LMFDB - Embedded newform 275.4.a.c.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	\(q+5.56155 q^{2} +3.56155 q^{3} +22.9309 q^{4} +19.8078 q^{6} -6.05398 q^{7} +83.0388 q^{8} -14.3153 q^{9} -11.0000 q^{11} +81.6695 q^{12} +4.38447 q^{13} -33.6695 q^{14} +278.378 q^{16} +110.546 q^{17} -79.6155 q^{18} -94.2699 q^{19} -21.5616 q^{21} -61.1771 q^{22} -15.7538 q^{23} +295.747 q^{24} +24.3845 q^{26} -147.147 q^{27} -138.823 q^{28} -256.870 q^{29} -170.702 q^{31} +883.902 q^{32} -39.1771 q^{33} +614.810 q^{34} -328.263 q^{36} +190.853 q^{37} -524.287 q^{38} +15.6155 q^{39} +249.602 q^{41} -119.916 q^{42} -291.602 q^{43} -252.240 q^{44} -87.6155 q^{46} -182.155 q^{47} +991.457 q^{48} -306.349 q^{49} +393.717 q^{51} +100.540 q^{52} +289.902 q^{53} -818.365 q^{54} -502.715 q^{56} -335.747 q^{57} -1428.60 q^{58} +282.725 q^{59} +167.825 q^{61} -949.366 q^{62} +86.6647 q^{63} +2688.85 q^{64} -217.885 q^{66} +176.233 q^{67} +2534.93 q^{68} -56.1080 q^{69} +919.255 q^{71} -1188.73 q^{72} -154.570 q^{73} +1061.44 q^{74} -2161.69 q^{76} +66.5937 q^{77} +86.8466 q^{78} -882.017 q^{79} -137.557 q^{81} +1388.18 q^{82} -277.619 q^{83} -494.425 q^{84} -1621.76 q^{86} -914.857 q^{87} -913.427 q^{88} -977.147 q^{89} -26.5435 q^{91} -361.248 q^{92} -607.963 q^{93} -1013.07 q^{94} +3148.07 q^{96} +1102.94 q^{97} -1703.78 q^{98} +157.469 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 7 q^{2} + 3 q^{3} + 17 q^{4} + 19 q^{6} + 25 q^{7} + 63 q^{8} - 41 q^{9} - 22 q^{11} + 85 q^{12} + 50 q^{13} + 11 q^{14} + 297 q^{16} + 151 q^{17} - 118 q^{18} - 3 q^{19} - 39 q^{21} - 77 q^{22} - 48 q^{23}+ \cdots + 451 q^{99}+O(q^{100}) $ 2 * q + 7 * q^2 + 3 * q^3 + 17 * q^4 + 19 * q^6 + 25 * q^7 + 63 * q^8 - 41 * q^9 - 22 * q^11 + 85 * q^12 + 50 * q^13 + 11 * q^14 + 297 * q^16 + 151 * q^17 - 118 * q^18 - 3 * q^19 - 39 * q^21 - 77 * q^22 - 48 * q^23 + 307 * q^24 + 90 * q^26 - 117 * q^27 - 323 * q^28 - 221 * q^29 + 141 * q^31 + 1071 * q^32 - 33 * q^33 + 673 * q^34 - 170 * q^36 + 559 * q^37 - 393 * q^38 - 10 * q^39 - 144 * q^41 - 145 * q^42 + 60 * q^43 - 187 * q^44 - 134 * q^46 + 48 * q^47 + 981 * q^48 + 315 * q^49 + 371 * q^51 - 170 * q^52 - 117 * q^53 - 775 * q^54 - 1125 * q^56 - 387 * q^57 - 1377 * q^58 - 86 * q^59 - 155 * q^61 - 501 * q^62 - 742 * q^63 + 2809 * q^64 - 209 * q^66 - 266 * q^67 + 2295 * q^68 - 38 * q^69 + 1587 * q^71 - 654 * q^72 - 70 * q^73 + 1591 * q^74 - 2703 * q^76 - 275 * q^77 + 50 * q^78 - 1294 * q^79 + 566 * q^81 + 822 * q^82 + 558 * q^83 - 391 * q^84 - 1116 * q^86 - 935 * q^87 - 693 * q^88 - 1777 * q^89 + 1390 * q^91 - 170 * q^92 - 783 * q^93 - 682 * q^94 + 3043 * q^96 + 334 * q^97 - 810 * q^98 + 451 * 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 $

$2$	5.56155	1.96631	0.983153	−	0.182785i	$-0.0585112\pi$
$2$	5.56155	1.96631	0.983153	+	0.182785i	$0.0585112\pi$
$3$	3.56155	0.685421	0.342711	−	0.939441i	$-0.388655\pi$
$3$	3.56155	0.685421	0.342711	+	0.939441i	$0.388655\pi$
$4$	22.9309	2.86636
$5$	0	0
$5$	0	0
$6$	19.8078	1.34775
$7$	−6.05398	−0.326884	−0.163442	−	0.986553i	$-0.552260\pi$
$7$	−6.05398	−0.326884	−0.163442	+	0.986553i	$0.552260\pi$
$8$	83.0388	3.66983
$9$	−14.3153	−0.530198
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	81.6695	1.96466
$13$	4.38447	0.0935411	0.0467705	−	0.998906i	$-0.485107\pi$
$13$	4.38447	0.0935411	0.0467705	+	0.998906i	$0.485107\pi$
$14$	−33.6695	−0.642754
$15$	0	0
$16$	278.378	4.34965
$17$	110.546	1.57714	0.788572	−	0.614943i	$-0.210821\pi$
$17$	110.546	1.57714	0.788572	+	0.614943i	$0.210821\pi$
$18$	−79.6155	−1.04253
$19$	−94.2699	−1.13826	−0.569131	−	0.822247i	$-0.692720\pi$
$19$	−94.2699	−1.13826	−0.569131	+	0.822247i	$0.692720\pi$
$20$	0	0
$21$	−21.5616	−0.224053
$22$	−61.1771	−0.592864
$23$	−15.7538	−0.142821	−0.0714107	−	0.997447i	$-0.522750\pi$
$23$	−15.7538	−0.142821	−0.0714107	+	0.997447i	$0.522750\pi$
$24$	295.747	2.51538
$25$	0	0
$26$	24.3845	0.183930
$27$	−147.147	−1.04883
$28$	−138.823	−0.936967
$29$	−256.870	−1.64481	−0.822407	−	0.568900i	$-0.807369\pi$
$29$	−256.870	−1.64481	−0.822407	+	0.568900i	$0.807369\pi$
$30$	0	0
$31$	−170.702	−0.988998	−0.494499	−	0.869178i	$-0.664648\pi$
$31$	−170.702	−0.988998	−0.494499	+	0.869178i	$0.664648\pi$
$32$	883.902	4.88292
$33$	−39.1771	−0.206662
$34$	614.810	3.10115
$35$	0	0
$36$	−328.263	−1.51974
$37$	190.853	0.848002	0.424001	−	0.905662i	$-0.360625\pi$
$37$	190.853	0.848002	0.424001	+	0.905662i	$0.360625\pi$
$38$	−524.287	−2.23817
$39$	15.6155	0.0641150
$40$	0	0
$41$	249.602	0.950764	0.475382	−	0.879780i	$-0.342310\pi$
$41$	249.602	0.950764	0.475382	+	0.879780i	$0.342310\pi$
$42$	−119.916	−0.440557
$43$	−291.602	−1.03416	−0.517081	−	0.855937i	$-0.672981\pi$
$43$	−291.602	−1.03416	−0.517081	+	0.855937i	$0.672981\pi$
$44$	−252.240	−0.864240
$45$	0	0
$46$	−87.6155	−0.280831
$47$	−182.155	−0.565321	−0.282660	−	0.959220i	$-0.591217\pi$
$47$	−182.155	−0.565321	−0.282660	+	0.959220i	$0.591217\pi$
$48$	991.457	2.98134
$49$	−306.349	−0.893147
$50$	0	0
$51$	393.717	1.08101
$52$	100.540	0.268122
$53$	289.902	0.751343	0.375671	−	0.926753i	$-0.377412\pi$
$53$	289.902	0.751343	0.375671	+	0.926753i	$0.377412\pi$
$54$	−818.365	−2.06232
$55$	0	0
$56$	−502.715	−1.19961
$57$	−335.747	−0.780189
$58$	−1428.60	−3.23421
$59$	282.725	0.623859	0.311930	−	0.950105i	$-0.399025\pi$
$59$	282.725	0.623859	0.311930	+	0.950105i	$0.399025\pi$
$60$	0	0
