720
$452$	−63.7668	−0.00663570
$453$	−8190.07	−0.849455
$454$	2851.02	0.294725
$455$	0	0
$456$	7226.98	0.742180
$457$	6359.77	0.650979	0.325489	−	0.945546i	$-0.394471\pi$
$457$	6359.77	0.650979	0.325489	+	0.945546i	$0.394471\pi$
$458$	16722.2	1.70606
$459$	9535.80	0.969702
$460$	0	0
$461$	−7640.04	−0.771870	−0.385935	−	0.922526i	$-0.626121\pi$
$461$	−7640.04	−0.771870	−0.385935	+	0.922526i	$0.626121\pi$
$462$	−2393.69	−0.241049
$463$	17220.5	1.72852	0.864259	−	0.503047i	$-0.167788\pi$
$463$	17220.5	1.72852	0.864259	+	0.503047i	$0.167788\pi$
$464$	−949.413	−0.0949901
$465$	0	0
$466$	15638.6	1.55461
$467$	−8990.52	−0.890860	−0.445430	−	0.895317i	$-0.646949\pi$
$467$	−8990.52	−0.890860	−0.445430	+	0.895317i	$0.646949\pi$
$468$	33.8302	0.00334146
$469$	9315.13	0.917127
$470$	0	0
$471$	−6164.79	−0.603097
$472$	11039.6	1.07657
$473$	3259.48	0.316852
$474$	11456.0	1.11010
$475$	0	0
$476$	−898.571	−0.0865251
$477$	5087.01	0.488298
$478$	−4523.22	−0.432819
$479$	−8495.45	−0.810369	−0.405185	−	0.914235i	$-0.632793\pi$
$479$	−8495.45	−0.810369	−0.405185	+	0.914235i	$0.632793\pi$
$480$	0	0
$481$	−298.754	−0.0283202
$482$	13190.1	1.24646
$483$	−13204.4	−1.24394
$484$	−93.3349	−0.00876549
$485$	0	0
$486$	5958.14	0.556105
$487$	5415.27	0.503880	0.251940	−	0.967743i	$-0.418932\pi$
$487$	5415.27	0.503880	0.251940	+	0.967743i	$0.418932\pi$
$488$	4273.64	0.396431
$489$	19702.7	1.82206
$490$	0	0
$491$	−2645.56	−0.243162	−0.121581	−	0.992582i	$-0.538796\pi$
$491$	−2645.56	−0.243162	−0.121581	+	0.992582i	$0.538796\pi$
$492$	324.825	0.0297647
$493$	1415.55	0.129317
$494$	−747.600	−0.0680893
$495$	0	0
$496$	−2042.27	−0.184881
$497$	−7917.24	−0.714561
$498$	23973.2	2.15716
$499$	15415.5	1.38295	0.691475	−	0.722400i	$-0.256961\pi$
$499$	15415.5	1.38295	0.691475	+	0.722400i	$0.256961\pi$
$500$	0	0
$501$	560.688	0.0499994
$502$	13766.0	1.22392
$503$	−16932.8	−1.50099	−0.750495	−	0.660876i	$-0.770185\pi$
$503$	−16932.8	−1.50099	−0.750495	+	0.660876i	$0.770185\pi$
$504$	−2624.54	−0.231957
$505$	0	0
$506$	4824.95	0.423904
$507$	12854.3	1.12600
$508$	−2130.44	−0.186069
$509$	16107.7	1.40267	0.701335	−	0.712832i	$-0.252588\pi$
$509$	16107.7	1.40267	0.701335	+	0.712832i	$0.252588\pi$
$510$	0	0
$511$	2599.40	0.225031
$512$	−12788.6	−1.10387
$513$	5775.85	0.497096
$514$	−18548.3	−1.59169
$515$	0	0
$516$	1355.17	0.115616
$517$	−2090.20	−0.177808
$518$	2038.23	0.172885
$519$	−12848.5	−1.08668
$520$	0	0
$521$	−9416.19	−0.791806	−0.395903	−	0.918292i	$-0.629568\pi$
$521$	−9416.19	−0.791806	−0.395903	+	0.918292i	$0.629568\pi$
$522$	363.595	0.0304868
$523$	−7246.02	−0.605825	−0.302912	−	0.953018i	$-0.597959\pi$
$523$	−7246.02	−0.605825	−0.302912	+	0.953018i	$0.597959\pi$
$524$	−1907.10	−0.158992
$525$	0	0
$526$	−13258.2	−1.09902
$527$	3044.98	0.251691
$528$	3732.72	0.307662
$529$	14449.1	1.18756
$530$	0	0
