LMFDB - Embedded newform 2075.4.a.k.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2075,4,Mod(1,2075)] 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("2075.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2075, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2075 = 5^{2} \cdot 83 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2075.a (trivial)

Newform invariants

comment:select newform

sage:traces = [47,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$122.428963262$
Analytic rank:	$0$
Dimension:	$47$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	2075.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.98801 q^{2} +4.43436 q^{3} +7.90426 q^{4} -17.6843 q^{6} -8.84879 q^{7} +0.381800 q^{8} -7.33643 q^{9} +52.4531 q^{11} +35.0504 q^{12} +0.962152 q^{13} +35.2891 q^{14} -64.7567 q^{16} +58.2046 q^{17} +29.2578 q^{18} +9.69079 q^{19} -39.2387 q^{21} -209.184 q^{22} +123.399 q^{23} +1.69304 q^{24} -3.83708 q^{26} -152.260 q^{27} -69.9432 q^{28} +33.7436 q^{29} +156.530 q^{31} +255.196 q^{32} +232.596 q^{33} -232.121 q^{34} -57.9891 q^{36} -76.7974 q^{37} -38.6470 q^{38} +4.26653 q^{39} +520.080 q^{41} +156.485 q^{42} -276.957 q^{43} +414.603 q^{44} -492.118 q^{46} -90.8164 q^{47} -287.155 q^{48} -264.699 q^{49} +258.100 q^{51} +7.60510 q^{52} +546.939 q^{53} +607.216 q^{54} -3.37846 q^{56} +42.9725 q^{57} -134.570 q^{58} +176.031 q^{59} -272.420 q^{61} -624.246 q^{62} +64.9185 q^{63} -499.673 q^{64} -927.597 q^{66} -883.433 q^{67} +460.064 q^{68} +547.196 q^{69} -1021.85 q^{71} -2.80105 q^{72} +322.584 q^{73} +306.269 q^{74} +76.5986 q^{76} -464.147 q^{77} -17.0150 q^{78} +329.789 q^{79} -477.093 q^{81} -2074.09 q^{82} +83.0000 q^{83} -310.153 q^{84} +1104.51 q^{86} +149.631 q^{87} +20.0266 q^{88} +1574.67 q^{89} -8.51388 q^{91} +975.379 q^{92} +694.112 q^{93} +362.177 q^{94} +1131.63 q^{96} -317.276 q^{97} +1055.62 q^{98} -384.819 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 47 q - q^{2} + 6 q^{3} + 229 q^{4} + 9 q^{6} + 6 q^{7} - 39 q^{8} + 531 q^{9} + 90 q^{11} - 6 q^{12} - 50 q^{13} + 115 q^{14} + 1149 q^{16} + 68 q^{17} - 8 q^{18} + 344 q^{19} + 625 q^{21} - 402 q^{22}+ \cdots + 13991 q^{99}+O(q^{100}) $ 47 * q - q^2 + 6 * q^3 + 229 * q^4 + 9 * q^6 + 6 * q^7 - 39 * q^8 + 531 * q^9 + 90 * q^11 - 6 * q^12 - 50 * q^13 + 115 * q^14 + 1149 * q^16 + 68 * q^17 - 8 * q^18 + 344 * q^19 + 625 * q^21 - 402 * q^22 + 469 * q^23 + 262 * q^24 + 1209 * q^26 - 12 * q^27 + 603 * q^28 + 480 * q^29 + 876 * q^31 - 1653 * q^32 + 757 * q^33 + 733 * q^34 + 4318 * q^36 - 634 * q^37 + 1190 * q^38 + 618 * q^39 + 1455 * q^41 - 2747 * q^42 + 652 * q^43 + 595 * q^44 + 1662 * q^46 - 446 * q^47 + 2145 * q^48 + 4099 * q^49 + 1681 * q^51 - 207 * q^52 + 126 * q^53 + 670 * q^54 + 3100 * q^56 - 650 * q^57 + 1082 * q^58 + 1144 * q^59 + 2762 * q^61 - 1916 * q^62 + 885 * q^63 + 6891 * q^64 + 2025 * q^66 + 100 * q^67 + 2052 * q^68 + 2556 * q^69 + 2546 * q^71 - 2940 * q^72 - 14 * q^73 + 1066 * q^74 + 4828 * q^76 + 210 * q^77 + 343 * q^78 + 1636 * q^79 + 8203 * q^81 - 480 * q^82 + 3901 * q^83 + 6209 * q^84 + 2706 * q^86 + 5479 * q^87 - 2178 * q^88 + 2558 * q^89 + 3342 * q^91 + 7501 * q^92 - 3353 * q^93 + 5769 * q^94 - 918 * q^96 + 3648 * q^97 - 8109 * q^98 + 13991 * 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 $

$2$	−3.98801	−1.40998	−0.704988	−	0.709219i	$-0.749048\pi$
$2$	−3.98801	−1.40998	−0.704988	+	0.709219i	$0.749048\pi$
$3$	4.43436	0.853393	0.426697	−	0.904395i	$-0.359677\pi$
$3$	4.43436	0.853393	0.426697	+	0.904395i	$0.359677\pi$
$4$	7.90426	0.988033
$5$	0	0
$5$	0	0
$6$	−17.6843	−1.20326
$7$	−8.84879	−0.477790	−0.238895	−	0.971045i	$-0.576785\pi$
$7$	−8.84879	−0.477790	−0.238895	+	0.971045i	$0.576785\pi$
$8$	0.381800	0.0168733
$9$	−7.33643	−0.271720
$10$	0	0
$11$	52.4531	1.43775	0.718873	−	0.695141i	$-0.244658\pi$
$11$	52.4531	1.43775	0.718873	+	0.695141i	$0.244658\pi$
$12$	35.0504	0.843181
$13$	0.962152	0.0205271	0.0102636	−	0.999947i	$-0.496733\pi$
$13$	0.962152	0.0205271	0.0102636	+	0.999947i	$0.496733\pi$
$14$	35.2891	0.673672
$15$	0	0
$16$	−64.7567	−1.01182
$17$	58.2046	0.830393	0.415197	−	0.909732i	$-0.363713\pi$
$17$	58.2046	0.830393	0.415197	+	0.909732i	$0.363713\pi$
$18$	29.2578	0.383118
$19$	9.69079	0.117012	0.0585058	−	0.998287i	$-0.481366\pi$
$19$	9.69079	0.117012	0.0585058	+	0.998287i	$0.481366\pi$
$20$	0	0
$21$	−39.2387	−0.407743
$22$	−209.184	−2.02719
$23$	123.399	1.11872	0.559359	−	0.828926i	$-0.311047\pi$
$23$	123.399	1.11872	0.559359	+	0.828926i	$0.311047\pi$
$24$	1.69304	0.0143996
$25$	0	0
$26$	−3.83708	−0.0289428
$27$	−152.260	−1.08528
$28$	−69.9432	−0.472072
$29$	33.7436	0.216070	0.108035	−	0.994147i	$-0.465544\pi$
$29$	33.7436	0.216070	0.108035	+	0.994147i	$0.465544\pi$
$30$	0	0
$31$	156.530	0.906893	0.453447	−	0.891283i	$-0.350194\pi$
$31$	156.530	0.906893	0.453447	+	0.891283i	$0.350194\pi$
$32$	255.196	1.40977
$33$	232.596	1.22696
$34$	−232.121	−1.17083
$35$	0	0
$36$	−57.9891	−0.268468
$37$	−76.7974	−0.341228	−0.170614	−	0.985338i	$-0.554575\pi$
$37$	−76.7974	−0.341228	−0.170614	+	0.985338i	$0.554575\pi$
$38$	−38.6470	−0.164984
$39$	4.26653	0.0175177
$40$	0	0
$41$	520.080	1.98104	0.990522	−	0.137356i	$-0.0438603\pi$
$41$	520.080	1.98104	0.990522	+	0.137356i	$0.0438603\pi$
$42$	156.485	0.574907
$43$	−276.957	−0.982223	−0.491111	−	0.871097i	$-0.663409\pi$
$43$	−276.957	−0.982223	−0.491111	+	0.871097i	$0.663409\pi$
$44$	414.603	1.42054
$45$	0	0
$46$	−492.118	−1.57736
