LMFDB - Embedded newform 1305.4.a.n.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,4,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1305.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1305.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:	$76.9974925575$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 37x^{5} + 55x^{4} + 336x^{3} - 227x^{2} - 824x - 166 $ x^7 - 2x^6 - 37x^5 + 55x^4 + 336x^3 - 227x^2 - 824x - 166
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.57720$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.57720 q^{2} +4.79637 q^{4} -5.00000 q^{5} +15.0281 q^{7} -11.4600 q^{8} -17.8860 q^{10} -70.8260 q^{11} +62.2117 q^{13} +53.7587 q^{14} -79.3658 q^{16} +67.1848 q^{17} +118.988 q^{19} -23.9818 q^{20} -253.359 q^{22} -72.5383 q^{23} +25.0000 q^{25} +222.544 q^{26} +72.0804 q^{28} -29.0000 q^{29} -180.227 q^{31} -192.227 q^{32} +240.334 q^{34} -75.1407 q^{35} -47.8439 q^{37} +425.646 q^{38} +57.3002 q^{40} -371.956 q^{41} -409.069 q^{43} -339.707 q^{44} -259.484 q^{46} -125.891 q^{47} -117.155 q^{49} +89.4300 q^{50} +298.390 q^{52} +215.764 q^{53} +354.130 q^{55} -172.223 q^{56} -103.739 q^{58} -356.919 q^{59} -466.115 q^{61} -644.707 q^{62} -52.7086 q^{64} -311.059 q^{65} -578.359 q^{67} +322.243 q^{68} -268.793 q^{70} -870.924 q^{71} +411.904 q^{73} -171.147 q^{74} +570.712 q^{76} -1064.38 q^{77} +1120.23 q^{79} +396.829 q^{80} -1330.56 q^{82} -1079.28 q^{83} -335.924 q^{85} -1463.32 q^{86} +811.668 q^{88} -388.993 q^{89} +934.926 q^{91} -347.920 q^{92} -450.339 q^{94} -594.942 q^{95} -1528.82 q^{97} -419.088 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 22 q^{4} - 35 q^{5} - 50 q^{7} + 33 q^{8} - 10 q^{10} - 76 q^{11} + 30 q^{13} - 89 q^{14} + 138 q^{16} + 140 q^{17} + 90 q^{19} - 110 q^{20} + 61 q^{22} - 34 q^{23} + 175 q^{25} + 241 q^{26}+ \cdots + 761 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 22 * q^4 - 35 * q^5 - 50 * q^7 + 33 * q^8 - 10 * q^10 - 76 * q^11 + 30 * q^13 - 89 * q^14 + 138 * q^16 + 140 * q^17 + 90 * q^19 - 110 * q^20 + 61 * q^22 - 34 * q^23 + 175 * q^25 + 241 * q^26 - 57 * q^28 - 203 * q^29 + 524 * q^31 + 6 * q^32 + 255 * q^34 + 250 * q^35 - 28 * q^37 - 222 * q^38 - 165 * q^40 - 1532 * q^41 - 464 * q^43 - 1475 * q^44 + 72 * q^46 + 360 * q^47 + 569 * q^49 + 50 * q^50 - 205 * q^52 - 282 * q^53 + 380 * q^55 - 1102 * q^56 - 58 * q^58 - 766 * q^59 + 1200 * q^61 - 2856 * q^62 + 701 * q^64 - 150 * q^65 + 1546 * q^67 - 1801 * q^68 + 445 * q^70 - 1802 * q^71 - 220 * q^73 - 1594 * q^74 + 1960 * q^76 - 3222 * q^77 + 1298 * q^79 - 690 * q^80 + 856 * q^82 - 1652 * q^83 - 700 * q^85 - 7628 * q^86 + 550 * q^88 - 2846 * q^89 - 816 * q^91 - 472 * q^92 + 745 * q^94 - 450 * q^95 + 1110 * q^97 + 761 * q^98