$531$	−3816.35	−0.311893
$532$	−544.266	−0.0443551
$533$	−382.094	−0.0310513
$534$	−12064.3	−0.977666
$535$	0	0
$536$	−16092.2	−1.29679
$537$	5616.94	0.451376
$538$	−14590.9	−1.16925
$539$	−1723.11	−0.137699
$540$	0	0
$541$	−23163.6	−1.84082	−0.920409	−	0.390957i	$-0.872144\pi$
$541$	−23163.6	−1.84082	−0.920409	+	0.390957i	$0.872144\pi$
$542$	4072.21	0.322724
$543$	−11426.6	−0.903058
$544$	2968.11	0.233928
$545$	0	0
$546$	1170.66	0.0917579
$547$	−7837.87	−0.612656	−0.306328	−	0.951926i	$-0.599100\pi$
$547$	−7837.87	−0.612656	−0.306328	+	0.951926i	$0.599100\pi$
$548$	989.114	0.0771037
$549$	−1477.38	−0.114850
$550$	0	0
$551$	857.401	0.0662913
$552$	22811.0	1.75888
$553$	−9810.57	−0.754408
$554$	7723.64	0.592321
$555$	0	0
$556$	−1642.09	−0.125252
$557$	5364.01	0.408043	0.204022	−	0.978966i	$-0.434599\pi$
$557$	5364.01	0.408043	0.204022	+	0.978966i	$0.434599\pi$
$558$	782.125	0.0593369
$559$	−1594.09	−0.120613
$560$	0	0
$561$	−5565.39	−0.418843
$562$	9254.25	0.694603
$563$	−6644.79	−0.497415	−0.248707	−	0.968579i	$-0.580006\pi$
$563$	−6644.79	−0.497415	−0.248707	+	0.968579i	$0.580006\pi$
$564$	−869.024	−0.0648803
$565$	0	0
$566$	892.913	0.0663108
$567$	−12049.2	−0.892452
$568$	13677.3	1.01036
$569$	10821.4	0.797285	0.398642	−	0.917107i	$-0.369482\pi$
$569$	10821.4	0.797285	0.398642	+	0.917107i	$0.369482\pi$
$570$	0	0
$571$	15659.1	1.14766	0.573829	−	0.818975i	$-0.305457\pi$
$571$	15659.1	1.14766	0.573829	+	0.818975i	$0.305457\pi$
$572$	45.6466	0.00333668
$573$	20447.7	1.49078
$574$	2606.82	0.189558
$575$	0	0
$576$	4495.17	0.325172
$577$	6311.86	0.455401	0.227700	−	0.973731i	$-0.426879\pi$
$577$	6311.86	0.455401	0.227700	+	0.973731i	$0.426879\pi$
$578$	6369.27	0.458351
$579$	19523.5	1.40133
$580$	0	0
$581$	−20530.0	−1.46597
$582$	−11276.3	−0.803125
$583$	6863.83	0.487600
$584$	−4490.55	−0.318186
$585$	0	0
$586$	6086.58	0.429069
$587$	10734.5	0.754787	0.377394	−	0.926053i	$-0.376820\pi$
$587$	10734.5	0.754787	0.377394	+	0.926053i	$0.376820\pi$
$588$	−716.402	−0.0502447
$589$	1844.35	0.129024
$590$	0	0
$591$	15319.7	1.06628
$592$	−3178.42	−0.220662
$593$	−5046.93	−0.349499	−0.174749	−	0.984613i	$-0.555911\pi$
$593$	−5046.93	−0.349499	−0.174749	+	0.984613i	$0.555911\pi$
$594$	3304.86	0.228283
$595$	0	0
$596$	−1508.23	−0.103657
$597$	21945.2	1.50445
$598$	−2359.70	−0.161364
$599$	−10485.6	−0.715244	−0.357622	−	0.933866i	$-0.616412\pi$
$599$	−10485.6	−0.715244	−0.357622	+	0.933866i	$0.616412\pi$
$600$	0	0
$601$	−9459.70	−0.642045	−0.321023	−	0.947072i	$-0.604027\pi$
$601$	−9459.70	−0.642045	−0.321023	+	0.947072i	$0.604027\pi$
$602$	10875.6	0.736306
$603$	5563.00	0.375693
$604$	−1065.54	−0.0717816
$605$	0	0
$606$	−31910.7	−2.13908
$607$	−6659.26	−0.445290	−0.222645	−	0.974900i	$-0.571469\pi$
$607$	−6659.26	−0.445290	−0.222645	+	0.974900i	$0.571469\pi$
$608$	1797.79	0.119918