$61$	167.825	0.352258	0.176129	−	0.984367i	$-0.443642\pi$
$61$	167.825	0.352258	0.176129	+	0.984367i	$0.443642\pi$
$62$	−949.366	−1.94467
$63$	86.6647	0.173313
$64$	2688.85	5.25166
$65$	0	0
$66$	−217.885	−0.406361
$67$	176.233	0.321347	0.160674	−	0.987008i	$-0.448633\pi$
$67$	176.233	0.321347	0.160674	+	0.987008i	$0.448633\pi$
$68$	2534.93	4.52066
$69$	−56.1080	−0.0978928
$70$	0	0
$71$	919.255	1.53656	0.768278	−	0.640116i	$-0.221114\pi$
$71$	919.255	1.53656	0.768278	+	0.640116i	$0.221114\pi$
$72$	−1188.73	−1.94574
$73$	−154.570	−0.247823	−0.123911	−	0.992293i	$-0.539544\pi$
$73$	−154.570	−0.247823	−0.123911	+	0.992293i	$0.539544\pi$
$74$	1061.44	1.66743
$75$	0	0
$76$	−2161.69	−3.26267
$77$	66.5937	0.0985592
$78$	86.8466	0.126070
$79$	−882.017	−1.25614	−0.628068	−	0.778159i	$-0.716154\pi$
$79$	−882.017	−1.25614	−0.628068	+	0.778159i	$0.716154\pi$
$80$	0	0
$81$	−137.557	−0.188692
$82$	1388.18	1.86949
$83$	−277.619	−0.367141	−0.183570	−	0.983007i	$-0.558766\pi$
$83$	−277.619	−0.367141	−0.183570	+	0.983007i	$0.558766\pi$
$84$	−494.425	−0.642217
$85$	0	0
$86$	−1621.76	−2.03348
$87$	−914.857	−1.12739
$88$	−913.427	−1.10650
$89$	−977.147	−1.16379	−0.581895	−	0.813264i	$-0.697689\pi$
$89$	−977.147	−1.16379	−0.581895	+	0.813264i	$0.697689\pi$
$90$	0	0
$91$	−26.5435	−0.0305771
$92$	−361.248	−0.409377
$93$	−607.963	−0.677880
$94$	−1013.07	−1.11159
$95$	0	0
$96$	3148.07	3.34685
$97$	1102.94	1.15451	0.577253	−	0.816565i	$-0.304125\pi$
$97$	1102.94	1.15451	0.577253	+	0.816565i	$0.304125\pi$
$98$	−1703.78	−1.75620
$99$	157.469	0.159861
$100$	0	0
$101$	484.314	0.477139	0.238570	−	0.971125i	$-0.423321\pi$
$101$	484.314	0.477139	0.238570	+	0.971125i	$0.423321\pi$
$102$	2189.68	2.12559
$103$	874.419	0.836495	0.418248	−	0.908333i	$-0.362644\pi$
$103$	874.419	0.836495	0.418248	+	0.908333i	$0.362644\pi$
$104$	364.081	0.343280
$105$	0	0
$106$	1612.31	1.47737
$107$	−119.845	−0.108279	−0.0541394	−	0.998533i	$-0.517242\pi$
$107$	−119.845	−0.108279	−0.0541394	+	0.998533i	$0.517242\pi$
$108$	−3374.20	−3.00632
$109$	414.621	0.364344	0.182172	−	0.983267i	$-0.441687\pi$
$109$	414.621	0.364344	0.182172	+	0.983267i	$0.441687\pi$
$110$	0	0
$111$	679.734	0.581239
$112$	−1685.29	−1.42183
$113$	−534.453	−0.444930	−0.222465	−	0.974941i	$-0.571410\pi$
$113$	−534.453	−0.444930	−0.222465	+	0.974941i	$0.571410\pi$
$114$	−1867.28	−1.53409
$115$	0	0
$116$	−5890.26	−4.71463
$117$	−62.7652	−0.0495953
$118$	1572.39	1.22670
$119$	−669.245	−0.515543
$120$	0	0
$121$	121.000	0.0909091
$122$	933.366	0.692648
$123$	888.972	0.651674
$124$	−3914.34	−2.83482
$125$	0	0
$126$	481.990	0.340787
$127$	−640.121	−0.447256	−0.223628	−	0.974675i	$-0.571790\pi$
$127$	−640.121	−0.447256	−0.223628	+	0.974675i	$0.571790\pi$
$128$	7882.95	5.44344
$129$	−1038.56	−0.708836
$130$	0	0
$131$	1051.05	0.700999	0.350499	−	0.936563i	$-0.386012\pi$
$131$	1051.05	0.700999	0.350499	+	0.936563i	$0.386012\pi$
$132$	−898.365	−0.592368
$133$	570.708	0.372080
$134$	980.129	0.631867
$135$	0	0
$136$	9179.64	5.78785
$137$	−1690.68	−1.05434	−0.527169	−	0.849761i	$-0.676746\pi$
$137$	−1690.68	−1.05434	−0.527169	+	0.849761i	$0.676746\pi$
$138$	−312.047	−0.192487
$139$	2789.43	1.70213	0.851067	−	0.525058i	$-0.175956\pi$
$139$	2789.43	1.70213	0.851067	+	0.525058i	$0.175956\pi$
$140$	0	0
$141$	−648.756	−0.387483
$142$	5112.48	3.02134
$143$	−48.2292	−0.0282037
$144$	−3985.07	−2.30618
$145$	0	0
$146$	−859.650	−0.487295
$147$	−1091.08	−0.612182
$148$	4376.43	2.43068
$149$	1090.62	0.599647	0.299823	−	0.953995i	$-0.403072\pi$
$149$	1090.62	0.599647	0.299823	+	0.953995i	$0.403072\pi$
$150$	0	0
$151$	623.574	0.336064	0.168032	−	0.985782i	$-0.446259\pi$
$151$	623.574	0.336064	0.168032	+	0.985782i	$0.446259\pi$
$152$	−7828.06	−4.17723
$153$	−1582.51	−0.836198
$154$	370.365	0.193798
$155$	0	0
$156$	358.078	0.183777
$157$	2114.96	1.07511	0.537555	−	0.843228i	$-0.319348\pi$
$157$	2114.96	1.07511	0.537555	+	0.843228i	$0.319348\pi$
$158$	−4905.38	−2.46995
$159$	1032.50	0.514986
$160$	0	0
$161$	95.3730	0.0466860
$162$	−765.029	−0.371027
$163$	−1153.53	−0.554301	−0.277151	−	0.960827i	$-0.589390\pi$
$163$	−1153.53	−0.554301	−0.277151	+	0.960827i	$0.589390\pi$
$164$	5723.60	2.72523
$165$	0	0
$166$	−1543.99	−0.721911
$167$	1100.93	0.510137	0.255068	−	0.966923i	$-0.417902\pi$
$167$	1100.93	0.510137	0.255068	+	0.966923i	$0.417902\pi$
$168$	−1790.45	−0.822238
$169$	−2177.78	−0.991250
$170$	0	0
$171$	1349.51	0.603504
$172$	−6686.69	−2.96428
$173$	2369.25	1.04122	0.520609	−	0.853795i	$-0.325705\pi$
$173$	2369.25	1.04122	0.520609	+	0.853795i	$0.325705\pi$
$174$	−5088.03	−2.21679
$175$	0	0
$176$	−3062.16	−1.31147
$177$	1006.94	0.427606
$178$	−5434.45	−2.28837
$179$	1226.77	0.512250	0.256125	−	0.966644i	$-0.417554\pi$
$179$	1226.77	0.512250	0.256125	+	0.966644i	$0.417554\pi$
$180$	0	0
$181$	−439.606	−0.180528	−0.0902642	−	0.995918i	$-0.528771\pi$
$181$	−439.606	−0.180528	−0.0902642	+	0.995918i	$0.528771\pi$
$182$	−147.623	−0.0601239
$183$	597.717	0.241445
$184$	−1308.18	−0.524131
$185$	0	0
$186$	−3381.22	−1.33292
$187$	−1216.01	−0.475527
$188$	−4176.98	−1.62041
$189$	890.823	0.342846
$190$	0	0
$191$	−4968.96	−1.88241	−0.941207	−	0.337829i	$-0.890307\pi$
$191$	−4968.96	−1.88241	−0.941207	+	0.337829i	$0.890307\pi$
$192$	9576.47	3.59960
$193$	1362.22	0.508054	0.254027	−	0.967197i	$-0.418245\pi$
$193$	1362.22	0.508054	0.254027	+	0.967197i	$0.418245\pi$
$194$	6134.09	2.27011
$195$	0	0