$47$	−90.8164	−0.281850	−0.140925	−	0.990020i	$-0.545008\pi$
$47$	−90.8164	−0.281850	−0.140925	+	0.990020i	$0.545008\pi$
$48$	−287.155	−0.863484
$49$	−264.699	−0.771717
$50$	0	0
$51$	258.100	0.708652
$52$	7.60510	0.0202815
$53$	546.939	1.41751	0.708754	−	0.705456i	$-0.249258\pi$
$53$	546.939	1.41751	0.708754	+	0.705456i	$0.249258\pi$
$54$	607.216	1.53022
$55$	0	0
$56$	−3.37846	−0.00806190
$57$	42.9725	0.0998569
$58$	−134.570	−0.304653
$59$	176.031	0.388429	0.194214	−	0.980959i	$-0.437784\pi$
$59$	176.031	0.388429	0.194214	+	0.980959i	$0.437784\pi$
$60$	0	0
$61$	−272.420	−0.571800	−0.285900	−	0.958260i	$-0.592292\pi$
$61$	−272.420	−0.571800	−0.285900	+	0.958260i	$0.592292\pi$
$62$	−624.246	−1.27870
$63$	64.9185	0.129825
$64$	−499.673	−0.975924
$65$	0	0
$66$	−927.597	−1.72999
$67$	−883.433	−1.61087	−0.805437	−	0.592682i	$-0.798069\pi$
$67$	−883.433	−1.61087	−0.805437	+	0.592682i	$0.798069\pi$
$68$	460.064	0.820456
$69$	547.196	0.954706
$70$	0	0
$71$	−1021.85	−1.70805	−0.854025	−	0.520232i	$-0.825845\pi$
$71$	−1021.85	−1.70805	−0.854025	+	0.520232i	$0.825845\pi$
$72$	−2.80105	−0.00458481
$73$	322.584	0.517200	0.258600	−	0.965985i	$-0.416739\pi$
$73$	322.584	0.517200	0.258600	+	0.965985i	$0.416739\pi$
$74$	306.269	0.481123
$75$	0	0
$76$	76.5986	0.115611
$77$	−464.147	−0.686941
$78$	−17.0150	−0.0246996
$79$	329.789	0.469672	0.234836	−	0.972035i	$-0.424545\pi$
$79$	329.789	0.469672	0.234836	+	0.972035i	$0.424545\pi$
$80$	0	0
$81$	−477.093	−0.654449
$82$	−2074.09	−2.79322
$83$	83.0000	0.109764
$83$	83.0000	0.109764
$84$	−310.153	−0.402863
$85$	0	0
$86$	1104.51	1.38491
$87$	149.631	0.184393
$88$	20.0266	0.0242596
$89$	1574.67	1.87545	0.937725	−	0.347380i	$-0.112929\pi$
$89$	1574.67	1.87545	0.937725	+	0.347380i	$0.112929\pi$
$90$	0	0
$91$	−8.51388	−0.00980766
$92$	975.379	1.10533
$93$	694.112	0.773937
$94$	362.177	0.397401
$95$	0	0
$96$	1131.63	1.20309
$97$	−317.276	−0.332108	−0.166054	−	0.986117i	$-0.553103\pi$
$97$	−317.276	−0.332108	−0.166054	+	0.986117i	$0.553103\pi$
$98$	1055.62	1.08810
$99$	−384.819	−0.390664
$100$	0	0
$101$	49.7506	0.0490136	0.0245068	−	0.999700i	$-0.492198\pi$
$101$	49.7506	0.0490136	0.0245068	+	0.999700i	$0.492198\pi$
$102$	−1029.31	−0.999183
$103$	−369.753	−0.353717	−0.176859	−	0.984236i	$-0.556593\pi$
$103$	−369.753	−0.353717	−0.176859	+	0.984236i	$0.556593\pi$
$104$	0.367349	0.000346361 0
$105$	0	0
$106$	−2181.20	−1.99865
$107$	−1235.42	−1.11619	−0.558096	−	0.829776i	$-0.688468\pi$
$107$	−1235.42	−1.11619	−0.558096	+	0.829776i	$0.688468\pi$
$108$	−1203.50	−1.07229
$109$	446.389	0.392260	0.196130	−	0.980578i	$-0.437163\pi$
$109$	446.389	0.392260	0.196130	+	0.980578i	$0.437163\pi$
$110$	0	0
$111$	−340.548	−0.291201
$112$	573.019	0.483439
$113$	661.570	0.550755	0.275377	−	0.961336i	$-0.411197\pi$
$113$	661.570	0.550755	0.275377	+	0.961336i	$0.411197\pi$
$114$	−171.375	−0.140796
$115$	0	0
$116$	266.718	0.213484
$117$	−7.05876	−0.00557763
$118$	−702.014	−0.547675
$119$	−515.040	−0.396753
$120$	0	0
$121$	1420.33	1.06712
$122$	1086.41	0.806224
$123$	2306.22	1.69061
$124$	1237.26	0.896040
$125$	0	0
$126$	−258.896	−0.183050
$127$	−190.871	−0.133363	−0.0666814	−	0.997774i	$-0.521241\pi$
$127$	−190.871	−0.133363	−0.0666814	+	0.997774i	$0.521241\pi$
$128$	−48.8668	−0.0337442
$129$	−1228.13	−0.838222
$130$	0	0
$131$	1194.90	0.796937	0.398469	−	0.917182i	$-0.369542\pi$
$131$	1194.90	0.796937	0.398469	+	0.917182i	$0.369542\pi$
$132$	1838.50	1.21228
$133$	−85.7518	−0.0559069
$134$	3523.14	2.27129
$135$	0	0
$136$	22.2225	0.0140115
$137$	1096.63	0.683878	0.341939	−	0.939722i	$-0.388916\pi$
$137$	1096.63	0.683878	0.341939	+	0.939722i	$0.388916\pi$
$138$	−2182.23	−1.34611
$139$	2227.82	1.35944	0.679718	−	0.733474i	$-0.262102\pi$
$139$	2227.82	1.35944	0.679718	+	0.733474i	$0.262102\pi$
$140$	0	0
$141$	−402.713	−0.240529
$142$	4075.16	2.40831
$143$	50.4679	0.0295128
$144$	475.083	0.274932
$145$	0	0
$146$	−1286.47	−0.729239
$147$	−1173.77	−0.658578
$148$	−607.027	−0.337144
$149$	591.441	0.325186	0.162593	−	0.986693i	$-0.448014\pi$
$149$	591.441	0.325186	0.162593	+	0.986693i	$0.448014\pi$
$150$	0	0
$151$	3631.82	1.95731	0.978653	−	0.205520i	$-0.0658886\pi$
$151$	3631.82	1.95731	0.978653	+	0.205520i	$0.0658886\pi$
$152$	3.69994	0.00197437
$153$	−427.014	−0.225634
$154$	1851.02	0.968570
$155$	0	0
$156$	33.7238	0.0173081
$157$	−1368.38	−0.695599	−0.347799	−	0.937569i	$-0.613071\pi$
$157$	−1368.38	−0.695599	−0.347799	+	0.937569i	$0.613071\pi$
$158$	−1315.20	−0.662227
$159$	2425.33	1.20969
$160$	0	0
$161$	−1091.93	−0.534512
$162$	1902.65	0.922757
$163$	−3709.95	−1.78274	−0.891368	−	0.453281i	$-0.850253\pi$
$163$	−3709.95	−1.78274	−0.891368	+	0.453281i	$0.850253\pi$
$164$	4110.85	1.95734
$165$	0	0
$166$	−331.005	−0.154765
$167$	−4094.91	−1.89745	−0.948723	−	0.316110i	$-0.897623\pi$
$167$	−4094.91	−1.89745	−0.948723	+	0.316110i	$0.897623\pi$
$168$	−14.9813	−0.00687997
$169$	−2196.07	−0.999579
$170$	0	0
$171$	−71.0958	−0.0317943
$172$	−2189.14	−0.970468
$173$	3230.14	1.41956	0.709778	−	0.704425i	$-0.248795\pi$
$173$	3230.14	1.41956	0.709778	+	0.704425i	$0.248795\pi$
$174$	−596.732	−0.259989
$175$	0	0
$176$	−3396.69	−1.45475
$177$	780.585	0.331482
$178$	−6279.82	−2.64434
$179$	2541.69	1.06131	0.530657	−	0.847587i	$-0.321945\pi$
$179$	2541.69	1.06131	0.530657	+	0.847587i	$0.321945\pi$
$180$	0	0
$181$	1652.46	0.678601	0.339300	−	0.940678i	$-0.389810\pi$
$181$	1652.46	0.678601	0.339300	+	0.940678i	$0.389810\pi$