$609$	−1342.60	−0.0893350
$610$	0	0
$611$	1022.24	0.0676848
$612$	−536.627	−0.0354442
$613$	11480.6	0.756438	0.378219	−	0.925716i	$-0.376537\pi$
$613$	11480.6	0.756438	0.378219	+	0.925716i	$0.376537\pi$
$614$	2799.31	0.183992
$615$	0	0
$616$	−3541.25	−0.231625
$617$	−17395.2	−1.13502	−0.567509	−	0.823368i	$-0.692093\pi$
$617$	−17395.2	−1.13502	−0.567509	+	0.823368i	$0.692093\pi$
$618$	17658.2	1.14938
$619$	26031.1	1.69027	0.845137	−	0.534550i	$-0.179519\pi$
$619$	26031.1	1.69027	0.845137	+	0.534550i	$0.179519\pi$
$620$	0	0
$621$	18230.7	1.17806
$622$	−16424.4	−1.05878
$623$	10331.6	0.664406
$624$	−1825.54	−0.117115
$625$	0	0
$626$	6037.48	0.385473
$627$	−3370.96	−0.214710
$628$	−802.046	−0.0509636
$629$	4738.93	0.300403
$630$	0	0
$631$	4770.62	0.300975	0.150488	−	0.988612i	$-0.451916\pi$
$631$	4770.62	0.300975	0.150488	+	0.988612i	$0.451916\pi$
$632$	16948.1	1.06671
$633$	13908.5	0.873321
$634$	−18281.1	−1.14517
$635$	0	0
$636$	2853.71	0.177920
$637$	842.709	0.0524165
$638$	490.593	0.0304432
$639$	−4728.18	−0.292713
$640$	0	0
$641$	−16635.8	−1.02508	−0.512540	−	0.858664i	$-0.671295\pi$
$641$	−16635.8	−1.02508	−0.512540	+	0.858664i	$0.671295\pi$
$642$	−943.451	−0.0579985
$643$	−91.0331	−0.00558320	−0.00279160	−	0.999996i	$-0.500889\pi$
$643$	−91.0331	−0.00558320	−0.00279160	+	0.999996i	$0.500889\pi$
$644$	−1717.91	−0.105116
$645$	0	0
$646$	11858.7	0.722250
$647$	−24876.7	−1.51160	−0.755798	−	0.654804i	$-0.772751\pi$
$647$	−24876.7	−1.51160	−0.755798	+	0.654804i	$0.772751\pi$
$648$	20815.4	1.26189
$649$	−5149.34	−0.311447
$650$	0	0
$651$	−2888.06	−0.173874
$652$	2563.34	0.153970
$653$	2684.49	0.160876	0.0804380	−	0.996760i	$-0.474368\pi$
$653$	2684.49	0.160876	0.0804380	+	0.996760i	$0.474368\pi$
$654$	7436.40	0.444627
$655$	0	0
$656$	−4065.07	−0.241942
$657$	1552.36	0.0921818
$658$	−6974.17	−0.413194
$659$	−10069.8	−0.595241	−0.297620	−	0.954684i	$-0.596193\pi$
$659$	−10069.8	−0.595241	−0.297620	+	0.954684i	$0.596193\pi$
$660$	0	0
$661$	−838.461	−0.0493379	−0.0246690	−	0.999696i	$-0.507853\pi$
$661$	−838.461	−0.0493379	−0.0246690	+	0.999696i	$0.507853\pi$
$662$	−3805.64	−0.223430
$663$	2721.82	0.159437
$664$	35466.3	2.07283
$665$	0	0
$666$	1217.23	0.0708209
$667$	2706.28	0.157103
$668$	72.9461	0.00422510
$669$	−37906.0	−2.19063
$670$	0	0
$671$	−1993.40	−0.114686
$672$	−2815.15	−0.161603
$673$	27709.3	1.58709	0.793547	−	0.608508i	$-0.208232\pi$
$673$	27709.3	1.58709	0.793547	+	0.608508i	$0.208232\pi$
$674$	21173.7	1.21006
$675$	0	0
$676$	1672.36	0.0951502
$677$	−6178.28	−0.350740	−0.175370	−	0.984503i	$-0.556112\pi$
$677$	−6178.28	−0.350740	−0.175370	+	0.984503i	$0.556112\pi$
$678$	−1317.78	−0.0746444
$679$	9656.74	0.545790
$680$	0	0
$681$	−6287.09	−0.353777
$682$	1055.31	0.0592521
$683$	−3258.19	−0.182535	−0.0912674	−	0.995826i	$-0.529092\pi$
$683$	−3258.19	−0.182535	−0.0912674	+	0.995826i	$0.529092\pi$