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 $

$2$	3.57720	1.26473	0.632366	−	0.774670i	$-0.282084\pi$
$2$	3.57720	1.26473	0.632366	+	0.774670i	$0.282084\pi$
$3$	0	0
$3$	0	0
$4$	4.79637	0.599546
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	15.0281	0.811443	0.405721	−	0.913997i	$-0.367020\pi$
$7$	15.0281	0.811443	0.405721	+	0.913997i	$0.367020\pi$
$8$	−11.4600	−0.506467
$9$	0	0
$10$	−17.8860	−0.565605
$11$	−70.8260	−1.94135	−0.970674	−	0.240399i	$-0.922722\pi$
$11$	−70.8260	−1.94135	−0.970674	+	0.240399i	$0.922722\pi$
$12$	0	0
$13$	62.2117	1.32726	0.663632	−	0.748059i	$-0.269014\pi$
$13$	62.2117	1.32726	0.663632	+	0.748059i	$0.269014\pi$
$14$	53.7587	1.02626
$15$	0	0
$16$	−79.3658	−1.24009
$17$	67.1848	0.958513	0.479256	−	0.877675i	$-0.340906\pi$
$17$	67.1848	0.958513	0.479256	+	0.877675i	$0.340906\pi$
$18$	0	0
$19$	118.988	1.43673	0.718364	−	0.695667i	$-0.244891\pi$
$19$	118.988	1.43673	0.718364	+	0.695667i	$0.244891\pi$
$20$	−23.9818	−0.268125
$21$	0	0
$22$	−253.359	−2.45528
$23$	−72.5383	−0.657621	−0.328810	−	0.944396i	$-0.606648\pi$
$23$	−72.5383	−0.657621	−0.328810	+	0.944396i	$0.606648\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	222.544	1.67863
$27$	0	0
$28$	72.0804	0.486497
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−180.227	−1.04418	−0.522092	−	0.852889i	$-0.674848\pi$
$31$	−180.227	−1.04418	−0.522092	+	0.852889i	$0.674848\pi$
$32$	−192.227	−1.06191
$33$	0	0
$34$	240.334	1.21226
$35$	−75.1407	−0.362888
$36$	0	0
$37$	−47.8439	−0.212581	−0.106290	−	0.994335i	$-0.533897\pi$
$37$	−47.8439	−0.212581	−0.106290	+	0.994335i	$0.533897\pi$
$38$	425.646	1.81708
$39$	0	0
$40$	57.3002	0.226499
$41$	−371.956	−1.41682	−0.708411	−	0.705800i	$-0.750588\pi$
$41$	−371.956	−1.41682	−0.708411	+	0.705800i	$0.750588\pi$
$42$	0	0
$43$	−409.069	−1.45075	−0.725377	−	0.688352i	$-0.758334\pi$
$43$	−409.069	−1.45075	−0.725377	+	0.688352i	$0.758334\pi$
$44$	−339.707	−1.16393
$45$	0	0
$46$	−259.484	−0.831714
$47$	−125.891	−0.390705	−0.195353	−	0.980733i	$-0.562585\pi$
$47$	−125.891	−0.390705	−0.195353	+	0.980733i	$0.562585\pi$
$48$	0	0
$49$	−117.155	−0.341560
$50$	89.4300	0.252946
$51$	0	0
$52$	298.390	0.795755
$53$	215.764	0.559198	0.279599	−	0.960117i	$-0.409798\pi$
$53$	215.764	0.559198	0.279599	+	0.960117i	$0.409798\pi$
$54$	0	0
$55$	354.130	0.868197
$56$	−172.223	−0.410969
$57$	0	0
$58$	−103.739	−0.234855
$59$	−356.919	−0.787575	−0.393788	−	0.919201i	$-0.628835\pi$