$196$	−7024.86	−2.56008
$197$	−2195.91	−0.794174	−0.397087	−	0.917781i	$-0.629979\pi$
$197$	−2195.91	−0.794174	−0.397087	+	0.917781i	$0.629979\pi$
$198$	875.771	0.314335
$199$	−558.189	−0.198839	−0.0994194	−	0.995046i	$-0.531699\pi$
$199$	−558.189	−0.198839	−0.0994194	+	0.995046i	$0.531699\pi$
$200$	0	0
$201$	627.663	0.220258
$202$	2693.54	0.938202
$203$	1555.09	0.537663
$204$	9028.27	3.09856
$205$	0	0
$206$	4863.12	1.64481
$207$	225.521	0.0757236
$208$	1220.54	0.406871
$209$	1036.97	0.343199
$210$	0	0
$211$	−3002.01	−0.979463	−0.489732	−	0.871873i	$-0.662905\pi$
$211$	−3002.01	−0.979463	−0.489732	+	0.871873i	$0.662905\pi$
$212$	6647.71	2.15362
$213$	3273.97	1.05319
$214$	−666.523	−0.212909
$215$	0	0
$216$	−12218.9	−3.84903
$217$	1033.42	0.323287
$218$	2305.94	0.716412
$219$	−550.509	−0.169863
$220$	0	0
$221$	484.688	0.147528
$222$	3780.38	1.14289
$223$	−854.595	−0.256627	−0.128314	−	0.991734i	$-0.540956\pi$
$223$	−854.595	−0.256627	−0.128314	+	0.991734i	$0.540956\pi$
$224$	−5351.12	−1.59615
$225$	0	0
$226$	−2972.39	−0.874868
$227$	−394.002	−0.115202	−0.0576010	−	0.998340i	$-0.518345\pi$
$227$	−394.002	−0.115202	−0.0576010	+	0.998340i	$0.518345\pi$
$228$	−7698.97	−2.23630
$229$	−491.822	−0.141924	−0.0709618	−	0.997479i	$-0.522607\pi$
$229$	−491.822	−0.141924	−0.0709618	+	0.997479i	$0.522607\pi$
$230$	0	0
$231$	237.177	0.0675546
$232$	−21330.2	−6.03619
$233$	6884.63	1.93574	0.967869	−	0.251456i	$-0.0809094\pi$
$233$	6884.63	1.93574	0.967869	+	0.251456i	$0.0809094\pi$
$234$	−349.072	−0.0975195
$235$	0	0
$236$	6483.14	1.78820
$237$	−3141.35	−0.860982
$238$	−3722.04	−1.01371
$239$	3012.77	0.815397	0.407699	−	0.913117i	$-0.366331\pi$
$239$	3012.77	0.815397	0.407699	+	0.913117i	$0.366331\pi$
$240$	0	0
$241$	3106.98	0.830448	0.415224	−	0.909719i	$-0.363703\pi$
$241$	3106.98	0.830448	0.415224	+	0.909719i	$0.363703\pi$
$242$	672.948	0.178755
$243$	3483.05	0.919496
$244$	3848.37	1.00970
$245$	0	0
$246$	4944.06	1.28139
$247$	−413.324	−0.106474
$248$	−14174.9	−3.62946
$249$	−988.756	−0.251646
$250$	0	0
$251$	−834.313	−0.209806	−0.104903	−	0.994482i	$-0.533453\pi$
$251$	−834.313	−0.209806	−0.104903	+	0.994482i	$0.533453\pi$
$252$	1987.30	0.496778
$253$	173.292	0.0430623
$254$	−3560.07	−0.879443
$255$	0	0
$256$	22330.6	5.45182
$257$	7536.63	1.82927	0.914635	−	0.404281i	$-0.132478\pi$
$257$	7536.63	1.82927	0.914635	+	0.404281i	$0.132478\pi$
$258$	−5775.99	−1.39379
$259$	−1155.42	−0.277198
$260$	0	0
$261$	3677.19	0.872077
$262$	5845.48	1.37838
$263$	6242.10	1.46351	0.731757	−	0.681565i	$-0.238700\pi$
$263$	6242.10	1.46351	0.731757	+	0.681565i	$0.238700\pi$
$264$	−3253.22	−0.758416
$265$	0	0
$266$	3174.02	0.731623
$267$	−3480.16	−0.797687
$268$	4041.17	0.921097
$269$	1636.95	0.371027	0.185514	−	0.982642i	$-0.440605\pi$
$269$	1636.95	0.371027	0.185514	+	0.982642i	$0.440605\pi$
$270$	0	0
$271$	787.212	0.176457	0.0882283	−	0.996100i	$-0.471880\pi$
$271$	787.212	0.176457	0.0882283	+	0.996100i	$0.471880\pi$
$272$	30773.7	6.86003
$273$	−94.5360	−0.0209582
$274$	−9402.78	−2.07315
$275$	0	0
$276$	−1286.60	−0.280596
$277$	−1954.96	−0.424052	−0.212026	−	0.977264i	$-0.568006\pi$
$277$	−1954.96	−0.424052	−0.212026	+	0.977264i	$0.568006\pi$
$278$	15513.6	3.34691
$279$	2443.65	0.524364
$280$	0	0
$281$	−5097.58	−1.08219	−0.541097	−	0.840960i	$-0.681991\pi$
$281$	−5097.58	−1.08219	−0.541097	+	0.840960i	$0.681991\pi$
$282$	−3608.09	−0.761910
$283$	7187.71	1.50977	0.754885	−	0.655857i	$-0.227693\pi$
$283$	7187.71	1.50977	0.754885	+	0.655857i	$0.227693\pi$
$284$	21079.3	4.40432
$285$	0	0
$286$	−268.229	−0.0554571
$287$	−1511.09	−0.310789
$288$	−12653.4	−2.58891
$289$	7307.51	1.48738
$290$	0	0
$291$	3928.20	0.791323
$292$	−3544.43	−0.710349
$293$	−6388.82	−1.27385	−0.636926	−	0.770925i	$-0.719794\pi$
$293$	−6388.82	−1.27385	−0.636926	+	0.770925i	$0.719794\pi$
$294$	−6068.10	−1.20374
$295$	0	0
$296$	15848.2	3.11203
$297$	1618.61	0.316234
$298$	6065.56	1.17909
$299$	−69.0720	−0.0133597
$300$	0	0
$301$	1765.35	0.338051
$302$	3468.04	0.660806
$303$	1724.91	0.327041
$304$	−26242.6	−4.95105
$305$	0	0
$306$	−8801.21	−1.64422
$307$	−4882.07	−0.907603	−0.453802	−	0.891103i	$-0.649933\pi$
$307$	−4882.07	−0.907603	−0.453802	+	0.891103i	$0.649933\pi$
$308$	1527.05	0.282506
$309$	3114.29	0.573352
$310$	0	0
$311$	2846.01	0.518914	0.259457	−	0.965755i	$-0.416456\pi$
$311$	2846.01	0.518914	0.259457	+	0.965755i	$0.416456\pi$
$312$	1296.70	0.235291
$313$	−8009.49	−1.44640	−0.723200	−	0.690639i	$-0.757330\pi$
$313$	−8009.49	−1.44640	−0.723200	+	0.690639i	$0.757330\pi$
$314$	11762.5	2.11400
$315$	0	0
$316$	−20225.4	−3.60053
$317$	−4668.89	−0.827227	−0.413613	−	0.910453i	$-0.635733\pi$
$317$	−4668.89	−0.827227	−0.413613	+	0.910453i	$0.635733\pi$
$318$	5742.32	1.01262
$319$	2825.57	0.495930
$320$	0	0
$321$	−426.833	−0.0742165
$322$	530.422	0.0917990
$323$	−10421.2	−1.79520
$324$	−3154.30	−0.540860
$325$	0	0
$326$	−6415.39	−1.08993
$327$	1476.70	0.249729
$328$	20726.7	3.48914
$329$	1102.76	0.184794
$330$	0	0
$331$	2581.31	0.428645	0.214323	−	0.976763i	$-0.431246\pi$
$331$	2581.31	0.428645	0.214323	+	0.976763i	$0.431246\pi$
$332$	−6366.05	−1.05236
$333$	−2732.13	−0.449609
$334$	6122.91	1.00309
$335$	0	0
$336$	−6002.26	−0.974554
$337$	8152.45	1.31778	0.658890	−	0.752239i	$-0.271026\pi$
$337$	8152.45	1.31778	0.658890	+	0.752239i	$0.271026\pi$
$338$	−12111.8	−1.94910
$339$	−1903.48	−0.304964
$340$	0	0