$684$	−325.036	−0.0181697
$685$	0	0
$686$	−18338.3	−1.02064
$687$	−36876.0	−2.04790
$688$	−16959.4	−0.939784
$689$	−3356.84	−0.185610
$690$	0	0
$691$	35334.6	1.94528	0.972641	−	0.232314i	$-0.0746295\pi$
$691$	35334.6	1.94528	0.972641	+	0.232314i	$0.0746295\pi$
$692$	−1671.60	−0.0918276
$693$	1224.20	0.0671044
$694$	−21963.1	−1.20131
$695$	0	0
$696$	2319.39	0.126316
$697$	6060.91	0.329373
$698$	−19283.2	−1.04567
$699$	−34486.5	−1.86609
$700$	0	0
$701$	−2724.54	−0.146797	−0.0733983	−	0.997303i	$-0.523384\pi$
$701$	−2724.54	−0.146797	−0.0733983	+	0.997303i	$0.523384\pi$
$702$	−1616.29	−0.0868985
$703$	2870.38	0.153995
$704$	6065.27	0.324707
$705$	0	0
$706$	25353.0	1.35152
$707$	27327.5	1.45368
$708$	−2140.89	−0.113644
$709$	−17884.3	−0.947335	−0.473667	−	0.880704i	$-0.657070\pi$
$709$	−17884.3	−0.947335	−0.473667	+	0.880704i	$0.657070\pi$
$710$	0	0
$711$	−5858.87	−0.309036
$712$	−17848.1	−0.939446
$713$	5821.45	0.305771
$714$	−18569.5	−0.973313
$715$	0	0
$716$	730.770	0.0381427
$717$	9974.65	0.519540
$718$	−9320.57	−0.484458
$719$	−25807.6	−1.33861	−0.669305	−	0.742988i	$-0.733408\pi$
$719$	−25807.6	−1.33861	−0.669305	+	0.742988i	$0.733408\pi$
$720$	0	0
$721$	−15122.0	−0.781101
$722$	−11258.4	−0.580323
$723$	−29086.9	−1.49620
$724$	−1486.61	−0.0763112
$725$	0	0
$726$	−1928.82	−0.0986022
$727$	−22723.4	−1.15924	−0.579618	−	0.814888i	$-0.696798\pi$
$727$	−22723.4	−1.15924	−0.579618	+	0.814888i	$0.696798\pi$
$728$	1731.90	0.0881708
$729$	10692.7	0.543246
$730$	0	0
$731$	25286.0	1.27940
$732$	−828.779	−0.0418478
$733$	6004.95	0.302589	0.151295	−	0.988489i	$-0.451656\pi$
$733$	6004.95	0.302589	0.151295	+	0.988489i	$0.451656\pi$
$734$	−16933.3	−0.851527
$735$	0	0
$736$	5674.49	0.284191
$737$	7506.07	0.375156
$738$	1556.79	0.0776507
$739$	15968.1	0.794852	0.397426	−	0.917634i	$-0.369904\pi$
$739$	15968.1	0.794852	0.397426	+	0.917634i	$0.369904\pi$
$740$	0	0
$741$	1648.61	0.0817319
$742$	22901.9	1.13309
$743$	13555.6	0.669321	0.334661	−	0.942339i	$-0.391378\pi$
$743$	13555.6	0.669321	0.334661	+	0.942339i	$0.391378\pi$
$744$	4989.22	0.245852
$745$	0	0
$746$	12418.1	0.609465
$747$	−12260.5	−0.600522
$748$	−724.062	−0.0353935
$749$	807.946	0.0394148
$750$	0	0
$751$	16613.1	0.807216	0.403608	−	0.914932i	$-0.367756\pi$
$751$	16613.1	0.807216	0.403608	+	0.914932i	$0.367756\pi$
$752$	10875.5	0.527380
$753$	−30357.0	−1.46915
$754$	−239.931	−0.0115886
$755$	0	0
$756$	−1176.68	−0.0566080
$757$	−22933.9	−1.10112	−0.550560	−	0.834795i	$-0.685586\pi$
$757$	−22933.9	−1.10112	−0.550560	+	0.834795i	$0.685586\pi$
$758$	−38414.6	−1.84074
$759$	−10640.0	−0.508838
$760$	0	0
$761$	−10297.4	−0.490512	−0.245256	−	0.969458i	$-0.578872\pi$
$761$	−10297.4	−0.490512	−0.245256	+	0.969458i	$0.578872\pi$
$762$	−44026.7	−2.09307
$763$	−6368.33	−0.302161
$764$	2660.27	0.125975
$765$	0	0