$59$	−356.919	−0.787575	−0.393788	+	0.919201i	$0.628835\pi$
$60$	0	0
$61$	−466.115	−0.978358	−0.489179	−	0.872183i	$-0.662704\pi$
$61$	−466.115	−0.978358	−0.489179	+	0.872183i	$0.662704\pi$
$62$	−644.707	−1.32061
$63$	0	0
$64$	−52.7086	−0.102946
$65$	−311.059	−0.593570
$66$	0	0
$67$	−578.359	−1.05459	−0.527297	−	0.849681i	$-0.676794\pi$
$67$	−578.359	−1.05459	−0.527297	+	0.849681i	$0.676794\pi$
$68$	322.243	0.574672
$69$	0	0
$70$	−268.793	−0.458956
$71$	−870.924	−1.45577	−0.727885	−	0.685699i	$-0.759497\pi$
$71$	−870.924	−1.45577	−0.727885	+	0.685699i	$0.759497\pi$
$72$	0	0
$73$	411.904	0.660407	0.330204	−	0.943910i	$-0.392883\pi$
$73$	411.904	0.660407	0.330204	+	0.943910i	$0.392883\pi$
$74$	−171.147	−0.268857
$75$	0	0
$76$	570.712	0.861384
$77$	−1064.38	−1.57529
$78$	0	0
$79$	1120.23	1.59538	0.797692	−	0.603065i	$-0.206054\pi$
$79$	1120.23	1.59538	0.797692	+	0.603065i	$0.206054\pi$
$80$	396.829	0.554585
$81$	0	0
$82$	−1330.56	−1.79190
$83$	−1079.28	−1.42730	−0.713652	−	0.700501i	$-0.752960\pi$
$83$	−1079.28	−1.42730	−0.713652	+	0.700501i	$0.752960\pi$
$84$	0	0
$85$	−335.924	−0.428660
$86$	−1463.32	−1.83481
$87$	0	0
$88$	811.668	0.983229
$89$	−388.993	−0.463294	−0.231647	−	0.972800i	$-0.574411\pi$
$89$	−388.993	−0.463294	−0.231647	+	0.972800i	$0.574411\pi$
$90$	0	0
$91$	934.926	1.07700
$92$	−347.920	−0.394274
$93$	0	0
$94$	−450.339	−0.494137
$95$	−594.942	−0.642524
$96$	0	0
$97$	−1528.82	−1.60029	−0.800147	−	0.599803i	$-0.795245\pi$
$97$	−1528.82	−1.60029	−0.800147	+	0.599803i	$0.795245\pi$
$98$	−419.088	−0.431982
$99$	0	0
$100$	119.909	0.119909
$101$	1425.50	1.40439	0.702193	−	0.711987i	$-0.252204\pi$
$101$	1425.50	1.40439	0.702193	+	0.711987i	$0.252204\pi$
$102$	0	0
$103$	−1595.68	−1.52648	−0.763240	−	0.646116i	$-0.776392\pi$
$103$	−1595.68	−1.52648	−0.763240	+	0.646116i	$0.776392\pi$
$104$	−712.949	−0.672215
$105$	0	0
$106$	771.832	0.707236
$107$	155.104	0.140135	0.0700675	−	0.997542i	$-0.477679\pi$
$107$	155.104	0.140135	0.0700675	+	0.997542i	$0.477679\pi$
$108$	0	0
$109$	1212.23	1.06524	0.532618	−	0.846356i	$-0.321208\pi$
$109$	1212.23	1.06524	0.532618	+	0.846356i	$0.321208\pi$
$110$	1266.79	1.09804
$111$	0	0
$112$	−1192.72	−1.00626
$113$	1152.01	0.959043	0.479521	−	0.877530i	$-0.340810\pi$
$113$	1152.01	0.959043	0.479521	+	0.877530i	$0.340810\pi$
$114$	0	0
$115$	362.691	0.294097
$116$	−139.095	−0.111333
$117$	0	0
$118$	−1276.77	−0.996072
$119$	1009.66	0.777778
$120$	0	0
$121$	3685.32	2.76883
$122$	−1667.39	−1.23736
$123$	0	0
$124$	−864.434	−0.626036
$125$	−125.000	−0.0894427
$126$	0	0
$127$	398.743	0.278604	0.139302	−	0.990250i	$-0.455514\pi$