$341$	1877.72	0.298194
$342$	7505.35	1.18667
$343$	3931.15	0.618839
$344$	−24214.3	−3.79520
$345$	0	0
$346$	13176.7	2.04735
$347$	3426.21	0.530054	0.265027	−	0.964241i	$-0.414619\pi$
$347$	3426.21	0.530054	0.265027	+	0.964241i	$0.414619\pi$
$348$	−20978.5	−3.23151
$349$	1334.33	0.204656	0.102328	−	0.994751i	$-0.467371\pi$
$349$	1334.33	0.204656	0.102328	+	0.994751i	$0.467371\pi$
$350$	0	0
$351$	−645.161	−0.0981087
$352$	−9722.93	−1.47225
$353$	−4406.21	−0.664360	−0.332180	−	0.943216i	$-0.607784\pi$
$353$	−4406.21	−0.664360	−0.332180	+	0.943216i	$0.607784\pi$
$354$	5600.16	0.840805
$355$	0	0
$356$	−22406.8	−3.33584
$357$	−2383.55	−0.353364
$358$	6822.72	1.00724
$359$	−8623.04	−1.26771	−0.633853	−	0.773453i	$-0.718528\pi$
$359$	−8623.04	−1.26771	−0.633853	+	0.773453i	$0.718528\pi$
$360$	0	0
$361$	2027.81	0.295642
$362$	−2444.89	−0.354974
$363$	430.948	0.0623110
$364$	−608.665	−0.0876448
$365$	0	0
$366$	3324.23	0.474755
$367$	3585.58	0.509989	0.254995	−	0.966942i	$-0.417926\pi$
$367$	3585.58	0.509989	0.254995	+	0.966942i	$0.417926\pi$
$368$	−4385.51	−0.621224
$369$	−3573.14	−0.504093
$370$	0	0
$371$	−1755.06	−0.245602
$372$	−13941.1	−1.94305
$373$	−9855.90	−1.36815	−0.684074	−	0.729413i	$-0.739793\pi$
$373$	−9855.90	−1.36815	−0.684074	+	0.729413i	$0.739793\pi$
$374$	−6762.91	−0.935031
$375$	0	0
$376$	−15126.0	−2.07463
$377$	−1126.24	−0.153858
$378$	4954.36	0.674139
$379$	−10837.8	−1.46887	−0.734435	−	0.678679i	$-0.762553\pi$
$379$	−10837.8	−1.46887	−0.734435	+	0.678679i	$0.762553\pi$
$380$	0	0
$381$	−2279.83	−0.306559
$382$	−27635.1	−3.70140
$383$	2025.55	0.270237	0.135119	−	0.990829i	$-0.456858\pi$
$383$	2025.55	0.270237	0.135119	+	0.990829i	$0.456858\pi$
$384$	28075.5	3.73105
$385$	0	0
$386$	7576.04	0.998990
$387$	4174.39	0.548310
$388$	25291.5	3.30923
$389$	−978.894	−0.127588	−0.0637942	−	0.997963i	$-0.520320\pi$
$389$	−978.894	−0.127588	−0.0637942	+	0.997963i	$0.520320\pi$
$390$	0	0
$391$	−1741.52	−0.225250
$392$	−25438.9	−3.27770
$393$	3743.38	0.480479
$394$	−12212.7	−1.56159
$395$	0	0
$396$	3610.90	0.458218
$397$	5008.28	0.633144	0.316572	−	0.948568i	$-0.397468\pi$
$397$	5008.28	0.633144	0.316572	+	0.948568i	$0.397468\pi$
$398$	−3104.39	−0.390978
$399$	2032.60	0.255031
$400$	0	0
$401$	−15584.1	−1.94073	−0.970366	−	0.241639i	$-0.922315\pi$
$401$	−15584.1	−1.94073	−0.970366	+	0.241639i	$0.922315\pi$
$402$	3490.78	0.433095
$403$	−748.437	−0.0925119
$404$	11105.7	1.36765
$405$	0	0
$406$	8648.69	1.05721
$407$	−2099.39	−0.255682
$408$	32693.8	3.96712
$409$	15106.6	1.82634	0.913171	−	0.407576i	$-0.133626\pi$
$409$	15106.6	1.82634	0.913171	+	0.407576i	$0.133626\pi$
$410$	0	0
$411$	−6021.43	−0.722665
$412$	20051.2	2.39770
$413$	−1711.61	−0.203930
$414$	1254.25	0.148896
$415$	0	0
$416$	3875.45	0.456753
$417$	9934.71	1.16668
$418$	5767.16	0.674834
$419$	−1518.17	−0.177011	−0.0885056	−	0.996076i	$-0.528209\pi$
$419$	−1518.17	−0.177011	−0.0885056	+	0.996076i	$0.528209\pi$
$420$	0	0
$421$	4637.05	0.536808	0.268404	−	0.963306i	$-0.413504\pi$
$421$	4637.05	0.536808	0.268404	+	0.963306i	$0.413504\pi$
$422$	−16695.8	−1.92592
$423$	2607.62	0.299732
$424$	24073.2	2.75730
$425$	0	0
$426$	18208.4	2.07089
$427$	−1016.01	−0.115148
$428$	−2748.14	−0.310366
$429$	−171.771	−0.0193314
$430$	0	0
$431$	−11477.9	−1.28276	−0.641380	−	0.767223i	$-0.721638\pi$
$431$	−11477.9	−1.28276	−0.641380	+	0.767223i	$0.721638\pi$
$432$	−40962.4	−4.56205
$433$	−10204.5	−1.13256	−0.566280	−	0.824213i	$-0.691618\pi$
$433$	−10204.5	−1.13256	−0.566280	+	0.824213i	$0.691618\pi$
$434$	5747.44	0.635682
$435$	0	0
$436$	9507.62	1.04434
$437$	1485.11	0.162568
$438$	−3061.69	−0.334002
$439$	−6919.06	−0.752229	−0.376115	−	0.926573i	$-0.622740\pi$
$439$	−6919.06	−0.752229	−0.376115	+	0.926573i	$0.622740\pi$
$440$	0	0
$441$	4385.50	0.473545
$442$	2695.62	0.290085
$443$	2912.53	0.312366	0.156183	−	0.987728i	$-0.450081\pi$
$443$	2912.53	0.312366	0.156183	+	0.987728i	$0.450081\pi$
$444$	15586.9	1.66604
$445$	0	0
$446$	−4752.87	−0.504608
$447$	3884.32	0.411011
$448$	−16278.2	−1.71668
$449$	−1155.53	−0.121454	−0.0607270	−	0.998154i	$-0.519342\pi$
$449$	−1155.53	−0.121454	−0.0607270	+	0.998154i	$0.519342\pi$
$450$	0	0
$451$	−2745.62	−0.286666
$452$	−12255.5	−1.27533
$453$	2220.89	0.230346
$454$	−2191.26	−0.226522
$455$	0	0
$456$	−27880.0	−2.86316
$457$	−2745.62	−0.281039	−0.140519	−	0.990078i	$-0.544877\pi$
$457$	−2745.62	−0.281039	−0.140519	+	0.990078i	$0.544877\pi$
$458$	−2735.29	−0.279065
$459$	−16266.5	−1.65416
$460$	0	0
$461$	11224.1	1.13397	0.566984	−	0.823729i	$-0.308110\pi$
$461$	11224.1	1.13397	0.566984	+	0.823729i	$0.308110\pi$
$462$	1319.07	0.132833
$463$	−15994.8	−1.60549	−0.802743	−	0.596325i	$-0.796627\pi$
$463$	−15994.8	−1.60549	−0.802743	+	0.596325i	$0.796627\pi$
$464$	−71507.0	−7.15437
$465$	0	0
$466$	38289.2	3.80625
$467$	−6674.60	−0.661378	−0.330689	−	0.943740i	$-0.607281\pi$
$467$	−6674.60	−0.661378	−0.330689	+	0.943740i	$0.607281\pi$
$468$	−1439.26	−0.142158
$469$	−1066.91	−0.105043
$470$	0	0
$471$	7532.55	0.736904
$472$	23477.2	2.28946
$473$	3207.62	0.311811
$474$	−17470.8	−1.69295
$475$	0	0
$476$	−15346.4	−1.47773
$477$	−4150.05	−0.398360
$478$	16755.7	1.60332
$479$	−11582.0	−1.10479	−0.552396	−	0.833582i	$-0.686286\pi$
$479$	−11582.0	−1.10479	−0.552396	+	0.833582i	$0.686286\pi$
$480$	0	0
$481$	836.791	0.0793230
$482$	17279.6	1.63292
$483$	339.676	0.0319996
$484$	2774.64	0.260578
$485$	0	0