$766$	−6930.09	−0.326886
$767$	2518.35	0.118556
$768$	6957.29	0.326888
$769$	20788.2	0.974826	0.487413	−	0.873172i	$-0.337941\pi$
$769$	20788.2	0.974826	0.487413	+	0.873172i	$0.337941\pi$
$770$	0	0
$771$	40902.9	1.91061
$772$	2540.03	0.118417
$773$	2934.95	0.136562	0.0682812	−	0.997666i	$-0.478249\pi$
$773$	2934.95	0.136562	0.0682812	+	0.997666i	$0.478249\pi$
$774$	6494.91	0.301621
$775$	0	0
$776$	−16682.3	−0.771728
$777$	−4494.72	−0.207525
$778$	−8534.91	−0.393305
$779$	3671.10	0.168846
$780$	0	0
$781$	−6379.66	−0.292295
$782$	37430.4	1.71165
$783$	1853.67	0.0846039
$784$	8965.51	0.408414
$785$	0	0
$786$	−39411.3	−1.78849
$787$	28424.2	1.28744	0.643718	−	0.765263i	$-0.277391\pi$
$787$	28424.2	1.28744	0.643718	+	0.765263i	$0.277391\pi$
$788$	1993.11	0.0901037
$789$	29237.1	1.31923
$790$	0	0
$791$	1128.51	0.0507271
$792$	−2114.84	−0.0948832
$793$	974.899	0.0436566
$794$	11951.1	0.534169
$795$	0	0
$796$	2855.10	0.127131
$797$	18507.8	0.822559	0.411280	−	0.911509i	$-0.365082\pi$
$797$	18507.8	0.822559	0.411280	+	0.911509i	$0.365082\pi$
$798$	−11247.6	−0.498947
$799$	−16215.1	−0.717959
$800$	0	0
$801$	6170.00	0.272168
$802$	3235.95	0.142476
$803$	2094.58	0.0920500
$804$	3120.73	0.136890
$805$	0	0
$806$	−516.113	−0.0225550
$807$	32175.9	1.40353
$808$	−47209.1	−2.05546
$809$	−27576.9	−1.19846	−0.599228	−	0.800578i	$-0.704526\pi$
$809$	−27576.9	−1.19846	−0.599228	+	0.800578i	$0.704526\pi$
$810$	0	0
$811$	−8553.71	−0.370360	−0.185180	−	0.982705i	$-0.559287\pi$
$811$	−8553.71	−0.370360	−0.185180	+	0.982705i	$0.559287\pi$
$812$	−174.674	−0.00754908
$813$	−8980.08	−0.387386
$814$	1642.39	0.0707197
$815$	0	0
$816$	28957.3	1.24229
$817$	15315.8	0.655853
$818$	−34237.1	−1.46341
$819$	−598.708	−0.0255440
$820$	0	0
$821$	27632.1	1.17462	0.587312	−	0.809360i	$-0.300186\pi$
$821$	27632.1	1.17462	0.587312	+	0.809360i	$0.300186\pi$
$822$	20440.6	0.867333
$823$	−644.423	−0.0272942	−0.0136471	−	0.999907i	$-0.504344\pi$
$823$	−644.423	−0.0272942	−0.0136471	+	0.999907i	$0.504344\pi$
$824$	26123.8	1.10445
$825$	0	0
$826$	−17181.3	−0.723746
$827$	−39394.2	−1.65643	−0.828216	−	0.560409i	$-0.810644\pi$
$827$	−39394.2	−1.65643	−0.828216	+	0.560409i	$0.810644\pi$
$828$	−1025.93	−0.0430600
$829$	−21915.9	−0.918179	−0.459089	−	0.888390i	$-0.651824\pi$
$829$	−21915.9	−0.918179	−0.459089	+	0.888390i	$0.651824\pi$
$830$	0	0
$831$	−17032.2	−0.711001
$832$	−2966.30	−0.123603
$833$	−13367.3	−0.556003
$834$	−33934.7	−1.40895
$835$	0	0
$836$	−438.566	−0.0181437
$837$	3987.42	0.164666
$838$	26021.2	1.07266
$839$	−525.642	−0.0216295	−0.0108148	−	0.999942i	$-0.503443\pi$
$839$	−525.642	−0.0216295	−0.0108148	+	0.999942i	$0.503443\pi$
$840$	0	0
$841$	−24113.8	−0.988717
$842$	3423.65	0.140127
$843$	−20407.5	−0.833776
$844$	1809.51	0.0737983
$845$	0	0
$846$	−4164.97	−0.169261
$847$	1651.79	0.0670084
$848$	−35713.2	−1.44622