$127$	398.743	0.278604	0.139302	+	0.990250i	$0.455514\pi$
$128$	1349.27	0.931715
$129$	0	0
$130$	−1112.72	−0.750707
$131$	385.241	0.256937	0.128468	−	0.991714i	$-0.458994\pi$
$131$	385.241	0.256937	0.128468	+	0.991714i	$0.458994\pi$
$132$	0	0
$133$	1788.17	1.16582
$134$	−2068.90	−1.33378
$135$	0	0
$136$	−769.941	−0.485455
$137$	351.877	0.219437	0.109719	−	0.993963i	$-0.465005\pi$
$137$	351.877	0.219437	0.109719	+	0.993963i	$0.465005\pi$
$138$	0	0
$139$	1138.48	0.694711	0.347355	−	0.937734i	$-0.387080\pi$
$139$	1138.48	0.694711	0.347355	+	0.937734i	$0.387080\pi$
$140$	−360.402	−0.217568
$141$	0	0
$142$	−3115.47	−1.84116
$143$	−4406.21	−2.57668
$144$	0	0
$145$	145.000	0.0830455
$146$	1473.46	0.835238
$147$	0	0
$148$	−229.477	−0.127452
$149$	−1200.59	−0.660107	−0.330053	−	0.943962i	$-0.607067\pi$
$149$	−1200.59	−0.660107	−0.330053	+	0.943962i	$0.607067\pi$
$150$	0	0
$151$	−1539.35	−0.829605	−0.414802	−	0.909911i	$-0.636149\pi$
$151$	−1539.35	−0.829605	−0.414802	+	0.909911i	$0.636149\pi$
$152$	−1363.61	−0.727655
$153$	0	0
$154$	−3807.51	−1.99232
$155$	901.134	0.466973
$156$	0	0
$157$	223.134	0.113427	0.0567136	−	0.998390i	$-0.481938\pi$
$157$	223.134	0.113427	0.0567136	+	0.998390i	$0.481938\pi$
$158$	4007.28	2.01773
$159$	0	0
$160$	961.136	0.474903
$161$	−1090.11	−0.533622
$162$	0	0
$163$	−3114.71	−1.49671	−0.748353	−	0.663301i	$-0.769155\pi$
$163$	−3114.71	−1.49671	−0.748353	+	0.663301i	$0.769155\pi$
$164$	−1784.04	−0.849450
$165$	0	0
$166$	−3860.79	−1.80516
$167$	1720.78	0.797351	0.398676	−	0.917092i	$-0.369470\pi$
$167$	1720.78	0.797351	0.398676	+	0.917092i	$0.369470\pi$
$168$	0	0
$169$	1673.30	0.761629
$170$	−1201.67	−0.542140
$171$	0	0
$172$	−1962.04	−0.869793
$173$	−2438.70	−1.07174	−0.535870	−	0.844301i	$-0.680016\pi$
$173$	−2438.70	−1.07174	−0.535870	+	0.844301i	$0.680016\pi$
$174$	0	0
$175$	375.703	0.162289
$176$	5621.16	2.40745
$177$	0	0
$178$	−1391.50	−0.585942
$179$	2248.00	0.938676	0.469338	−	0.883018i	$-0.344493\pi$
$179$	2248.00	0.938676	0.469338	+	0.883018i	$0.344493\pi$
$180$	0	0
$181$	−2357.13	−0.967977	−0.483989	−	0.875074i	$-0.660812\pi$
$181$	−2357.13	−0.967977	−0.483989	+	0.875074i	$0.660812\pi$
$182$	3344.42	1.36211
$183$	0	0
$184$	831.292	0.333063
$185$	239.219	0.0950689
$186$	0	0
$187$	−4758.43	−1.86081
$188$	−603.821	−0.234246
$189$	0	0
$190$	−2128.23	−0.812621
$191$	−1922.56	−0.728331	−0.364166	−	0.931334i	$-0.618646\pi$
$191$	−1922.56	−0.728331	−0.364166	+	0.931334i	$0.618646\pi$
$192$	0	0
$193$	1840.68	0.686504	0.343252	−	0.939243i	$-0.388472\pi$
$193$	1840.68	0.686504	0.343252	+	0.939243i	$0.388472\pi$