$486$	19371.2	1.80801
$487$	10618.7	0.988047	0.494024	−	0.869448i	$-0.335526\pi$
$487$	10618.7	0.988047	0.494024	+	0.869448i	$0.335526\pi$
$488$	13936.0	1.29273
$489$	−4108.34	−0.379930
$490$	0	0
$491$	−17948.0	−1.64966	−0.824829	−	0.565382i	$-0.808729\pi$
$491$	−17948.0	−1.64966	−0.824829	+	0.565382i	$0.808729\pi$
$492$	20384.9	1.86793
$493$	−28396.1	−2.59411
$494$	−2298.72	−0.209361
$495$	0	0
$496$	−47519.6	−4.30180
$497$	−5565.15	−0.502275
$498$	−5499.02	−0.494813
$499$	−10409.6	−0.933865	−0.466932	−	0.884293i	$-0.654641\pi$
$499$	−10409.6	−0.933865	−0.466932	+	0.884293i	$0.654641\pi$
$500$	0	0
$501$	3921.04	0.349659
$502$	−4640.07	−0.412543
$503$	−7319.98	−0.648870	−0.324435	−	0.945908i	$-0.605174\pi$
$503$	−7319.98	−0.648870	−0.324435	+	0.945908i	$0.605174\pi$
$504$	7196.54	0.636030
$505$	0	0
$506$	963.771	0.0846736
$507$	−7756.27	−0.679424
$508$	−14678.5	−1.28200
$509$	7619.94	0.663552	0.331776	−	0.943358i	$-0.392352\pi$
$509$	7619.94	0.663552	0.331776	+	0.943358i	$0.392352\pi$
$510$	0	0
$511$	935.763	0.0810093
$512$	61129.5	5.27650
$513$	13871.5	1.19384
$514$	41915.4	3.59690
$515$	0	0
$516$	−23815.0	−2.03178
$517$	2003.71	0.170451
$518$	−6425.93	−0.545057
$519$	8438.20	0.713672
$520$	0	0
$521$	−12413.4	−1.04384	−0.521921	−	0.852994i	$-0.674785\pi$
$521$	−12413.4	−1.04384	−0.521921	+	0.852994i	$0.674785\pi$
$522$	20450.9	1.71477
$523$	2524.30	0.211051	0.105526	−	0.994417i	$-0.466347\pi$
$523$	2524.30	0.211051	0.105526	+	0.994417i	$0.466347\pi$
$524$	24101.5	2.00931
$525$	0	0
$526$	34715.8	2.87772
$527$	−18870.5	−1.55979
$528$	−10906.0	−0.898909
$529$	−11918.8	−0.979602
$530$	0	0
$531$	−4047.31	−0.330769
$532$	13086.8	1.06651
$533$	1094.37	0.0889355
$534$	−19355.1	−1.56850
$535$	0	0
$536$	14634.2	1.17929
$537$	4369.19	0.351107
$538$	9103.96	0.729553
$539$	3369.84	0.269294
$540$	0	0
$541$	10271.4	0.816269	0.408135	−	0.912922i	$-0.366179\pi$
$541$	10271.4	0.816269	0.408135	+	0.912922i	$0.366179\pi$
$542$	4378.12	0.346968
$543$	−1565.68	−0.123738
$544$	97712.2	7.70106
$545$	0	0
$546$	−525.767	−0.0412102
$547$	−7810.11	−0.610487	−0.305243	−	0.952274i	$-0.598738\pi$
$547$	−7810.11	−0.610487	−0.305243	+	0.952274i	$0.598738\pi$
$548$	−38768.7	−3.02211
$549$	−2402.47	−0.186767
$550$	0	0
$551$	24215.1	1.87223
$552$	−4659.14	−0.359250
$553$	5339.71	0.410610
$554$	−10872.6	−0.833815
$555$	0	0
$556$	63964.1	4.87892
$557$	18348.8	1.39580	0.697902	−	0.716193i	$-0.254117\pi$
$557$	18348.8	1.39580	0.697902	+	0.716193i	$0.254117\pi$
$558$	13590.5	1.03106
$559$	−1278.52	−0.0967365
$560$	0	0
$561$	−4330.89	−0.325936
$562$	−28350.5	−2.12792
$563$	174.680	0.0130761	0.00653807	−	0.999979i	$-0.497919\pi$
$563$	174.680	0.0130761	0.00653807	+	0.999979i	$0.497919\pi$
$564$	−14876.5	−1.11066
$565$	0	0
$566$	39974.8	2.96867
$567$	832.765	0.0616805
$568$	76333.8	5.63890
$569$	−3208.08	−0.236362	−0.118181	−	0.992992i	$-0.537706\pi$
$569$	−3208.08	−0.236362	−0.118181	+	0.992992i	$0.537706\pi$
$570$	0	0
$571$	−11660.4	−0.854592	−0.427296	−	0.904112i	$-0.640534\pi$
$571$	−11660.4	−0.854592	−0.427296	+	0.904112i	$0.640534\pi$
$572$	−1105.94	−0.0808419
$573$	−17697.2	−1.29025
$574$	−8403.98	−0.611107
$575$	0	0
$576$	−38491.8	−2.78442
$577$	12906.9	0.931233	0.465617	−	0.884987i	$-0.345833\pi$
$577$	12906.9	0.931233	0.465617	+	0.884987i	$0.345833\pi$
$578$	40641.1	2.92465
$579$	4851.61	0.348231
$580$	0	0
$581$	1680.70	0.120012
$582$	21846.9	1.55598
$583$	−3188.93	−0.226538
$584$	−12835.3	−0.909468
$585$	0	0
$586$	−35531.8	−2.50478
$587$	−27427.0	−1.92850	−0.964252	−	0.264986i	$-0.914633\pi$
$587$	−27427.0	−1.92850	−0.964252	+	0.264986i	$0.914633\pi$
$588$	−25019.4	−1.75473
$589$	16092.0	1.12574
$590$	0	0
$591$	−7820.86	−0.544344
$592$	53129.3	3.68852
$593$	−5332.11	−0.369247	−0.184623	−	0.982809i	$-0.559107\pi$
$593$	−5332.11	−0.369247	−0.184623	+	0.982809i	$0.559107\pi$
$594$	9002.01	0.621813
$595$	0	0
$596$	25009.0	1.71880
$597$	−1988.02	−0.136288
$598$	−384.148	−0.0262692
$599$	−22329.2	−1.52312	−0.761558	−	0.648097i	$-0.775565\pi$
$599$	−22329.2	−1.52312	−0.761558	+	0.648097i	$0.775565\pi$
$600$	0	0
$601$	15511.8	1.05281	0.526405	−	0.850234i	$-0.323540\pi$
$601$	15511.8	1.05281	0.526405	+	0.850234i	$0.323540\pi$
$602$	9818.10	0.664711
$603$	−2522.83	−0.170378
$604$	14299.1	0.963281
$605$	0	0
$606$	9593.18	0.643063
$607$	−7205.25	−0.481799	−0.240900	−	0.970550i	$-0.577442\pi$
$607$	−7205.25	−0.481799	−0.240900	+	0.970550i	$0.577442\pi$
$608$	−83325.4	−5.55804
$609$	5538.52	0.368526
$610$	0	0
$611$	−798.655	−0.0528807
$612$	−36288.3	−2.39684
$613$	2837.16	0.186936	0.0934682	−	0.995622i	$-0.470205\pi$
$613$	2837.16	0.186936	0.0934682	+	0.995622i	$0.470205\pi$
$614$	−27151.9	−1.78463
$615$	0	0
$616$	5529.86	0.361696
$617$	7423.58	0.484379	0.242190	−	0.970229i	$-0.422134\pi$
$617$	7423.58	0.484379	0.242190	+	0.970229i	$0.422134\pi$
$618$	17320.3	1.12738
$619$	9747.62	0.632940	0.316470	−	0.948603i	$-0.397502\pi$
$619$	9747.62	0.632940	0.316470	+	0.948603i	$0.397502\pi$
$620$	0	0
$621$	2318.12	0.149795
$622$	15828.2	1.02034
$623$	5915.62	0.380424
$624$	4347.02	0.278878
$625$	0	0
$626$	−44545.2	−2.84407
$627$	3693.22	0.235236
$628$	48497.9	3.08165
$629$	21098.1	1.33742
$630$	0	0
$631$	5914.75	0.373157	0.186579	−	0.982440i	$-0.440260\pi$
$631$	5914.75	0.373157	0.186579	+	0.982440i	$0.440260\pi$
$632$	−73241.7	−4.60980
$633$	−10691.8	−0.671345
$634$	−25966.3	−1.62658
$635$	0	0