$182$	33.9535	0.0138286
$183$	−1208.01	−0.487970
$184$	47.1137	0.0188765
$185$	0	0
$186$	−2768.13	−1.09123
$187$	3053.01	1.19390
$188$	−717.837	−0.278477
$189$	1347.32	0.518534
$190$	0	0
$191$	3585.92	1.35847	0.679236	−	0.733919i	$-0.262311\pi$
$191$	3585.92	1.35847	0.679236	+	0.733919i	$0.262311\pi$
$192$	−2215.73	−0.832847
$193$	−3254.75	−1.21390	−0.606949	−	0.794741i	$-0.707607\pi$
$193$	−3254.75	−1.21390	−0.606949	+	0.794741i	$0.707607\pi$
$194$	1265.30	0.468264
$195$	0	0
$196$	−2092.25	−0.762482
$197$	−1456.19	−0.526646	−0.263323	−	0.964708i	$-0.584819\pi$
$197$	−1456.19	−0.526646	−0.263323	+	0.964708i	$0.584819\pi$
$198$	1534.66	0.550827
$199$	5103.56	1.81800	0.909000	−	0.416796i	$-0.136847\pi$
$199$	5103.56	1.81800	0.909000	+	0.416796i	$0.136847\pi$
$200$	0	0
$201$	−3917.46	−1.37471
$202$	−198.406	−0.0691080
$203$	−298.590	−0.103236
$204$	2040.09	0.700172
$205$	0	0
$206$	1474.58	0.498733
$207$	−905.309	−0.303977
$208$	−62.3058	−0.0207699
$209$	508.312	0.168233
$210$	0	0
$211$	−4576.00	−1.49301	−0.746505	−	0.665380i	$-0.768269\pi$
$211$	−4576.00	−1.49301	−0.746505	+	0.665380i	$0.768269\pi$
$212$	4323.15	1.40054
$213$	−4531.26	−1.45764
$214$	4926.87	1.57380
$215$	0	0
$216$	−58.1329	−0.0183122
$217$	−1385.10	−0.433304
$218$	−1780.21	−0.553077
$219$	1430.45	0.441375
$220$	0	0
$221$	56.0016	0.0170456
$222$	1358.11	0.410587
$223$	−1949.48	−0.585413	−0.292706	−	0.956202i	$-0.594556\pi$
$223$	−1949.48	−0.585413	−0.292706	+	0.956202i	$0.594556\pi$
$224$	−2258.18	−0.673576
$225$	0	0
$226$	−2638.35	−0.776551
$227$	1886.41	0.551565	0.275782	−	0.961220i	$-0.411063\pi$
$227$	1886.41	0.551565	0.275782	+	0.961220i	$0.411063\pi$
$228$	339.666	0.0986619
$229$	−5157.91	−1.48840	−0.744202	−	0.667954i	$-0.767170\pi$
$229$	−5157.91	−1.48840	−0.744202	+	0.667954i	$0.767170\pi$
$230$	0	0
$231$	−2058.19	−0.586231
$232$	12.8833	0.00364581
$233$	−4228.22	−1.18884	−0.594420	−	0.804155i	$-0.702618\pi$
$233$	−4228.22	−1.18884	−0.594420	+	0.804155i	$0.702618\pi$
$234$	28.1504	0.00786433
$235$	0	0
$236$	1391.40	0.383780
$237$	1462.40	0.400815
$238$	2053.99	0.559413
$239$	−2363.15	−0.639580	−0.319790	−	0.947488i	$-0.603612\pi$
$239$	−2363.15	−0.639580	−0.319790	+	0.947488i	$0.603612\pi$
$240$	0	0
$241$	5839.13	1.56071	0.780355	−	0.625336i	$-0.215038\pi$
$241$	5839.13	1.56071	0.780355	+	0.625336i	$0.215038\pi$
$242$	−5664.30	−1.50461
$243$	1995.42	0.526775
$244$	−2153.28	−0.564957
$245$	0	0
$246$	−9197.25	−2.38372
$247$	9.32401	0.00240191
$248$	59.7632	0.0153023
$249$	368.052	0.0936721
$250$	0	0
$251$	−3673.23	−0.923714	−0.461857	−	0.886954i	$-0.652817\pi$
$251$	−3673.23	−0.923714	−0.461857	+	0.886954i	$0.652817\pi$
$252$	513.133	0.128271
$253$	6472.67	1.60843
$254$	761.197	0.188038
$255$	0	0
$256$	4192.27	1.02350
$257$	5757.67	1.39749	0.698743	−	0.715373i	$-0.253743\pi$
$257$	5757.67	1.39749	0.698743	+	0.715373i	$0.253743\pi$
$258$	4897.79	1.18187
$259$	679.564	0.163035
$260$	0	0
$261$	−247.557	−0.0587104
$262$	−4765.27	−1.12366
$263$	−4166.66	−0.976910	−0.488455	−	0.872589i	$-0.662439\pi$
$263$	−4166.66	−0.976910	−0.488455	+	0.872589i	$0.662439\pi$
$264$	88.8051	0.0207029
$265$	0	0
$266$	341.979	0.0788274
$267$	6982.67	1.60050
$268$	−6982.89	−1.59160
$269$	−2790.13	−0.632406	−0.316203	−	0.948692i	$-0.602408\pi$
$269$	−2790.13	−0.632406	−0.316203	+	0.948692i	$0.602408\pi$
$270$	0	0
$271$	919.598	0.206131	0.103066	−	0.994675i	$-0.467135\pi$
$271$	919.598	0.206131	0.103066	+	0.994675i	$0.467135\pi$
$272$	−3769.14	−0.840212
$273$	−37.7536	−0.00836979
$274$	−4373.37	−0.964251
$275$	0	0
$276$	4325.18	0.943281
$277$	1601.53	0.347389	0.173695	−	0.984800i	$-0.444429\pi$
$277$	1601.53	0.347389	0.173695	+	0.984800i	$0.444429\pi$
$278$	−8884.60	−1.91677
$279$	−1148.37	−0.246421
$280$	0	0
$281$	5330.71	1.13168	0.565842	−	0.824514i	$-0.308551\pi$
$281$	5330.71	1.13168	0.565842	+	0.824514i	$0.308551\pi$
$282$	1606.02	0.339140
$283$	4816.61	1.01172	0.505862	−	0.862615i	$-0.331175\pi$
$283$	4816.61	1.01172	0.505862	+	0.862615i	$0.331175\pi$
$284$	−8076.99	−1.68761
$285$	0	0
$286$	−201.267	−0.0416124
$287$	−4602.08	−0.946522
$288$	−1872.23	−0.383063
$289$	−1525.23	−0.310447
$290$	0	0
$291$	−1406.91	−0.283419
$292$	2549.79	0.511010
$293$	3376.51	0.673234	0.336617	−	0.941642i	$-0.390717\pi$
$293$	3376.51	0.673234	0.336617	+	0.941642i	$0.390717\pi$
$294$	4681.02	0.928580
$295$	0	0
$296$	−29.3212	−0.00575764
$297$	−7986.52	−1.56035
$298$	−2358.68	−0.458505
$299$	118.729	0.0229641
$300$	0	0
$301$	2450.74	0.469296
$302$	−14483.7	−2.75975
$303$	220.612	0.0418279
$304$	−627.544	−0.118395
$305$	0	0
$306$	1702.94	0.318139
$307$	2455.51	0.456493	0.228247	−	0.973603i	$-0.426701\pi$
$307$	2455.51	0.456493	0.228247	+	0.973603i	$0.426701\pi$
$308$	−3668.74	−0.678720
$309$	−1639.62	−0.301860
$310$	0	0
$311$	1162.36	0.211933	0.105967	−	0.994370i	$-0.466206\pi$
$311$	1162.36	0.211933	0.105967	+	0.994370i	$0.466206\pi$
$312$	1.62896	0.000295582 0
$313$	−3780.35	−0.682677	−0.341338	−	0.939940i	$-0.610880\pi$
$313$	−3780.35	−0.682677	−0.341338	+	0.939940i	$0.610880\pi$
$314$	5457.14	0.980778
$315$	0	0
$316$	2606.74	0.464052
$317$	678.717	0.120254	0.0601270	−	0.998191i	$-0.480849\pi$
$317$	678.717	0.120254	0.0601270	+	0.998191i	$0.480849\pi$
$318$	−9672.24	−1.70564
$319$	1769.96	0.310654
$320$	0	0
$321$	−5478.30	−0.952551
$322$	4354.65	0.753649
$323$	564.048	0.0971656
$324$	−3771.07	−0.646617
$325$	0	0
$326$	14795.3	2.51361
$327$	1979.45	0.334752
$328$	198.566	0.0334268