$194$	−5468.91	−2.02394
$195$	0	0
$196$	−561.919	−0.204781
$197$	2712.79	0.981109	0.490555	−	0.871410i	$-0.336794\pi$
$197$	2712.79	0.981109	0.490555	+	0.871410i	$0.336794\pi$
$198$	0	0
$199$	224.631	0.0800185	0.0400093	−	0.999199i	$-0.487261\pi$
$199$	224.631	0.0800185	0.0400093	+	0.999199i	$0.487261\pi$
$200$	−286.501	−0.101293
$201$	0	0
$202$	5099.31	1.77617
$203$	−435.816	−0.150681
$204$	0	0
$205$	1859.78	0.633622
$206$	−5708.08	−1.93059
$207$	0	0
$208$	−4937.48	−1.64593
$209$	−8427.48	−2.78919
$210$	0	0
$211$	−4160.10	−1.35731	−0.678656	−	0.734456i	$-0.737437\pi$
$211$	−4160.10	−1.35731	−0.678656	+	0.734456i	$0.737437\pi$
$212$	1034.88	0.335265
$213$	0	0
$214$	554.837	0.177233
$215$	2045.34	0.648797
$216$	0	0
$217$	−2708.47	−0.847295
$218$	4336.40	1.34724
$219$	0	0
$220$	1698.54	0.520524
$221$	4179.68	1.27220
$222$	0	0
$223$	−1746.41	−0.524432	−0.262216	−	0.965009i	$-0.584453\pi$
$223$	−1746.41	−0.524432	−0.262216	+	0.965009i	$0.584453\pi$
$224$	−2888.81	−0.861683
$225$	0	0
$226$	4120.97	1.21293
$227$	−129.508	−0.0378666	−0.0189333	−	0.999821i	$-0.506027\pi$
$227$	−129.508	−0.0378666	−0.0189333	+	0.999821i	$0.506027\pi$
$228$	0	0
$229$	−682.166	−0.196851	−0.0984253	−	0.995144i	$-0.531381\pi$
$229$	−682.166	−0.196851	−0.0984253	+	0.995144i	$0.531381\pi$
$230$	1297.42	0.371954
$231$	0	0
$232$	332.341	0.0940486
$233$	−1731.11	−0.486732	−0.243366	−	0.969935i	$-0.578252\pi$
$233$	−1731.11	−0.486732	−0.243366	+	0.969935i	$0.578252\pi$
$234$	0	0
$235$	629.457	0.174729
$236$	−1711.92	−0.472188
$237$	0	0
$238$	3611.77	0.983681
$239$	−3308.38	−0.895402	−0.447701	−	0.894183i	$-0.647757\pi$
$239$	−3308.38	−0.895402	−0.447701	+	0.894183i	$0.647757\pi$
$240$	0	0
$241$	3586.39	0.958588	0.479294	−	0.877654i	$-0.340893\pi$
$241$	3586.39	0.958588	0.479294	+	0.877654i	$0.340893\pi$
$242$	13183.1	3.50183
$243$	0	0
$244$	−2235.66	−0.586571
$245$	585.776	0.152750
$246$	0	0
$247$	7402.48	1.90692
$248$	2065.41	0.528844
$249$	0	0
$250$	−447.150	−0.113121
$251$	2956.22	0.743406	0.371703	−	0.928352i	$-0.378774\pi$
$251$	2956.22	0.743406	0.371703	+	0.928352i	$0.378774\pi$
$252$	0	0
$253$	5137.59	1.27667
$254$	1426.39	0.352360
$255$	0	0
$256$	5248.27	1.28132
$257$	−2258.12	−0.548084	−0.274042	−	0.961718i	$-0.588361\pi$
$257$	−2258.12	−0.548084	−0.274042	+	0.961718i	$0.588361\pi$
$258$	0	0
$259$	−719.004	−0.172497
$260$	−1491.95	−0.355873
$261$	0	0
$262$	1378.09	0.324956
$263$	−2908.67	−0.681963	−0.340981	−	0.940070i	$-0.610759\pi$
$263$	−2908.67	−0.681963	−0.340981	+	0.940070i	$0.610759\pi$
$264$	0	0
$265$	−1078.82	−0.250081