$636$	23676.2	1.47614
$637$	−1343.18	−0.0835459
$638$	15714.6	0.975150
$639$	−13159.4	−0.814679
$640$	0	0
$641$	−25438.0	−1.56746	−0.783728	−	0.621104i	$-0.786684\pi$
$641$	−25438.0	−1.56746	−0.783728	+	0.621104i	$0.786684\pi$
$642$	−2373.86	−0.145932
$643$	−769.253	−0.0471794	−0.0235897	−	0.999722i	$-0.507510\pi$
$643$	−769.253	−0.0471794	−0.0235897	+	0.999722i	$0.507510\pi$
$644$	2186.99	0.133819
$645$	0	0
$646$	−57958.0	−3.52992
$647$	−25813.2	−1.56850	−0.784250	−	0.620445i	$-0.786952\pi$
$647$	−25813.2	−1.56850	−0.784250	+	0.620445i	$0.786952\pi$
$648$	−11422.6	−0.692469
$649$	−3109.98	−0.188101
$650$	0	0
$651$	3680.59	0.221588
$652$	−26451.3	−1.58883
$653$	13138.6	0.787372	0.393686	−	0.919245i	$-0.371200\pi$
$653$	13138.6	0.787372	0.393686	+	0.919245i	$0.371200\pi$
$654$	8212.72	0.491044
$655$	0	0
$656$	69483.7	4.13549
$657$	2212.72	0.131395
$658$	6133.08	0.363362
$659$	−19105.0	−1.12933	−0.564663	−	0.825322i	$-0.690994\pi$
$659$	−19105.0	−1.12933	−0.564663	+	0.825322i	$0.690994\pi$
$660$	0	0
$661$	−31694.3	−1.86500	−0.932502	−	0.361166i	$-0.882379\pi$
$661$	−31694.3	−1.86500	−0.932502	+	0.361166i	$0.882379\pi$
$662$	14356.1	0.842848
$663$	1726.24	0.101119
$664$	−23053.2	−1.34734
$665$	0	0
$666$	−15194.9	−0.884069
$667$	4046.68	0.234915
$668$	25245.4	1.46224
$669$	−3043.68	−0.175898
$670$	0	0
$671$	−1846.07	−0.106210
$672$	−19058.3	−1.09403
$673$	23110.6	1.32370	0.661848	−	0.749638i	$-0.269772\pi$
$673$	23110.6	1.32370	0.661848	+	0.749638i	$0.269772\pi$
$674$	45340.3	2.59116
$675$	0	0
$676$	−49938.3	−2.84128
$677$	17052.4	0.968062	0.484031	−	0.875051i	$-0.339172\pi$
$677$	17052.4	0.968062	0.484031	+	0.875051i	$0.339172\pi$
$678$	−10586.3	−0.599653
$679$	−6677.20	−0.377390
$680$	0	0
$681$	−1403.26	−0.0789618
$682$	10443.0	0.586341
$683$	−28542.8	−1.59907	−0.799533	−	0.600623i	$-0.794919\pi$
$683$	−28542.8	−1.59907	−0.799533	+	0.600623i	$0.794919\pi$
$684$	30945.3	1.72986
$685$	0	0
$686$	21863.3	1.21683
$687$	−1751.65	−0.0972774
$688$	−81175.6	−4.49824
$689$	1271.07	0.0702814
$690$	0	0
$691$	6479.20	0.356701	0.178351	−	0.983967i	$-0.442924\pi$
$691$	6479.20	0.356701	0.178351	+	0.983967i	$0.442924\pi$
$692$	54328.9	2.98450
$693$	−953.312	−0.0522559
$694$	19055.1	1.04225
$695$	0	0
$696$	−75968.6	−4.13733
$697$	27592.6	1.49949
$698$	7420.94	0.402417
$699$	24520.0	1.32680
$700$	0	0
$701$	21118.6	1.13786	0.568929	−	0.822386i	$-0.307358\pi$
$701$	21118.6	1.13786	0.568929	+	0.822386i	$0.307358\pi$
$702$	−3588.10	−0.192912
$703$	−17991.7	−0.965249
$704$	−29577.3	−1.58343
$705$	0	0
$706$	−24505.4	−1.30633
$707$	−2932.03	−0.155969
$708$	23090.0	1.22567
$709$	7072.53	0.374632	0.187316	−	0.982300i	$-0.440021\pi$
$709$	7072.53	0.374632	0.187316	+	0.982300i	$0.440021\pi$
$710$	0	0
$711$	12626.4	0.666000
$712$	−81141.1	−4.27092
$713$	2689.20	0.141250
$714$	−13256.3	−0.694822
$715$	0	0
$716$	28130.8	1.46829
$717$	10730.1	0.558890
$718$	−47957.5	−2.49270
$719$	22177.9	1.15034	0.575170	−	0.818034i	$-0.304936\pi$
$719$	22177.9	1.15034	0.575170	+	0.818034i	$0.304936\pi$
$720$	0	0
$721$	−5293.71	−0.273437
$722$	11277.8	0.581323
$723$	11065.7	0.569207
$724$	−10080.5	−0.517459
$725$	0	0
$726$	2396.74	0.122523
$727$	17390.7	0.887186	0.443593	−	0.896228i	$-0.353704\pi$
$727$	17390.7	0.887186	0.443593	+	0.896228i	$0.353704\pi$
$728$	−2204.14	−0.112213
$729$	16119.1	0.818935
$730$	0	0
$731$	−32235.6	−1.63102
$732$	13706.2	0.692069
$733$	21877.0	1.10238	0.551191	−	0.834379i	$-0.314174\pi$
$733$	21877.0	1.10238	0.551191	+	0.834379i	$0.314174\pi$
$734$	19941.4	1.00279
$735$	0	0
$736$	−13924.8	−0.697385
$737$	−1938.56	−0.0968899
$738$	−19872.2	−0.991201
$739$	14203.1	0.706994	0.353497	−	0.935436i	$-0.384992\pi$
$739$	14203.1	0.706994	0.353497	+	0.935436i	$0.384992\pi$
$740$	0	0
$741$	−1472.07	−0.0729797
$742$	−9760.87	−0.482928
$743$	3933.68	0.194230	0.0971148	−	0.995273i	$-0.469039\pi$
$743$	3933.68	0.194230	0.0971148	+	0.995273i	$0.469039\pi$
$744$	−50484.5	−2.48771
$745$	0	0
$746$	−54814.1	−2.69020
$747$	3974.21	0.194657
$748$	−27884.2	−1.36303
$749$	725.537	0.0353946
$750$	0	0
$751$	22554.3	1.09590	0.547949	−	0.836512i	$-0.315409\pi$
$751$	22554.3	1.09590	0.547949	+	0.836512i	$0.315409\pi$
$752$	−50708.0	−2.45895
$753$	−2971.45	−0.143806
$754$	−6263.65	−0.302531
$755$	0	0
$756$	20427.3	0.982719
$757$	−11432.0	−0.548883	−0.274441	−	0.961604i	$-0.588493\pi$
$757$	−11432.0	−0.548883	−0.274441	+	0.961604i	$0.588493\pi$
$758$	−60275.2	−2.88825
$759$	617.187	0.0295158
$760$	0	0
$761$	−32660.1	−1.55575	−0.777877	−	0.628416i	$-0.783704\pi$
$761$	−32660.1	−1.55575	−0.777877	+	0.628416i	$0.783704\pi$
$762$	−12679.4	−0.602789
$763$	−2510.11	−0.119098
$764$	−113943.	−5.39568
$765$	0	0
$766$	11265.2	0.531369
$767$	1239.60	0.0583565
$768$	79531.8	3.73679
$769$	−8569.93	−0.401872	−0.200936	−	0.979604i	$-0.564398\pi$
$769$	−8569.93	−0.401872	−0.200936	+	0.979604i	$0.564398\pi$
$770$	0	0
$771$	26842.1	1.25382
$772$	31236.8	1.45627
$773$	29158.0	1.35671	0.678357	−	0.734733i	$-0.262693\pi$
$773$	29158.0	1.35671	0.678357	+	0.734733i	$0.262693\pi$
$774$	23216.1	1.07815
$775$	0	0
$776$	91587.2	4.23684
$777$	−4115.09	−0.189998
$778$	−5444.17	−0.250878
$779$	−23530.0	−1.08222
$780$	0	0
$781$	−10111.8	−0.463289
$782$	−9685.58	−0.442910
$783$	37797.6	1.72513
$784$	−85280.9	−3.88488
$785$	0	0
$786$	20819.0	0.944770
$787$	8501.30	0.385055	0.192528	−	0.981292i	$-0.438332\pi$