$329$	803.615	0.134665
$330$	0	0
$331$	−7990.08	−1.32681	−0.663406	−	0.748260i	$-0.730890\pi$
$331$	−7990.08	−1.32681	−0.663406	+	0.748260i	$0.730890\pi$
$332$	656.054	0.108451
$333$	563.419	0.0927182
$334$	16330.5	2.67535
$335$	0	0
$336$	2540.97	0.412564
$337$	3045.42	0.492269	0.246134	−	0.969236i	$-0.420840\pi$
$337$	3045.42	0.492269	0.246134	+	0.969236i	$0.420840\pi$
$338$	8757.98	1.40938
$339$	2933.64	0.470011
$340$	0	0
$341$	8210.51	1.30388
$342$	283.531	0.0448293
$343$	5377.40	0.846508
$344$	−105.742	−0.0165734
$345$	0	0
$346$	−12881.9	−2.00154
$347$	8796.74	1.36090	0.680452	−	0.732792i	$-0.261783\pi$
$347$	8796.74	1.36090	0.680452	+	0.732792i	$0.261783\pi$
$348$	1182.72	0.182186
$349$	−2491.17	−0.382089	−0.191045	−	0.981581i	$-0.561187\pi$
$349$	−2491.17	−0.382089	−0.191045	+	0.981581i	$0.561187\pi$
$350$	0	0
$351$	−146.497	−0.0222776
$352$	13385.8	2.02690
$353$	7917.03	1.19371	0.596857	−	0.802348i	$-0.296416\pi$
$353$	7917.03	1.19371	0.596857	+	0.802348i	$0.296416\pi$
$354$	−3112.99	−0.467382
$355$	0	0
$356$	12446.6	1.85301
$357$	−2283.87	−0.338587
$358$	−10136.3	−1.49643
$359$	6736.53	0.990363	0.495182	−	0.868790i	$-0.335102\pi$
$359$	6736.53	0.990363	0.495182	+	0.868790i	$0.335102\pi$
$360$	0	0
$361$	−6765.09	−0.986308
$362$	−6590.05	−0.956811
$363$	6298.26	0.910669
$364$	−67.2960	−0.00969029
$365$	0	0
$366$	4817.55	0.688026
$367$	10789.4	1.53461	0.767306	−	0.641281i	$-0.221597\pi$
$367$	10789.4	1.53461	0.767306	+	0.641281i	$0.221597\pi$
$368$	−7990.92	−1.13194
$369$	−3815.53	−0.538289
$370$	0	0
$371$	−4839.75	−0.677270
$372$	5486.45	0.764675
$373$	2053.75	0.285092	0.142546	−	0.989788i	$-0.454471\pi$
$373$	2053.75	0.285092	0.142546	+	0.989788i	$0.454471\pi$
$374$	−12175.5	−1.68336
$375$	0	0
$376$	−34.6737	−0.00475574
$377$	32.4664	0.00443530
$378$	−5373.13	−0.731121
$379$	4048.58	0.548711	0.274356	−	0.961628i	$-0.411535\pi$
$379$	4048.58	0.548711	0.274356	+	0.961628i	$0.411535\pi$
$380$	0	0
$381$	−846.392	−0.113811
$382$	−14300.7	−1.91541
$383$	5601.58	0.747330	0.373665	−	0.927564i	$-0.378101\pi$
$383$	5601.58	0.747330	0.373665	+	0.927564i	$0.378101\pi$
$384$	−216.693	−0.0287971
$385$	0	0
$386$	12980.0	1.71157
$387$	2031.88	0.266889
$388$	−2507.83	−0.328133
$389$	4603.99	0.600081	0.300040	−	0.953927i	$-0.403000\pi$
$389$	4603.99	0.600081	0.300040	+	0.953927i	$0.403000\pi$
$390$	0	0
$391$	7182.39	0.928975
$392$	−101.062	−0.0130214
$393$	5298.61	0.680101
$394$	5807.31	0.742559
$395$	0	0
$396$	−3041.71	−0.385989
$397$	5899.76	0.745845	0.372922	−	0.927863i	$-0.378356\pi$
$397$	5899.76	0.745845	0.372922	+	0.927863i	$0.378356\pi$
$398$	−20353.1	−2.56334
$399$	−380.254	−0.0477106
$400$	0	0
$401$	−3806.44	−0.474026	−0.237013	−	0.971506i	$-0.576168\pi$
$401$	−3806.44	−0.474026	−0.237013	+	0.971506i	$0.576168\pi$
$402$	15622.9	1.93831
$403$	150.606	0.0186159
$404$	393.242	0.0484271
$405$	0	0
$406$	1190.78	0.145560
$407$	−4028.27	−0.490599
$408$	98.5425	0.0119573
$409$	−9086.49	−1.09853	−0.549264	−	0.835649i	$-0.685092\pi$
$409$	−9086.49	−1.09853	−0.549264	+	0.835649i	$0.685092\pi$
$410$	0	0
$411$	4862.85	0.583617
$412$	−2922.63	−0.349484
$413$	−1557.66	−0.185587
$414$	3610.39	0.428601
$415$	0	0
$416$	245.538	0.0289386
$417$	9878.98	1.16013
$418$	−2027.16	−0.237204
$419$	8179.37	0.953671	0.476836	−	0.878992i	$-0.341784\pi$
$419$	8179.37	0.953671	0.476836	+	0.878992i	$0.341784\pi$
$420$	0	0
$421$	7592.19	0.878909	0.439455	−	0.898265i	$-0.355172\pi$
$421$	7592.19	0.878909	0.439455	+	0.898265i	$0.355172\pi$
$422$	18249.2	2.10511
$423$	666.268	0.0765841
$424$	208.821	0.0239180
$425$	0	0
$426$	18070.7	2.05524
$427$	2410.59	0.273200
$428$	−9765.09	−1.10283
$429$	223.793	0.0251861
$430$	0	0
$431$	4847.63	0.541768	0.270884	−	0.962612i	$-0.412684\pi$
$431$	4847.63	0.541768	0.270884	+	0.962612i	$0.412684\pi$
$432$	9859.87	1.09811
$433$	11155.3	1.23808	0.619041	−	0.785359i	$-0.287521\pi$
$433$	11155.3	1.23808	0.619041	+	0.785359i	$0.287521\pi$
$434$	5523.82	0.610949
$435$	0	0
$436$	3528.37	0.387565
$437$	1195.83	0.130903
$438$	−5704.67	−0.622328
$439$	11673.2	1.26910	0.634548	−	0.772883i	$-0.281186\pi$
$439$	11673.2	1.26910	0.634548	+	0.772883i	$0.281186\pi$
$440$	0	0
$441$	1941.95	0.209691
$442$	−223.335	−0.0240339
$443$	749.078	0.0803380	0.0401690	−	0.999193i	$-0.487210\pi$
$443$	749.078	0.0803380	0.0401690	+	0.999193i	$0.487210\pi$
$444$	−2691.78	−0.287717
$445$	0	0
$446$	7774.57	0.825418
$447$	2622.66	0.277512
$448$	4421.50	0.466287
$449$	4931.16	0.518298	0.259149	−	0.965837i	$-0.416558\pi$
$449$	4931.16	0.518298	0.259149	+	0.965837i	$0.416558\pi$
$450$	0	0
$451$	27279.8	2.84824
$452$	5229.23	0.544164
$453$	16104.8	1.67035
$454$	−7523.02	−0.777693
$455$	0	0
$456$	16.4069	0.00168492
$457$	9351.22	0.957181	0.478590	−	0.878038i	$-0.341148\pi$
$457$	9351.22	0.957181	0.478590	+	0.878038i	$0.341148\pi$
$458$	20569.8	2.09861
$459$	−8862.24	−0.901207
$460$	0	0
$461$	13291.2	1.34281	0.671404	−	0.741091i	$-0.265691\pi$
$461$	13291.2	1.34281	0.671404	+	0.741091i	$0.265691\pi$
$462$	8208.11	0.826571
$463$	−17624.6	−1.76908	−0.884539	−	0.466467i	$-0.845527\pi$
$463$	−17624.6	−1.76908	−0.884539	+	0.466467i	$0.845527\pi$
$464$	−2185.12	−0.218625
$465$	0	0
$466$	16862.2	1.67624
$467$	6703.38	0.664230	0.332115	−	0.943239i	$-0.392238\pi$
$467$	6703.38	0.664230	0.332115	+	0.943239i	$0.392238\pi$
$468$	−55.7943	−0.00551088
$469$	7817.31	0.769659
$470$	0	0
$471$	−6067.91	−0.593619
$472$	67.2086	0.00655408
$473$	−14527.3	−1.41219