$266$	6396.66	1.47445
$267$	0	0
$268$	−2774.02	−0.632277
$269$	297.910	0.0675238	0.0337619	−	0.999430i	$-0.489251\pi$
$269$	297.910	0.0675238	0.0337619	+	0.999430i	$0.489251\pi$
$270$	0	0
$271$	6993.04	1.56752	0.783758	−	0.621067i	$-0.213300\pi$
$271$	6993.04	1.56752	0.783758	+	0.621067i	$0.213300\pi$
$272$	−5332.18	−1.18864
$273$	0	0
$274$	1258.73	0.277529
$275$	−1770.65	−0.388270
$276$	0	0
$277$	5701.84	1.23679	0.618394	−	0.785868i	$-0.287784\pi$
$277$	5701.84	1.23679	0.618394	+	0.785868i	$0.287784\pi$
$278$	4072.58	0.878622
$279$	0	0
$280$	861.115	0.183791
$281$	1216.34	0.258223	0.129112	−	0.991630i	$-0.458787\pi$
$281$	1216.34	0.258223	0.129112	+	0.991630i	$0.458787\pi$
$282$	0	0
$283$	4096.17	0.860396	0.430198	−	0.902735i	$-0.358444\pi$
$283$	4096.17	0.860396	0.430198	+	0.902735i	$0.358444\pi$
$284$	−4177.27	−0.872801
$285$	0	0
$286$	−15761.9	−3.25881
$287$	−5589.80	−1.14967
$288$	0	0
$289$	−399.197	−0.0812532
$290$	518.694	0.105030
$291$	0	0
$292$	1975.64	0.395944
$293$	−5854.29	−1.16727	−0.583637	−	0.812015i	$-0.698371\pi$
$293$	−5854.29	−1.16727	−0.583637	+	0.812015i	$0.698371\pi$
$294$	0	0
$295$	1784.60	0.352214
$296$	548.292	0.107665
$297$	0	0
$298$	−4294.74	−0.834858
$299$	−4512.73	−0.872836
$300$	0	0
$301$	−6147.54	−1.17720
$302$	−5506.56	−1.04923
$303$	0	0
$304$	−9443.62	−1.78167
$305$	2330.57	0.437535
$306$	0	0
$307$	−2748.38	−0.510940	−0.255470	−	0.966817i	$-0.582230\pi$
$307$	−2748.38	−0.510940	−0.255470	+	0.966817i	$0.582230\pi$
$308$	−5105.17	−0.944461
$309$	0	0
$310$	3223.54	0.590595
$311$	10289.0	1.87600	0.938000	−	0.346634i	$-0.112675\pi$
$311$	10289.0	1.87600	0.938000	+	0.346634i	$0.112675\pi$
$312$	0	0
$313$	8540.93	1.54237	0.771185	−	0.636611i	$-0.219664\pi$
$313$	8540.93	1.54237	0.771185	+	0.636611i	$0.219664\pi$
$314$	798.196	0.143455
$315$	0	0
$316$	5373.02	0.956506
$317$	131.153	0.0232375	0.0116187	−	0.999933i	$-0.496302\pi$
$317$	131.153	0.0232375	0.0116187	+	0.999933i	$0.496302\pi$
$318$	0	0
$319$	2053.95	0.360499
$320$	263.543	0.0460391
$321$	0	0
$322$	−3899.56	−0.674888
$323$	7994.22	1.37712
$324$	0	0
$325$	1555.29	0.265453
$326$	−11142.0	−1.89293
$327$	0	0
$328$	4262.63	0.717574
$329$	−1891.91	−0.317035
$330$	0	0
$331$	−7274.79	−1.20803	−0.604016	−	0.796972i	$-0.706434\pi$
$331$	−7274.79	−1.20803	−0.604016	+	0.796972i	$0.706434\pi$
$332$	−5176.61	−0.855734
$333$	0	0
$334$	6155.56	1.00844
$335$	2891.79	0.471628
$336$	0	0
$337$	4663.56	0.753830	0.376915	−	0.926248i	$-0.376985\pi$
$337$	4663.56	0.753830	0.376915	+	0.926248i	$0.376985\pi$
$338$	5985.72	0.963256
$339$	0	0
$340$	−1611.22	−0.257001