$849$	−1969.06	−0.0795971
$850$	0	0
$851$	9059.99	0.364950
$852$	−2652.41	−0.106655
$853$	−8465.47	−0.339803	−0.169902	−	0.985461i	$-0.554345\pi$
$853$	−8465.47	−0.339803	−0.169902	+	0.985461i	$0.554345\pi$
$854$	−6651.19	−0.266510
$855$	0	0
$856$	−1395.75	−0.0557311
$857$	−3954.48	−0.157622	−0.0788112	−	0.996890i	$-0.525112\pi$
$857$	−3954.48	−0.157622	−0.0788112	+	0.996890i	$0.525112\pi$
$858$	943.314	0.0375340
$859$	18581.2	0.738047	0.369024	−	0.929420i	$-0.379692\pi$
$859$	18581.2	0.738047	0.369024	+	0.929420i	$0.379692\pi$
$860$	0	0
$861$	−5748.57	−0.227539
$862$	−39797.9	−1.57253
$863$	40385.4	1.59297	0.796487	−	0.604656i	$-0.206689\pi$
$863$	40385.4	1.59297	0.796487	+	0.604656i	$0.206689\pi$
$864$	3886.76	0.153044
$865$	0	0
$866$	25573.0	1.00347
$867$	−14045.6	−0.550187
$868$	−375.740	−0.0146929
$869$	−7905.29	−0.308594
$870$	0	0
$871$	−3670.94	−0.142807
$872$	11001.5	0.427245
$873$	5767.01	0.223578
$874$	22671.7	0.877438
$875$	0	0
$876$	870.845	0.0335881
$877$	16920.0	0.651481	0.325740	−	0.945459i	$-0.394386\pi$
$877$	16920.0	0.651481	0.325740	+	0.945459i	$0.394386\pi$
$878$	9484.68	0.364570
$879$	−13422.2	−0.515038
$880$	0	0
$881$	10081.2	0.385522	0.192761	−	0.981246i	$-0.438256\pi$
$881$	10081.2	0.385522	0.192761	+	0.981246i	$0.438256\pi$
$882$	−3433.50	−0.131079
$883$	−10340.6	−0.394098	−0.197049	−	0.980394i	$-0.563136\pi$
$883$	−10340.6	−0.394098	−0.197049	+	0.980394i	$0.563136\pi$
$884$	354.112	0.0134729
$885$	0	0
$886$	−41153.4	−1.56047
$887$	−17722.7	−0.670880	−0.335440	−	0.942062i	$-0.608885\pi$
$887$	−17722.7	−0.670880	−0.335440	+	0.942062i	$0.608885\pi$
$888$	7764.78	0.293433
$889$	37703.3	1.42242
$890$	0	0
$891$	−9709.19	−0.365062
$892$	−4931.61	−0.185115
$893$	−9821.52	−0.368046
$894$	−31168.5	−1.16603
$895$	0	0
$896$	16438.9	0.612929
$897$	5203.64	0.193695
$898$	−5004.46	−0.185970
$899$	591.916	0.0219594
$900$	0	0
$901$	53247.4	1.96884
$902$	2100.55	0.0775397
$903$	−23983.0	−0.883835
$904$	−1949.53	−0.0717263
$905$	0	0
$906$	−22019.9	−0.807465
$907$	16489.0	0.603646	0.301823	−	0.953364i	$-0.402405\pi$
$907$	16489.0	0.603646	0.301823	+	0.953364i	$0.402405\pi$
$908$	−817.957	−0.0298952
$909$	16320.0	0.595488
$910$	0	0
$911$	29428.2	1.07025	0.535127	−	0.844772i	$-0.320264\pi$
$911$	29428.2	1.07025	0.535127	+	0.844772i	$0.320264\pi$
$912$	17539.5	0.636831
$913$	−16543.0	−0.599663
$914$	17099.0	0.618800
$915$	0	0
$916$	−4797.60	−0.173054
$917$	33750.8	1.21543
$918$	25638.1	0.921767
$919$	−22961.8	−0.824200	−0.412100	−	0.911139i	$-0.635205\pi$
$919$	−22961.8	−0.824200	−0.412100	+	0.911139i	$0.635205\pi$
$920$	0	0
$921$	−6173.07	−0.220857
$922$	−20541.1	−0.733715
$923$	3120.06	0.111265
$924$	686.749	0.0244506
$925$	0	0
$926$	46299.2	1.64307
$927$	−9030.88	−0.319971
$928$	576.973	0.0204096
$929$	2036.28	0.0719142	0.0359571	−	0.999353i	$-0.488552\pi$