$474$	−5832.08	−0.565140
$475$	0	0
$476$	−4071.01	−0.392005
$477$	−4012.58	−0.385165
$478$	9424.29	0.901792
$479$	7332.12	0.699401	0.349701	−	0.936862i	$-0.386283\pi$
$479$	7332.12	0.699401	0.349701	+	0.936862i	$0.386283\pi$
$480$	0	0
$481$	−73.8908	−0.00700443
$482$	−23286.5	−2.20056
$483$	−4842.03	−0.456149
$484$	11226.7	1.05434
$485$	0	0
$486$	−7957.77	−0.742740
$487$	−15681.5	−1.45913	−0.729564	−	0.683913i	$-0.760277\pi$
$487$	−15681.5	−1.45913	−0.729564	+	0.683913i	$0.760277\pi$
$488$	−104.010	−0.00964815
$489$	−16451.3	−1.52137
$490$	0	0
$491$	−8749.24	−0.804171	−0.402085	−	0.915602i	$-0.631715\pi$
$491$	−8749.24	−0.804171	−0.402085	+	0.915602i	$0.631715\pi$
$492$	18229.0	1.67038
$493$	1964.03	0.179423
$494$	−37.1843	−0.00338664
$495$	0	0
$496$	−10136.4	−0.917616
$497$	9042.15	0.816089
$498$	−1467.80	−0.132075
$499$	16330.3	1.46502	0.732511	−	0.680755i	$-0.238348\pi$
$499$	16330.3	1.46502	0.732511	+	0.680755i	$0.238348\pi$
$500$	0	0
$501$	−18158.3	−1.61927
$502$	14648.9	1.30241
$503$	12246.5	1.08558	0.542788	−	0.839870i	$-0.317369\pi$
$503$	12246.5	1.08558	0.542788	+	0.839870i	$0.317369\pi$
$504$	24.7859	0.00219058
$505$	0	0
$506$	−25813.1	−2.26785
$507$	−9738.19	−0.853034
$508$	−1508.70	−0.131767
$509$	−5589.79	−0.486764	−0.243382	−	0.969930i	$-0.578257\pi$
$509$	−5589.79	−0.486764	−0.243382	+	0.969930i	$0.578257\pi$
$510$	0	0
$511$	−2854.48	−0.247113
$512$	−16327.9	−1.40937
$513$	−1475.52	−0.126990
$514$	−22961.7	−1.97042
$515$	0	0
$516$	−9707.45	−0.828191
$517$	−4763.60	−0.405228
$518$	−2710.11	−0.229876
$519$	14323.6	1.21144
$520$	0	0
$521$	−5897.21	−0.495896	−0.247948	−	0.968773i	$-0.579756\pi$
$521$	−5897.21	−0.495896	−0.247948	+	0.968773i	$0.579756\pi$
$522$	987.263	0.0827803
$523$	18408.9	1.53913	0.769565	−	0.638568i	$-0.220473\pi$
$523$	18408.9	1.53913	0.769565	+	0.638568i	$0.220473\pi$
$524$	9444.80	0.787400
$525$	0	0
$526$	16616.7	1.37742
$527$	9110.79	0.753078
$528$	−15062.2	−1.24147
$529$	3060.34	0.251528
$530$	0	0
$531$	−1291.44	−0.105544
$532$	−677.805	−0.0552379
$533$	500.396	0.0406652
$534$	−27847.0	−2.25666
$535$	0	0
$536$	−337.294	−0.0271808
$537$	11270.8	0.905718
$538$	11127.1	0.891678
$539$	−13884.3	−1.10953
$540$	0	0
$541$	15499.8	1.23177	0.615885	−	0.787836i	$-0.288799\pi$
$541$	15499.8	1.23177	0.615885	+	0.787836i	$0.288799\pi$
$542$	−3667.37	−0.290640
$543$	7327.62	0.579113
$544$	14853.6	1.17067
$545$	0	0
$546$	150.562	0.0118012
$547$	15497.4	1.21138	0.605688	−	0.795702i	$-0.292898\pi$
$547$	15497.4	1.21138	0.605688	+	0.795702i	$0.292898\pi$
$548$	8668.04	0.675694
$549$	1998.59	0.155369
$550$	0	0
$551$	327.002	0.0252827
$552$	208.919	0.0161091
$553$	−2918.23	−0.224405
$554$	−6386.94	−0.489811
$555$	0	0
$556$	17609.3	1.34317
$557$	−1240.98	−0.0944025	−0.0472012	−	0.998885i	$-0.515030\pi$
$557$	−1240.98	−0.0944025	−0.0472012	+	0.998885i	$0.515030\pi$
$558$	4579.73	0.347447
$559$	−266.475	−0.0201622
$560$	0	0
$561$	13538.2	1.01886
$562$	−21258.9	−1.59565
$563$	5459.99	0.408723	0.204362	−	0.978895i	$-0.434488\pi$
$563$	5459.99	0.408723	0.204362	+	0.978895i	$0.434488\pi$
$564$	−3183.15	−0.237650
$565$	0	0
$566$	−19208.7	−1.42651
$567$	4221.70	0.312689
$568$	−390.143	−0.0288205
$569$	3868.41	0.285013	0.142506	−	0.989794i	$-0.454484\pi$
$569$	3868.41	0.285013	0.142506	+	0.989794i	$0.454484\pi$
$570$	0	0
$571$	20731.5	1.51942	0.759708	−	0.650264i	$-0.225342\pi$
$571$	20731.5	1.51942	0.759708	+	0.650264i	$0.225342\pi$
$572$	398.911	0.0291597
$573$	15901.3	1.15931
$574$	18353.1	1.33457
$575$	0	0
$576$	3665.82	0.265178
$577$	−1557.94	−0.112406	−0.0562028	−	0.998419i	$-0.517899\pi$
$577$	−1557.94	−0.112406	−0.0562028	+	0.998419i	$0.517899\pi$
$578$	6082.63	0.437723
$579$	−14432.8	−1.03593
$580$	0	0
$581$	−734.450	−0.0524442
$582$	5610.80	0.399613
$583$	28688.7	2.03802
$584$	123.162	0.00872687
$585$	0	0
$586$	−13465.6	−0.949244
$587$	1514.60	0.106498	0.0532488	−	0.998581i	$-0.483042\pi$
$587$	1514.60	0.106498	0.0532488	+	0.998581i	$0.483042\pi$
$588$	−9277.79	−0.650697
$589$	1516.90	0.106117
$590$	0	0
$591$	−6457.28	−0.449437
$592$	4973.15	0.345262
$593$	−7811.48	−0.540943	−0.270471	−	0.962728i	$-0.587180\pi$
$593$	−7811.48	−0.540943	−0.270471	+	0.962728i	$0.587180\pi$
$594$	31850.4	2.20006
$595$	0	0
$596$	4674.91	0.321295
$597$	22631.1	1.55147
$598$	−473.492	−0.0323788
$599$	−22845.7	−1.55835	−0.779175	−	0.626806i	$-0.784362\pi$
$599$	−22845.7	−1.55835	−0.779175	+	0.626806i	$0.784362\pi$
$600$	0	0
$601$	−4707.20	−0.319486	−0.159743	−	0.987159i	$-0.551066\pi$
$601$	−4707.20	−0.319486	−0.159743	+	0.987159i	$0.551066\pi$
$602$	−9773.57	−0.661696
$603$	6481.24	0.437706
$604$	28706.8	1.93388
$605$	0	0
$606$	−879.806	−0.0589763
$607$	−16745.2	−1.11971	−0.559856	−	0.828590i	$-0.689144\pi$
$607$	−16745.2	−1.11971	−0.559856	+	0.828590i	$0.689144\pi$
$608$	2473.05	0.164960
$609$	−1324.06	−0.0881009
$610$	0	0
$611$	−87.3792	−0.00578557
$612$	−3375.23	−0.222934
$613$	−4682.36	−0.308513	−0.154257	−	0.988031i	$-0.549298\pi$
$613$	−4682.36	−0.308513	−0.154257	+	0.988031i	$0.549298\pi$
$614$	−9792.61	−0.643644
$615$	0	0
$616$	−177.211	−0.0115910
$617$	−12369.8	−0.807112	−0.403556	−	0.914955i	$-0.632226\pi$
$617$	−12369.8	−0.807112	−0.403556	+	0.914955i	$0.632226\pi$
$618$	6538.83	0.425615
$619$	22280.3	1.44672	0.723362	−	0.690469i	$-0.242596\pi$
$619$	22280.3	1.44672	0.723362	+	0.690469i	$0.242596\pi$
$620$	0	0
$621$	−18788.8	−1.21412
$622$	−4635.50	−0.298821
$623$	−13933.9	−0.896070
$624$	−276.287	−0.0177249