$341$	12764.7	2.02712
$342$	0	0
$343$	−6915.27	−1.08860
$344$	4687.94	0.734759
$345$	0	0
$346$	−8723.72	−1.35546
$347$	−6050.81	−0.936093	−0.468046	−	0.883704i	$-0.655042\pi$
$347$	−6050.81	−0.936093	−0.468046	+	0.883704i	$0.655042\pi$
$348$	0	0
$349$	12824.0	1.96691	0.983455	−	0.181152i	$-0.0579825\pi$
$349$	12824.0	1.96691	0.983455	+	0.181152i	$0.0579825\pi$
$350$	1343.97	0.205251
$351$	0	0
$352$	13614.7	2.06155
$353$	−12514.5	−1.88691	−0.943457	−	0.331494i	$-0.892447\pi$
$353$	−12514.5	−1.88691	−0.943457	+	0.331494i	$0.892447\pi$
$354$	0	0
$355$	4354.62	0.651040
$356$	−1865.75	−0.277766
$357$	0	0
$358$	8041.53	1.18717
$359$	−3394.70	−0.499068	−0.249534	−	0.968366i	$-0.580277\pi$
$359$	−3394.70	−0.499068	−0.249534	+	0.968366i	$0.580277\pi$
$360$	0	0
$361$	7299.26	1.06419
$362$	−8431.92	−1.22423
$363$	0	0
$364$	4484.25	0.645710
$365$	−2059.52	−0.295343
$366$	0	0
$367$	−9991.93	−1.42118	−0.710592	−	0.703604i	$-0.751573\pi$
$367$	−9991.93	−1.42118	−0.710592	+	0.703604i	$0.751573\pi$
$368$	5757.06	0.815509
$369$	0	0
$370$	855.735	0.120237
$371$	3242.54	0.453758
$372$	0	0
$373$	7219.40	1.00216	0.501081	−	0.865400i	$-0.332936\pi$
$373$	7219.40	1.00216	0.501081	+	0.865400i	$0.332936\pi$
$374$	−17021.9	−2.35342
$375$	0	0
$376$	1442.72	0.197879
$377$	−1804.14	−0.246467
$378$	0	0
$379$	10797.8	1.46345	0.731724	−	0.681601i	$-0.238716\pi$
$379$	10797.8	1.46345	0.731724	+	0.681601i	$0.238716\pi$
$380$	−2853.56	−0.385223
$381$	0	0
$382$	−6877.37	−0.921144
$383$	3396.75	0.453174	0.226587	−	0.973991i	$-0.427243\pi$
$383$	3396.75	0.453174	0.226587	+	0.973991i	$0.427243\pi$
$384$	0	0
$385$	5321.91	0.704493
$386$	6584.49	0.868243
$387$	0	0
$388$	−7332.80	−0.959450
$389$	4927.07	0.642192	0.321096	−	0.947047i	$-0.395949\pi$
$389$	4927.07	0.642192	0.321096	+	0.947047i	$0.395949\pi$
$390$	0	0
$391$	−4873.47	−0.630338
$392$	1342.60	0.172989
$393$	0	0
$394$	9704.21	1.24084
$395$	−5601.13	−0.713477
$396$	0	0
$397$	−7777.55	−0.983234	−0.491617	−	0.870812i	$-0.663594\pi$
$397$	−7777.55	−0.983234	−0.491617	+	0.870812i	$0.663594\pi$
$398$	803.551	0.101202
$399$	0	0
$400$	−1984.14	−0.248018
$401$	−6058.22	−0.754446	−0.377223	−	0.926122i	$-0.623121\pi$
$401$	−6058.22	−0.754446	−0.377223	+	0.926122i	$0.623121\pi$
$402$	0	0
$403$	−11212.2	−1.38591
$404$	6837.24	0.841994
$405$	0	0
$406$	−1559.00	−0.190571
$407$	3388.59	0.412693
$408$	0	0
$409$	−15078.2	−1.82291	−0.911454	−	0.411403i	$-0.865039\pi$
$409$	−15078.2	−1.82291	−0.911454	+	0.411403i	$0.865039\pi$
$410$	6652.80	0.801362
$411$	0	0
$412$	−7653.48	−0.915194
$413$	−5363.83	−0.639073
$414$	0	0
$415$	5396.39	0.638309