$787$	8501.30	0.385055	0.192528	+	0.981292i	$0.438332\pi$
$788$	−50354.2	−2.27639
$789$	22231.6	1.00312
$790$	0	0
$791$	3235.56	0.145440
$792$	13076.0	0.586662
$793$	735.823	0.0329506
$794$	27853.8	1.24496
$795$	0	0
$796$	−12799.7	−0.569944
$797$	−37459.5	−1.66485	−0.832424	−	0.554139i	$-0.813047\pi$
$797$	−37459.5	−1.66485	−0.832424	+	0.554139i	$0.813047\pi$
$798$	11304.4	0.501470
$799$	−20136.6	−0.891592
$800$	0	0
$801$	13988.2	0.617039
$802$	−86671.9	−3.81607
$803$	1700.27	0.0747214
$804$	14392.9	0.631339
$805$	0	0
$806$	−4162.47	−0.181907
$807$	5830.07	0.254310
$808$	40216.9	1.75102
$809$	18045.8	0.784246	0.392123	−	0.919913i	$-0.371741\pi$
$809$	18045.8	0.784246	0.392123	+	0.919913i	$0.371741\pi$
$810$	0	0
$811$	914.961	0.0396161	0.0198080	−	0.999804i	$-0.493694\pi$
$811$	914.961	0.0396161	0.0198080	+	0.999804i	$0.493694\pi$
$812$	35659.5	1.54114
$813$	2803.70	0.120947
$814$	−11675.8	−0.502750
$815$	0	0
$816$	109602.	4.70201
$817$	27489.3	1.17715
$818$	84016.3	3.59115
$819$	379.979	0.0162119
$820$	0	0
$821$	15188.9	0.645672	0.322836	−	0.946455i	$-0.395364\pi$
$821$	15188.9	0.645672	0.322836	+	0.946455i	$0.395364\pi$
$822$	−33488.5	−1.42098
$823$	37930.3	1.60652	0.803261	−	0.595628i	$-0.203097\pi$
$823$	37930.3	1.60652	0.803261	+	0.595628i	$0.203097\pi$
$824$	72610.7	3.06980
$825$	0	0
$826$	−9519.22	−0.400988
$827$	−29679.9	−1.24797	−0.623985	−	0.781437i	$-0.714487\pi$
$827$	−29679.9	−1.24797	−0.623985	+	0.781437i	$0.714487\pi$
$828$	5171.39	0.217051
$829$	29094.2	1.21892	0.609458	−	0.792818i	$-0.291387\pi$
$829$	29094.2	1.21892	0.609458	+	0.792818i	$0.291387\pi$
$830$	0	0
$831$	−6962.70	−0.290654
$832$	11789.2	0.491245
$833$	−33865.8	−1.40862
$834$	55252.4	2.29405
$835$	0	0
$836$	23778.6	0.983732
$837$	25118.2	1.03729
$838$	−8443.41	−0.348058
$839$	38791.2	1.59621	0.798105	−	0.602519i	$-0.205836\pi$
$839$	38791.2	1.59621	0.798105	+	0.602519i	$0.205836\pi$
$840$	0	0
$841$	41593.3	1.70541
$842$	25789.2	1.05553
$843$	−18155.3	−0.741758
$844$	−68838.7	−2.80749
$845$	0	0
$846$	14502.4	0.589365
$847$	−732.531	−0.0297167
$848$	80702.4	3.26808
$849$	25599.4	1.03483
$850$	0	0
$851$	−3006.66	−0.121113
$852$	75075.1	3.01881
$853$	42933.7	1.72335	0.861677	−	0.507458i	$-0.169415\pi$
$853$	42933.7	1.72335	0.861677	+	0.507458i	$0.169415\pi$
$854$	−5650.58	−0.226415
$855$	0	0
$856$	−9951.76	−0.397365
$857$	2664.36	0.106199	0.0530997	−	0.998589i	$-0.483090\pi$
$857$	2664.36	0.106199	0.0530997	+	0.998589i	$0.483090\pi$
$858$	−955.312	−0.0380115
$859$	−25002.7	−0.993111	−0.496556	−	0.868005i	$-0.665402\pi$
$859$	−25002.7	−0.993111	−0.496556	+	0.868005i	$0.665402\pi$
$860$	0	0
$861$	−5381.81	−0.213022
$862$	−63834.8	−2.52230
$863$	−27509.8	−1.08510	−0.542551	−	0.840023i	$-0.682541\pi$
$863$	−27509.8	−1.08510	−0.542551	+	0.840023i	$0.682541\pi$
$864$	−130063.	−5.12135
$865$	0	0
$866$	−56753.1	−2.22696
$867$	26026.1	1.01948
$868$	23697.3	0.926658
$869$	9702.19	0.378739
$870$	0	0
$871$	772.688	0.0300592
$872$	34429.6	1.33708
$873$	−15789.0	−0.612117
$874$	8259.51	0.319659
$875$	0	0
$876$	−12623.7	−0.486888
$877$	25993.8	1.00085	0.500426	−	0.865779i	$-0.333177\pi$
$877$	25993.8	1.00085	0.500426	+	0.865779i	$0.333177\pi$
$878$	−38480.7	−1.47911
$879$	−22754.1	−0.873126
$880$	0	0
$881$	−29528.7	−1.12922	−0.564612	−	0.825357i	$-0.690974\pi$
$881$	−29528.7	−1.12922	−0.564612	+	0.825357i	$0.690974\pi$
$882$	24390.2	0.931133
$883$	50497.5	1.92455	0.962274	−	0.272081i	$-0.0877119\pi$
$883$	50497.5	1.92455	0.962274	+	0.272081i	$0.0877119\pi$
$884$	11114.3	0.422867
$885$	0	0
$886$	16198.2	0.614208
$887$	−36471.9	−1.38062	−0.690309	−	0.723515i	$-0.742525\pi$
$887$	−36471.9	−1.38062	−0.690309	+	0.723515i	$0.742525\pi$
$888$	56444.3	2.13305
$889$	3875.28	0.146201
$890$	0	0
$891$	1513.12	0.0568929
$892$	−19596.6	−0.735586
$893$	17171.8	0.643484
$894$	21602.8	0.808173
$895$	0	0
$896$	−47723.2	−1.77937
$897$	−246.004	−0.00915700
$898$	−6426.54	−0.238816
$899$	43848.2	1.62672
$900$	0	0
$901$	32047.7	1.18498
$902$	−15269.9	−0.563673
$903$	6287.40	0.231707
$904$	−44380.3	−1.63282
$905$	0	0
$906$	12351.6	0.452930
$907$	−15130.2	−0.553902	−0.276951	−	0.960884i	$-0.589324\pi$
$907$	−15130.2	−0.553902	−0.276951	+	0.960884i	$0.589324\pi$
$908$	−9034.81	−0.330210
$909$	−6933.12	−0.252978
$910$	0	0
$911$	13937.0	0.506864	0.253432	−	0.967353i	$-0.418441\pi$
$911$	13937.0	0.506864	0.253432	+	0.967353i	$0.418441\pi$
$912$	−93464.6	−3.39355
$913$	3053.81	0.110697
$914$	−15269.9	−0.552608
$915$	0	0
$916$	−11277.9	−0.406804
$917$	−6363.04	−0.229145
$918$	−90467.3	−3.25258
$919$	−40897.5	−1.46799	−0.733996	−	0.679153i	$-0.762347\pi$
$919$	−40897.5	−1.46799	−0.733996	+	0.679153i	$0.762347\pi$
$920$	0	0
$921$	−17387.7	−0.622091
$922$	62423.5	2.22973
$923$	4030.45	0.143731
$924$	5438.68	0.193636
$925$	0	0
$926$	−88955.7	−3.15688
$927$	−12517.6	−0.443508
$928$	−227048.	−8.03149
$929$	−12154.1	−0.429239	−0.214620	−	0.976698i	$-0.568851\pi$
$929$	−12154.1	−0.429239	−0.214620	+	0.976698i	$0.568851\pi$
$930$	0	0
$931$	28879.5	1.01664
$932$	157870.	5.54852
$933$	10136.2	0.355675
$934$	−37121.1	−1.30047
$935$	0	0
$936$	−5211.95	−0.182006
$937$	15754.1	0.549267	0.274634	−	0.961549i	$-0.411443\pi$
$937$	15754.1	0.549267	0.274634	+	0.961549i	$0.411443\pi$
$938$	−5933.67	−0.206547
$939$	−28526.2	−0.991393
$940$	0	0
$941$	−4217.53	−0.146108	−0.0730539	−	0.997328i	$-0.523275\pi$
$941$	−4217.53	−0.146108	−0.0730539	+	0.997328i	$0.523275\pi$