$625$	0	0
$626$	15076.1	0.962558
$627$	2254.04	0.143569
$628$	−10816.1	−0.687274
$629$	−4469.96	−0.283353
$630$	0	0
$631$	6018.99	0.379734	0.189867	−	0.981810i	$-0.439194\pi$
$631$	6018.99	0.379734	0.189867	+	0.981810i	$0.439194\pi$
$632$	125.913	0.00792493
$633$	−20291.6	−1.27412
$634$	−2706.73	−0.169555
$635$	0	0
$636$	19170.4	1.19521
$637$	−254.681	−0.0158411
$638$	−7058.61	−0.438014
$639$	7496.75	0.464111
$640$	0	0
$641$	−2732.44	−0.168370	−0.0841848	−	0.996450i	$-0.526829\pi$
$641$	−2732.44	−0.168370	−0.0841848	+	0.996450i	$0.526829\pi$
$642$	21847.5	1.34307
$643$	2452.93	0.150442	0.0752209	−	0.997167i	$-0.476034\pi$
$643$	2452.93	0.150442	0.0752209	+	0.997167i	$0.476034\pi$
$644$	−8630.93	−0.528115
$645$	0	0
$646$	−2249.43	−0.137001
$647$	21675.7	1.31709	0.658546	−	0.752541i	$-0.271172\pi$
$647$	21675.7	1.31709	0.658546	+	0.752541i	$0.271172\pi$
$648$	−182.154	−0.0110427
$649$	9233.38	0.558462
$650$	0	0
$651$	−6142.06	−0.369779
$652$	−29324.4	−1.76140
$653$	26735.5	1.60221	0.801104	−	0.598525i	$-0.204246\pi$
$653$	26735.5	1.60221	0.801104	+	0.598525i	$0.204246\pi$
$654$	−7894.07	−0.471992
$655$	0	0
$656$	−33678.7	−2.00447
$657$	−2366.61	−0.140533
$658$	−3204.83	−0.189874
$659$	−9354.29	−0.552946	−0.276473	−	0.961022i	$-0.589166\pi$
$659$	−9354.29	−0.552946	−0.276473	+	0.961022i	$0.589166\pi$
$660$	0	0
$661$	−17127.5	−1.00784	−0.503921	−	0.863749i	$-0.668110\pi$
$661$	−17127.5	−1.00784	−0.503921	+	0.863749i	$0.668110\pi$
$662$	31864.6	1.87077
$663$	248.332	0.0145466
$664$	31.6894	0.00185209
$665$	0	0
$666$	−2246.92	−0.130731
$667$	4163.93	0.241721
$668$	−32367.2	−1.87474
$669$	−8644.72	−0.499587
$670$	0	0
$671$	−14289.3	−0.822103
$672$	−10013.6	−0.574825
$673$	18329.4	1.04985	0.524923	−	0.851150i	$-0.324094\pi$
$673$	18329.4	1.04985	0.524923	+	0.851150i	$0.324094\pi$
$674$	−12145.2	−0.694087
$675$	0	0
$676$	−17358.3	−0.987617
$677$	−25116.7	−1.42587	−0.712934	−	0.701231i	$-0.752634\pi$
$677$	−25116.7	−1.42587	−0.712934	+	0.701231i	$0.752634\pi$
$678$	−11699.4	−0.662704
$679$	2807.50	0.158678
$680$	0	0
$681$	8365.01	0.470702
$682$	−32743.6	−1.83844
$683$	−18557.8	−1.03967	−0.519834	−	0.854268i	$-0.674006\pi$
$683$	−18557.8	−1.03967	−0.519834	+	0.854268i	$0.674006\pi$
$684$	−561.960	−0.0314139
$685$	0	0
$686$	−21445.2	−1.19356
$687$	−22872.1	−1.27019
$688$	17934.8	0.993836
$689$	526.239	0.0290974
$690$	0	0
$691$	32891.9	1.81081	0.905403	−	0.424554i	$-0.139569\pi$
$691$	32891.9	1.81081	0.905403	+	0.424554i	$0.139569\pi$
$692$	25531.9	1.40257
$693$	3405.18	0.186655
$694$	−35081.5	−1.91884
$695$	0	0
$696$	57.1291	0.00311131
$697$	30271.0	1.64505
$698$	9934.81	0.538737
$699$	−18749.4	−1.01455
$700$	0	0
$701$	17939.4	0.966567	0.483283	−	0.875464i	$-0.339444\pi$
$701$	17939.4	0.966567	0.483283	+	0.875464i	$0.339444\pi$
$702$	584.234	0.0314109
$703$	−744.228	−0.0399276
$704$	−26209.4	−1.40313
$705$	0	0
$706$	−31573.2	−1.68311
$707$	−440.233	−0.0234182
$708$	6169.95	0.327515
$709$	−2147.18	−0.113736	−0.0568681	−	0.998382i	$-0.518111\pi$
$709$	−2147.18	−0.113736	−0.0568681	+	0.998382i	$0.518111\pi$
$710$	0	0
$711$	−2419.47	−0.127619
$712$	601.209	0.0316450
$713$	19315.7	1.01456
$714$	9108.13	0.477399
$715$	0	0
$716$	20090.2	1.04861
$717$	−10479.1	−0.545813
$718$	−26865.4	−1.39639
$719$	−27518.8	−1.42737	−0.713683	−	0.700469i	$-0.752974\pi$
$719$	−27518.8	−1.42737	−0.713683	+	0.700469i	$0.752974\pi$
$720$	0	0
$721$	3271.87	0.169002
$722$	26979.3	1.39067
$723$	25892.8	1.33190
$724$	13061.5	0.670480
$725$	0	0
$726$	−25117.6	−1.28402
$727$	−5077.35	−0.259021	−0.129511	−	0.991578i	$-0.541341\pi$
$727$	−5077.35	−0.259021	−0.129511	+	0.991578i	$0.541341\pi$
$728$	−3.25060	−0.000165488 0
$729$	21729.9	1.10400
$730$	0	0
$731$	−16120.2	−0.815631
$732$	−9548.41	−0.482130
$733$	21133.2	1.06490	0.532450	−	0.846462i	$-0.321272\pi$
$733$	21133.2	1.06490	0.532450	+	0.846462i	$0.321272\pi$
$734$	−43028.3	−2.16377
$735$	0	0
$736$	31491.0	1.57714
$737$	−46338.8	−2.31603
$738$	15216.4	0.758974
$739$	−23956.5	−1.19249	−0.596247	−	0.802801i	$-0.703342\pi$
$739$	−23956.5	−1.19249	−0.596247	+	0.802801i	$0.703342\pi$
$740$	0	0
$741$	41.3460	0.00204978
$742$	19301.0	0.954935
$743$	−6517.42	−0.321805	−0.160902	−	0.986970i	$-0.551440\pi$
$743$	−6517.42	−0.321805	−0.160902	+	0.986970i	$0.551440\pi$
$744$	265.012	0.0130589
$745$	0	0
$746$	−8190.40	−0.401973
$747$	−608.924	−0.0298251
$748$	24131.8	1.17961
$749$	10932.0	0.533305
$750$	0	0
$751$	−34284.7	−1.66587	−0.832934	−	0.553372i	$-0.813341\pi$
$751$	−34284.7	−1.66587	−0.832934	+	0.553372i	$0.813341\pi$
$752$	5880.97	0.285182
$753$	−16288.4	−0.788291
$754$	−129.477	−0.00625366
$755$	0	0
$756$	10649.6	0.512329
$757$	28705.1	1.37821	0.689106	−	0.724661i	$-0.258004\pi$
$757$	28705.1	1.37821	0.689106	+	0.724661i	$0.258004\pi$
$758$	−16145.8	−0.773670
$759$	28702.2	1.37263
$760$	0	0
$761$	−28492.3	−1.35722	−0.678609	−	0.734499i	$-0.737417\pi$
$761$	−28492.3	−1.35722	−0.678609	+	0.734499i	$0.737417\pi$
$762$	3375.42	0.160471
$763$	−3950.00	−0.187418
$764$	28344.1	1.34222
$765$	0	0
$766$	−22339.2	−1.05372
$767$	169.369	0.00797333
$768$	18590.0	0.873451
$769$	35907.2	1.68381	0.841904	−	0.539627i	$-0.181435\pi$
$769$	35907.2	1.68381	0.841904	+	0.539627i	$0.181435\pi$
$770$	0	0
$771$	25531.6	1.19261
$772$	−25726.4	−1.19937
$773$	20291.4	0.944154	0.472077	−	0.881557i	$-0.343504\pi$
$773$	20291.4	0.944154	0.472077	+	0.881557i	$0.343504\pi$
$774$	−8103.16	−0.376307
$775$	0	0
$776$	−121.136	−0.00560376