$929$	2036.28	0.0719142	0.0359571	+	0.999353i	$0.488552\pi$
$930$	0	0
$931$	−8096.62	−0.285022
$932$	−4486.72	−0.157690
$933$	36219.3	1.27092
$934$	−24172.0	−0.846823
$935$	0	0
$936$	1034.29	0.0361183
$937$	−37879.9	−1.32069	−0.660344	−	0.750964i	$-0.729589\pi$
$937$	−37879.9	−1.32069	−0.660344	+	0.750964i	$0.729589\pi$
$938$	25044.8	0.871792
$939$	−13313.9	−0.462708
$940$	0	0
$941$	37440.0	1.29703	0.648517	−	0.761200i	$-0.275390\pi$
$941$	37440.0	1.29703	0.648517	+	0.761200i	$0.275390\pi$
$942$	−16574.7	−0.573285
$943$	11587.4	0.400145
$944$	26792.5	0.923752
$945$	0	0
$946$	8763.49	0.301190
$947$	7494.83	0.257180	0.128590	−	0.991698i	$-0.458955\pi$
$947$	7494.83	0.257180	0.128590	+	0.991698i	$0.458955\pi$
$948$	−3286.71	−0.112603
$949$	−1024.38	−0.0350399
$950$	0	0
$951$	40313.6	1.37462
$952$	−27471.9	−0.935263
$953$	26703.2	0.907662	0.453831	−	0.891088i	$-0.350057\pi$
$953$	26703.2	0.907662	0.453831	+	0.891088i	$0.350057\pi$
$954$	13677.0	0.464160
$955$	0	0
$956$	1297.71	0.0439027
$957$	−1081.86	−0.0365429
$958$	−22841.0	−0.770311
$959$	−17504.8	−0.589425
$960$	0	0
$961$	−28517.7	−0.957260
$962$	−803.233	−0.0269202
$963$	482.505	0.0161459
$964$	−3784.24	−0.126434
$965$	0	0
$966$	−35501.5	−1.18245
$967$	−28221.4	−0.938511	−0.469256	−	0.883062i	$-0.655478\pi$
$967$	−28221.4	−0.938511	−0.469256	+	0.883062i	$0.655478\pi$
$968$	−2853.52	−0.0947475
$969$	−26150.9	−0.866963
$970$	0	0
$971$	35882.6	1.18592	0.592959	−	0.805233i	$-0.297960\pi$
$971$	35882.6	1.18592	0.592959	+	0.805233i	$0.297960\pi$
$972$	−1709.39	−0.0564082
$973$	29060.8	0.957498
$974$	14559.6	0.478972
$975$	0	0
$976$	10371.9	0.340159
$977$	−2339.87	−0.0766214	−0.0383107	−	0.999266i	$-0.512198\pi$
$977$	−2339.87	−0.0766214	−0.0383107	+	0.999266i	$0.512198\pi$
$978$	52972.9	1.73199
$979$	8325.09	0.271779
$980$	0	0
$981$	−3803.17	−0.123778
$982$	−7112.89	−0.231142
$983$	2328.54	0.0755534	0.0377767	−	0.999286i	$-0.487972\pi$
$983$	2328.54	0.0755534	0.0377767	+	0.999286i	$0.487972\pi$
$984$	9930.85	0.321731
$985$	0	0
$986$	3805.87	0.122924
$987$	15379.5	0.495983
$988$	214.486	0.00690660
$989$	48342.4	1.55430
$990$	0	0
$991$	−43619.2	−1.39819	−0.699097	−	0.715027i	$-0.746414\pi$
$991$	−43619.2	−1.39819	−0.699097	+	0.715027i	$0.746414\pi$
$992$	1241.12	0.0397235
$993$	8392.23	0.268197
$994$	−21286.4	−0.679239
$995$	0	0
$996$	−6877.92	−0.218810
$997$	13765.9	0.437281	0.218641	−	0.975805i	$-0.429838\pi$
$997$	13765.9	0.437281	0.218641	+	0.975805i	$0.429838\pi$
$998$	41446.3	1.31459
$999$	6205.66	0.196535

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.4.a.i.1.4	yes	5
3.2	odd	2		2475.4.a.bg.1.2		5
5.2	odd	4		275.4.b.g.199.8		10
5.3	odd	4		275.4.b.g.199.3		10
5.4	even	2		275.4.a.f.1.2	✓	5
15.14	odd	2		2475.4.a.bk.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.4.a.f.1.2	✓	5	5.4	even	2
275.4.a.i.1.4	yes	5	1.1	even	1	trivial