$942$	41892.7	1.44898
$943$	−3932.18	−0.135789
$944$	78704.5	2.71357
$945$	0	0
$946$	17839.4	0.613116
$947$	−49839.0	−1.71019	−0.855095	−	0.518471i	$-0.826502\pi$
$947$	−49839.0	−1.71019	−0.855095	+	0.518471i	$0.826502\pi$
$948$	−72033.9	−2.46788
$949$	−677.708	−0.0231816
$950$	0	0
$951$	−16628.5	−0.566999
$952$	−55573.3	−1.89196
$953$	12845.7	0.436635	0.218318	−	0.975878i	$-0.429943\pi$
$953$	12845.7	0.436635	0.218318	+	0.975878i	$0.429943\pi$
$954$	−23080.7	−0.783298
$955$	0	0
$956$	69085.4	2.33722
$957$	10063.4	0.339921
$958$	−64413.9	−2.17236
$959$	10235.3	0.344646
$960$	0	0
$961$	−651.937	−0.0218837
$962$	4653.86	0.155973
$963$	1715.62	0.0574092
$964$	71245.7	2.38036
$965$	0	0
$966$	1889.13	0.0629210
$967$	38829.6	1.29129	0.645645	−	0.763638i	$-0.276589\pi$
$967$	38829.6	1.29129	0.645645	+	0.763638i	$0.276589\pi$
$968$	10047.7	0.333621
$969$	−37115.6	−1.23047
$970$	0	0
$971$	−11438.8	−0.378053	−0.189026	−	0.981972i	$-0.560533\pi$
$971$	−11438.8	−0.378053	−0.189026	+	0.981972i	$0.560533\pi$
$972$	79869.3	2.63561
$973$	−16887.1	−0.556400
$974$	59056.4	1.94280
$975$	0	0
$976$	46718.7	1.53220
$977$	12084.1	0.395706	0.197853	−	0.980232i	$-0.436603\pi$
$977$	12084.1	0.395706	0.197853	+	0.980232i	$0.436603\pi$
$978$	−22848.8	−0.747058
$979$	10748.6	0.350896
$980$	0	0
$981$	−5935.44	−0.193174
$982$	−99818.8	−3.24373
$983$	23502.2	0.762569	0.381284	−	0.924458i	$-0.375482\pi$
$983$	23502.2	0.762569	0.381284	+	0.924458i	$0.375482\pi$
$984$	73819.1	2.39153
$985$	0	0
$986$	−157926.	−5.10081
$987$	3927.55	0.126662
$988$	−9477.87	−0.305194
$989$	4593.84	0.147700
$990$	0	0
$991$	−18664.1	−0.598268	−0.299134	−	0.954211i	$-0.596698\pi$
$991$	−18664.1	−0.598268	−0.299134	+	0.954211i	$0.596698\pi$
$992$	−150884.	−4.82919
$993$	9193.47	0.293803
$994$	−30950.9	−0.987627
$995$	0	0
$996$	−22673.0	−0.721308
$997$	−24528.4	−0.779158	−0.389579	−	0.920993i	$-0.627380\pi$
$997$	−24528.4	−0.779158	−0.389579	+	0.920993i	$0.627380\pi$
$998$	−57893.7	−1.83626
$999$	−28083.4	−0.889410

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.4.a.c.1.2		2
3.2	odd	2		2475.4.a.l.1.1		2
5.2	odd	4		275.4.b.b.199.4		4
5.3	odd	4		275.4.b.b.199.1		4
5.4	even	2		55.4.a.b.1.1	✓	2
15.14	odd	2		495.4.a.e.1.2		2
20.19	odd	2		880.4.a.r.1.2		2
55.54	odd	2		605.4.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.a.b.1.1	✓	2	5.4	even	2
275.4.a.c.1.2		2	1.1	even	1	trivial
275.4.b.b.199.1		4	5.3	odd	4
275.4.b.b.199.4		4	5.2	odd	4
495.4.a.e.1.2		2	15.14	odd	2
605.4.a.g.1.2		2	55.54	odd	2
880.4.a.r.1.2		2	20.19	odd	2
2475.4.a.l.1.1		2	3.2	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)

\(f(q)\)	\(=\)	\(q+5.56155 q^{2} +3.56155 q^{3} +22.9309 q^{4} +19.8078 q^{6} -6.05398 q^{7} +83.0388 q^{8} -14.3153 q^{9} -11.0000 q^{11} +81.6695 q^{12} +4.38447 q^{13} -33.6695 q^{14} +278.378 q^{16} +110.546 q^{17} -79.6155 q^{18} -94.2699 q^{19} -21.5616 q^{21} -61.1771 q^{22} -15.7538 q^{23} +295.747 q^{24} +24.3845 q^{26} -147.147 q^{27} -138.823 q^{28} -256.870 q^{29} -170.702 q^{31} +883.902 q^{32} -39.1771 q^{33} +614.810 q^{34} -328.263 q^{36} +190.853 q^{37} -524.287 q^{38} +15.6155 q^{39} +249.602 q^{41} -119.916 q^{42} -291.602 q^{43} -252.240 q^{44} -87.6155 q^{46} -182.155 q^{47} +991.457 q^{48} -306.349 q^{49} +393.717 q^{51} +100.540 q^{52} +289.902 q^{53} -818.365 q^{54} -502.715 q^{56} -335.747 q^{57} -1428.60 q^{58} +282.725 q^{59} +167.825 q^{61} -949.366 q^{62} +86.6647 q^{63} +2688.85 q^{64} -217.885 q^{66} +176.233 q^{67} +2534.93 q^{68} -56.1080 q^{69} +919.255 q^{71} -1188.73 q^{72} -154.570 q^{73} +1061.44 q^{74} -2161.69 q^{76} +66.5937 q^{77} +86.8466 q^{78} -882.017 q^{79} -137.557 q^{81} +1388.18 q^{82} -277.619 q^{83} -494.425 q^{84} -1621.76 q^{86} -914.857 q^{87} -913.427 q^{88} -977.147 q^{89} -26.5435 q^{91} -361.248 q^{92} -607.963 q^{93} -1013.07 q^{94} +3148.07 q^{96} +1102.94 q^{97} -1703.78 q^{98} +157.469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 7 q^{2} + 3 q^{3} + 17 q^{4} + 19 q^{6} + 25 q^{7} + 63 q^{8} - 41 q^{9} - 22 q^{11} + 85 q^{12} + 50 q^{13} + 11 q^{14} + 297 q^{16} + 151 q^{17} - 118 q^{18} - 3 q^{19} - 39 q^{21} - 77 q^{22} - 48 q^{23}+ \cdots + 451 q^{99}+O(q^{100}) \) 2 * q + 7 * q^2 + 3 * q^3 + 17 * q^4 + 19 * q^6 + 25 * q^7 + 63 * q^8 - 41 * q^9 - 22 * q^11 + 85 * q^12 + 50 * q^13 + 11 * q^14 + 297 * q^16 + 151 * q^17 - 118 * q^18 - 3 * q^19 - 39 * q^21 - 77 * q^22 - 48 * q^23 + 307 * q^24 + 90 * q^26 - 117 * q^27 - 323 * q^28 - 221 * q^29 + 141 * q^31 + 1071 * q^32 - 33 * q^33 + 673 * q^34 - 170 * q^36 + 559 * q^37 - 393 * q^38 - 10 * q^39 - 144 * q^41 - 145 * q^42 + 60 * q^43 - 187 * q^44 - 134 * q^46 + 48 * q^47 + 981 * q^48 + 315 * q^49 + 371 * q^51 - 170 * q^52 - 117 * q^53 - 775 * q^54 - 1125 * q^56 - 387 * q^57 - 1377 * q^58 - 86 * q^59 - 155 * q^61 - 501 * q^62 - 742 * q^63 + 2809 * q^64 - 209 * q^66 - 266 * q^67 + 2295 * q^68 - 38 * q^69 + 1587 * q^71 - 654 * q^72 - 70 * q^73 + 1591 * q^74 - 2703 * q^76 - 275 * q^77 + 50 * q^78 - 1294 * q^79 + 566 * q^81 + 822 * q^82 + 558 * q^83 - 391 * q^84 - 1116 * q^86 - 935 * q^87 - 693 * q^88 - 1777 * q^89 + 1390 * q^91 - 170 * q^92 - 783 * q^93 - 682 * q^94 + 3043 * q^96 + 334 * q^97 - 810 * q^98 + 451 * q^99

Introduction

L-functions

Modular forms

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Database

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