$777$	3013.43	0.139133
$778$	−18360.8	−0.846099
$779$	5039.98	0.231805
$780$	0	0
$781$	−53599.3	−2.45574
$782$	−28643.5	−1.30983
$783$	−5137.80	−0.234496
$784$	17141.0	0.780842
$785$	0	0
$786$	−21131.0	−0.958926
$787$	16582.6	0.751087	0.375544	−	0.926805i	$-0.377456\pi$
$787$	16582.6	0.751087	0.375544	+	0.926805i	$0.377456\pi$
$788$	−11510.1	−0.520344
$789$	−18476.5	−0.833689
$790$	0	0
$791$	−5854.10	−0.263145
$792$	−146.924	−0.00659180
$793$	−262.109	−0.0117374
$794$	−23528.3	−1.05162
$795$	0	0
$796$	40339.9	1.79624
$797$	28736.7	1.27717	0.638585	−	0.769551i	$-0.279520\pi$
$797$	28736.7	1.27717	0.638585	+	0.769551i	$0.279520\pi$
$798$	1516.46	0.0672708
$799$	−5285.93	−0.234046
$800$	0	0
$801$	−11552.5	−0.509596
$802$	15180.1	0.668366
$803$	16920.5	0.743602
$804$	−30964.6	−1.35826
$805$	0	0
$806$	−600.619	−0.0262480
$807$	−12372.4	−0.539691
$808$	18.9948	0.000827022 0
$809$	−12129.9	−0.527149	−0.263574	−	0.964639i	$-0.584901\pi$
$809$	−12129.9	−0.527149	−0.263574	+	0.964639i	$0.584901\pi$
$810$	0	0
$811$	−15010.3	−0.649918	−0.324959	−	0.945728i	$-0.605351\pi$
$811$	−15010.3	−0.649918	−0.324959	+	0.945728i	$0.605351\pi$
$812$	−2360.13	−0.102000
$813$	4077.83	0.175911
$814$	16064.8	0.691733
$815$	0	0
$816$	−16713.7	−0.717031
$817$	−2683.93	−0.114931
$818$	36237.0	1.54890
$819$	62.4615	0.00266493
$820$	0	0
$821$	33180.3	1.41048	0.705238	−	0.708970i	$-0.250840\pi$
$821$	33180.3	1.41048	0.705238	+	0.708970i	$0.250840\pi$
$822$	−19393.1	−0.822886
$823$	−36445.9	−1.54365	−0.771826	−	0.635834i	$-0.780656\pi$
$823$	−36445.9	−1.54365	−0.771826	+	0.635834i	$0.780656\pi$
$824$	−141.172	−0.00596838
$825$	0	0
$826$	6211.98	0.261674
$827$	−27615.0	−1.16115	−0.580573	−	0.814208i	$-0.697172\pi$
$827$	−27615.0	−1.16115	−0.580573	+	0.814208i	$0.697172\pi$
$828$	−7155.80	−0.300340
$829$	20221.4	0.847188	0.423594	−	0.905852i	$-0.360768\pi$
$829$	20221.4	0.847188	0.423594	+	0.905852i	$0.360768\pi$
$830$	0	0
$831$	7101.78	0.296460
$832$	−480.762	−0.0200329
$833$	−15406.7	−0.640829
$834$	−39397.5	−1.63576
$835$	0	0
$836$	4017.83	0.166220
$837$	−23833.3	−0.984231
$838$	−32619.4	−1.34465
$839$	34370.3	1.41430	0.707148	−	0.707066i	$-0.249982\pi$
$839$	34370.3	1.41430	0.707148	+	0.707066i	$0.249982\pi$
$840$	0	0
$841$	−23250.4	−0.953314
$842$	−30277.8	−1.23924
$843$	23638.3	0.965772
$844$	−36169.9	−1.47514
$845$	0	0
$846$	−2657.09	−0.107982
$847$	−12568.2	−0.509857
$848$	−35418.0	−1.43427
$849$	21358.6	0.863398
$850$	0	0
$851$	−9476.74	−0.381737
$852$	−35816.3	−1.44019
$853$	−8431.95	−0.338458	−0.169229	−	0.985577i	$-0.554128\pi$
$853$	−8431.95	−0.338458	−0.169229	+	0.985577i	$0.554128\pi$
$854$	−9613.45	−0.385206
$855$	0	0
$856$	−471.683	−0.0188339
$857$	−14228.6	−0.567142	−0.283571	−	0.958951i	$-0.591519\pi$
$857$	−14228.6	−0.567142	−0.283571	+	0.958951i	$0.591519\pi$
$858$	−892.489	−0.0355117
$859$	−4495.67	−0.178568	−0.0892842	−	0.996006i	$-0.528458\pi$
$859$	−4495.67	−0.178568	−0.0892842	+	0.996006i	$0.528458\pi$
$860$	0	0
$861$	−20407.3	−0.807756
$862$	−19332.4	−0.763880
$863$	4782.34	0.188636	0.0943179	−	0.995542i	$-0.469933\pi$
$863$	4782.34	0.188636	0.0943179	+	0.995542i	$0.469933\pi$
$864$	−38856.2	−1.53000
$865$	0	0
$866$	−44487.5	−1.74567
$867$	−6763.41	−0.264933
$868$	−10948.2	−0.428119
$869$	17298.4	0.675270
$870$	0	0
$871$	−849.997	−0.0330666
$872$	170.431	0.00661872
$873$	2327.67	0.0902402
$874$	−4769.01	−0.184570
$875$	0	0
$876$	11306.7	0.436093
$877$	−36418.0	−1.40222	−0.701112	−	0.713051i	$-0.747313\pi$
$877$	−36418.0	−1.40222	−0.701112	+	0.713051i	$0.747313\pi$
$878$	−46553.1	−1.78940
$879$	14972.7	0.574534
$880$	0	0
$881$	18898.1	0.722695	0.361347	−	0.932431i	$-0.382317\pi$
$881$	18898.1	0.722695	0.361347	+	0.932431i	$0.382317\pi$
$882$	−7744.51	−0.295659
$883$	10411.0	0.396781	0.198391	−	0.980123i	$-0.436429\pi$
$883$	10411.0	0.396781	0.198391	+	0.980123i	$0.436429\pi$
$884$	442.652	0.0168416
$885$	0	0
$886$	−2987.33	−0.113275
$887$	24716.3	0.935615	0.467808	−	0.883830i	$-0.345044\pi$
$887$	24716.3	0.935615	0.467808	+	0.883830i	$0.345044\pi$
$888$	−130.021	−0.00491353
$889$	1688.98	0.0637194
$890$	0	0
$891$	−25025.0	−0.940931
$892$	−15409.2	−0.578407
$893$	−880.083	−0.0329797
$894$	−10459.2	−0.391285
$895$	0	0
$896$	432.412	0.0161226
$897$	526.486	0.0195974
$898$	−19665.5	−0.730788
$899$	5281.89	0.195952
$900$	0	0
$901$	31834.4	1.17709
$902$	−108792.	−4.01595
$903$	10867.5	0.400494
$904$	252.587	0.00929306
$905$	0	0
$906$	−64226.2	−2.35516
$907$	−17074.3	−0.625075	−0.312537	−	0.949905i	$-0.601179\pi$
$907$	−17074.3	−0.625075	−0.312537	+	0.949905i	$0.601179\pi$
$908$	14910.7	0.544964
$909$	−364.992	−0.0133180
$910$	0	0
$911$	−39449.4	−1.43470	−0.717352	−	0.696710i	$-0.754646\pi$
$911$	−39449.4	−1.43470	−0.717352	+	0.696710i	$0.754646\pi$
$912$	−2782.76	−0.101038
$913$	4353.61	0.157813
$914$	−37292.8	−1.34960
$915$	0	0
$916$	−40769.5	−1.47059
$917$	−10573.4	−0.380769
$918$	35342.7	1.27068
$919$	−27032.6	−0.970321	−0.485160	−	0.874425i	$-0.661239\pi$
$919$	−27032.6	−0.970321	−0.485160	+	0.874425i	$0.661239\pi$
$920$	0	0
$921$	10888.6	0.389568
$922$	−53005.7	−1.89333
$923$	−983.177	−0.0350614
$924$	−16268.5	−0.579215
$925$	0	0
$926$	70287.0	2.49436
$927$	2712.67	0.0961119
$928$	8611.24	0.304610
$929$	−11054.3	−0.390398	−0.195199	−	0.980764i	$-0.562535\pi$
$929$	−11054.3	−0.390398	−0.195199	+	0.980764i	$0.562535\pi$
$930$	0	0
$931$	−2565.14	−0.0902998
$932$	−33420.9	−1.17461
$933$	5154.31	0.180862
$934$	−26733.2	−0.936549