275.4.b.g.199.3		10	5.3	odd	4
275.4.b.g.199.8		10	5.2	odd	4
2475.4.a.bg.1.2		5	3.2	odd	2
2475.4.a.bk.1.4		5	15.14	odd	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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q+2.68861 q^{2} -5.92895 q^{3} -0.771363 q^{4} -15.9407 q^{6} +13.6511 q^{7} -23.5828 q^{8} +8.15246 q^{9} +11.0000 q^{11} +4.57337 q^{12} -5.37969 q^{13} +36.7026 q^{14} -57.2341 q^{16} +85.3345 q^{17} +21.9188 q^{18} +51.6873 q^{19} -80.9370 q^{21} +29.5747 q^{22} +163.144 q^{23} +139.821 q^{24} -14.4639 q^{26} +111.746 q^{27} -10.5300 q^{28} +16.5882 q^{29} +35.6828 q^{31} +34.7821 q^{32} -65.2185 q^{33} +229.431 q^{34} -6.28851 q^{36} +55.5336 q^{37} +138.967 q^{38} +31.8959 q^{39} +71.0253 q^{41} -217.608 q^{42} +296.317 q^{43} -8.48499 q^{44} +438.632 q^{46} -190.018 q^{47} +339.338 q^{48} -156.646 q^{49} -505.944 q^{51} +4.14969 q^{52} +623.984 q^{53} +300.442 q^{54} -321.932 q^{56} -306.451 q^{57} +44.5994 q^{58} -468.122 q^{59} -181.218 q^{61} +95.9373 q^{62} +111.290 q^{63} +551.388 q^{64} -175.347 q^{66} +682.370 q^{67} -65.8239 q^{68} -967.275 q^{69} -579.969 q^{71} -192.258 q^{72} +190.416 q^{73} +149.308 q^{74} -39.8696 q^{76} +150.163 q^{77} +85.7558 q^{78} -718.663 q^{79} -882.654 q^{81} +190.960 q^{82} -1503.91 q^{83} +62.4318 q^{84} +796.681 q^{86} -98.3509 q^{87} -259.411 q^{88} +756.827 q^{89} -73.4390 q^{91} -125.843 q^{92} -211.562 q^{93} -510.885 q^{94} -206.221 q^{96} +707.394 q^{97} -421.161 q^{98} +89.6771 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 12 q^{3} + 12 q^{4} + 8 q^{6} + 24 q^{7} + 27 q^{8} + 31 q^{9} + 55 q^{11} + 3 q^{12} + 111 q^{13} + 47 q^{14} - 56 q^{16} + 40 q^{17} + 217 q^{18} + 205 q^{19} - 94 q^{21} + 22 q^{22} + 287 q^{23}+ \cdots + 341 q^{99}+O(q^{100}) \) 5 * q + 2 * q^2 + 12 * q^3 + 12 * q^4 + 8 * q^6 + 24 * q^7 + 27 * q^8 + 31 * q^9 + 55 * q^11 + 3 * q^12 + 111 * q^13 + 47 * q^14 - 56 * q^16 + 40 * q^17 + 217 * q^18 + 205 * q^19 - 94 * q^21 + 22 * q^22 + 287 * q^23 + 273 * q^24 - 354 * q^26 + 270 * q^27 + 460 * q^28 + 251 * q^29 - 289 * q^31 + 248 * q^32 + 132 * q^33 + 522 * q^34 - 722 * q^36 + 224 * q^37 + 540 * q^38 + 538 * q^39 - 462 * q^41 + 175 * q^42 + 593 * q^43 + 132 * q^44 - 972 * q^46 + 766 * q^47 + 992 * q^48 + 75 * q^49 - 820 * q^51 + 696 * q^53 + 281 * q^54 - 527 * q^56 + 1170 * q^57 + 1461 * q^58 - 22 * q^59 + 720 * q^61 - 998 * q^62 - 168 * q^63 - 317 * q^64 + 88 * q^66 + 1230 * q^67 - 109 * q^68 - 514 * q^69 + 951 * q^71 - 1590 * q^72 + 666 * q^73 + 873 * q^74 + 290 * q^76 + 264 * q^77 + 56 * q^78 - 588 * q^79 + 1885 * q^81 - 1807 * q^82 - 867 * q^83 + 1321 * q^84 - 411 * q^86 + 766 * q^87 + 297 * q^88 - 51 * q^89 + 2172 * q^91 - 4137 * q^92 - 1916 * q^93 - 865 * q^94 + 603 * q^96 + 2849 * q^97 - 4104 * q^98 + 341 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more