$935$	0	0
$936$	−2.69503	−9.41131e−5 0
$937$	−13156.1	−0.458687	−0.229343	−	0.973346i	$-0.573658\pi$
$937$	−13156.1	−0.458687	−0.229343	+	0.973346i	$0.573658\pi$
$938$	−31175.6	−1.08520
$939$	−16763.4	−0.582592
$940$	0	0
$941$	37520.5	1.29982	0.649912	−	0.760010i	$-0.274806\pi$
$941$	37520.5	1.29982	0.649912	+	0.760010i	$0.274806\pi$
$942$	24198.9	0.836989
$943$	64177.4	2.21623
$944$	−11399.2	−0.393021
$945$	0	0
$946$	57935.0	1.99115
$947$	1142.67	0.0392099	0.0196050	−	0.999808i	$-0.493759\pi$
$947$	1142.67	0.0392099	0.0196050	+	0.999808i	$0.493759\pi$
$948$	11559.2	0.396019
$949$	310.375	0.0106166
$950$	0	0
$951$	3009.68	0.102624
$952$	−196.642	−0.00669455
$953$	−20516.0	−0.697354	−0.348677	−	0.937243i	$-0.613369\pi$
$953$	−20516.0	−0.697354	−0.348677	+	0.937243i	$0.613369\pi$
$954$	16002.2	0.543073
$955$	0	0
$956$	−18679.0	−0.631926
$957$	7848.63	0.265110
$958$	−29240.6	−0.986139
$959$	−9703.83	−0.326750
$960$	0	0
$961$	−5289.23	−0.177545
$962$	294.678	0.00987608
$963$	9063.58	0.303291
$964$	46154.0	1.54203
$965$	0	0
$966$	19310.1	0.643159
$967$	−10935.2	−0.363651	−0.181826	−	0.983331i	$-0.558201\pi$
$967$	−10935.2	−0.363651	−0.181826	+	0.983331i	$0.558201\pi$
$968$	542.281	0.0180058
$969$	2501.19	0.0829205
$970$	0	0
$971$	−14188.5	−0.468929	−0.234465	−	0.972125i	$-0.575334\pi$
$971$	−14188.5	−0.468929	−0.234465	+	0.972125i	$0.575334\pi$
$972$	15772.3	0.520471
$973$	−19713.6	−0.649525
$974$	62537.9	2.05734
$975$	0	0
$976$	17641.0	0.578560
$977$	7643.75	0.250302	0.125151	−	0.992138i	$-0.460058\pi$
$977$	7643.75	0.250302	0.125151	+	0.992138i	$0.460058\pi$
$978$	65607.9	2.14510
$979$	82596.5	2.69642
$980$	0	0
$981$	−3274.90	−0.106585
$982$	34892.1	1.13386
$983$	−20447.2	−0.663442	−0.331721	−	0.943378i	$-0.607629\pi$
$983$	−20447.2	−0.663442	−0.331721	+	0.943378i	$0.607629\pi$
$984$	880.514	0.0285262
$985$	0	0
$986$	−7832.58	−0.252982
$987$	3563.52	0.114922
$988$	73.6994	0.00237317
$989$	−34176.3	−1.09883
$990$	0	0
$991$	30015.1	0.962119	0.481059	−	0.876688i	$-0.340252\pi$
$991$	30015.1	0.962119	0.481059	+	0.876688i	$0.340252\pi$
$992$	39946.0	1.27851
$993$	−35430.9	−1.13229
$994$	−36060.2	−1.15067
$995$	0	0
$996$	2909.18	0.0925511
$997$	−10921.0	−0.346912	−0.173456	−	0.984842i	$-0.555493\pi$
$997$	−10921.0	−0.346912	−0.173456	+	0.984842i	$0.555493\pi$
$998$	−65125.6	−2.06565
$999$	11693.2	0.370326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2075.4.a.k.1.9	✓	47
5.4	even	2		2075.4.a.l.1.39	yes	47

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2075.4.a.k.1.9	✓	47	1.1	even	1	trivial
2075.4.a.l.1.39	yes	47	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2075 = 5^{2} \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2075.a (trivial)

\(f(q)\)	\(=\)	\(q-3.98801 q^{2} +4.43436 q^{3} +7.90426 q^{4} -17.6843 q^{6} -8.84879 q^{7} +0.381800 q^{8} -7.33643 q^{9} +52.4531 q^{11} +35.0504 q^{12} +0.962152 q^{13} +35.2891 q^{14} -64.7567 q^{16} +58.2046 q^{17} +29.2578 q^{18} +9.69079 q^{19} -39.2387 q^{21} -209.184 q^{22} +123.399 q^{23} +1.69304 q^{24} -3.83708 q^{26} -152.260 q^{27} -69.9432 q^{28} +33.7436 q^{29} +156.530 q^{31} +255.196 q^{32} +232.596 q^{33} -232.121 q^{34} -57.9891 q^{36} -76.7974 q^{37} -38.6470 q^{38} +4.26653 q^{39} +520.080 q^{41} +156.485 q^{42} -276.957 q^{43} +414.603 q^{44} -492.118 q^{46} -90.8164 q^{47} -287.155 q^{48} -264.699 q^{49} +258.100 q^{51} +7.60510 q^{52} +546.939 q^{53} +607.216 q^{54} -3.37846 q^{56} +42.9725 q^{57} -134.570 q^{58} +176.031 q^{59} -272.420 q^{61} -624.246 q^{62} +64.9185 q^{63} -499.673 q^{64} -927.597 q^{66} -883.433 q^{67} +460.064 q^{68} +547.196 q^{69} -1021.85 q^{71} -2.80105 q^{72} +322.584 q^{73} +306.269 q^{74} +76.5986 q^{76} -464.147 q^{77} -17.0150 q^{78} +329.789 q^{79} -477.093 q^{81} -2074.09 q^{82} +83.0000 q^{83} -310.153 q^{84} +1104.51 q^{86} +149.631 q^{87} +20.0266 q^{88} +1574.67 q^{89} -8.51388 q^{91} +975.379 q^{92} +694.112 q^{93} +362.177 q^{94} +1131.63 q^{96} -317.276 q^{97} +1055.62 q^{98} -384.819 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 47 q - q^{2} + 6 q^{3} + 229 q^{4} + 9 q^{6} + 6 q^{7} - 39 q^{8} + 531 q^{9} + 90 q^{11} - 6 q^{12} - 50 q^{13} + 115 q^{14} + 1149 q^{16} + 68 q^{17} - 8 q^{18} + 344 q^{19} + 625 q^{21} - 402 q^{22}+ \cdots + 13991 q^{99}+O(q^{100}) \) 47 * q - q^2 + 6 * q^3 + 229 * q^4 + 9 * q^6 + 6 * q^7 - 39 * q^8 + 531 * q^9 + 90 * q^11 - 6 * q^12 - 50 * q^13 + 115 * q^14 + 1149 * q^16 + 68 * q^17 - 8 * q^18 + 344 * q^19 + 625 * q^21 - 402 * q^22 + 469 * q^23 + 262 * q^24 + 1209 * q^26 - 12 * q^27 + 603 * q^28 + 480 * q^29 + 876 * q^31 - 1653 * q^32 + 757 * q^33 + 733 * q^34 + 4318 * q^36 - 634 * q^37 + 1190 * q^38 + 618 * q^39 + 1455 * q^41 - 2747 * q^42 + 652 * q^43 + 595 * q^44 + 1662 * q^46 - 446 * q^47 + 2145 * q^48 + 4099 * q^49 + 1681 * q^51 - 207 * q^52 + 126 * q^53 + 670 * q^54 + 3100 * q^56 - 650 * q^57 + 1082 * q^58 + 1144 * q^59 + 2762 * q^61 - 1916 * q^62 + 885 * q^63 + 6891 * q^64 + 2025 * q^66 + 100 * q^67 + 2052 * q^68 + 2556 * q^69 + 2546 * q^71 - 2940 * q^72 - 14 * q^73 + 1066 * q^74 + 4828 * q^76 + 210 * q^77 + 343 * q^78 + 1636 * q^79 + 8203 * q^81 - 480 * q^82 + 3901 * q^83 + 6209 * q^84 + 2706 * q^86 + 5479 * q^87 - 2178 * q^88 + 2558 * q^89 + 3342 * q^91 + 7501 * q^92 - 3353 * q^93 + 5769 * q^94 - 918 * q^96 + 3648 * q^97 - 8109 * q^98 + 13991 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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