$416$	−11958.8	−1.40944
$417$	0	0
$418$	−30146.8	−3.52758
$419$	−5698.22	−0.664383	−0.332192	−	0.943212i	$-0.607788\pi$
$419$	−5698.22	−0.664383	−0.332192	+	0.943212i	$0.607788\pi$
$420$	0	0
$421$	846.838	0.0980341	0.0490171	−	0.998798i	$-0.484391\pi$
$421$	846.838	0.0980341	0.0490171	+	0.998798i	$0.484391\pi$
$422$	−14881.5	−1.71664
$423$	0	0
$424$	−2472.67	−0.283215
$425$	1679.62	0.191703
$426$	0	0
$427$	−7004.83	−0.793882
$428$	743.934	0.0840173
$429$	0	0
$430$	7316.61	0.820554
$431$	7518.27	0.840237	0.420119	−	0.907469i	$-0.361988\pi$
$431$	7518.27	0.840237	0.420119	+	0.907469i	$0.361988\pi$
$432$	0	0
$433$	12686.0	1.40797	0.703987	−	0.710213i	$-0.251401\pi$
$433$	12686.0	1.40797	0.703987	+	0.710213i	$0.251401\pi$
$434$	−9688.75	−1.07160
$435$	0	0
$436$	5814.31	0.638658
$437$	−8631.22	−0.944822
$438$	0	0
$439$	10530.4	1.14485	0.572424	−	0.819958i	$-0.306003\pi$
$439$	10530.4	1.14485	0.572424	+	0.819958i	$0.306003\pi$
$440$	−4058.34	−0.439713
$441$	0	0
$442$	14951.6	1.60899
$443$	5199.17	0.557608	0.278804	−	0.960348i	$-0.410062\pi$
$443$	5199.17	0.557608	0.278804	+	0.960348i	$0.410062\pi$
$444$	0	0
$445$	1944.96	0.207191
$446$	−6247.27	−0.663266
$447$	0	0
$448$	−792.112	−0.0835352
$449$	11836.7	1.24412	0.622058	−	0.782971i	$-0.286297\pi$
$449$	11836.7	1.24412	0.622058	+	0.782971i	$0.286297\pi$
$450$	0	0
$451$	26344.1	2.75055
$452$	5525.46	0.574990
$453$	0	0
$454$	−463.274	−0.0478911
$455$	−4674.63	−0.481648
$456$	0	0
$457$	404.451	0.0413992	0.0206996	−	0.999786i	$-0.493411\pi$
$457$	404.451	0.0413992	0.0206996	+	0.999786i	$0.493411\pi$
$458$	−2440.24	−0.248963
$459$	0	0
$460$	1739.60	0.176325
$461$	−7072.59	−0.714541	−0.357270	−	0.934001i	$-0.616293\pi$
$461$	−7072.59	−0.714541	−0.357270	+	0.934001i	$0.616293\pi$
$462$	0	0
$463$	−6500.22	−0.652464	−0.326232	−	0.945290i	$-0.605779\pi$
$463$	−6500.22	−0.652464	−0.326232	+	0.945290i	$0.605779\pi$
$464$	2301.61	0.230279
$465$	0	0
$466$	−6192.51	−0.615585
$467$	8850.61	0.876997	0.438498	−	0.898732i	$-0.355511\pi$
$467$	8850.61	0.876997	0.438498	+	0.898732i	$0.355511\pi$
$468$	0	0
$469$	−8691.65	−0.855742
$470$	2251.69	0.220985
$471$	0	0
$472$	4090.31	0.398881
$473$	28972.7	2.81642
$474$	0	0
$475$	2974.71	0.287346
$476$	4842.71	0.466314
$477$	0	0
$478$	−11834.7	−1.13244
$479$	8582.91	0.818712	0.409356	−	0.912375i	$-0.365753\pi$
$479$	8582.91	0.818712	0.409356	+	0.912375i	$0.365753\pi$
$480$	0	0
$481$	−2976.45	−0.282150
$482$	12829.2	1.21236
$483$	0	0
$484$	17676.1	1.66004
$485$	7644.12	0.715674
$486$	0	0
$487$	−2823.62	−0.262732	−0.131366	−	0.991334i	$-0.541936\pi$
$487$	−2823.62	−0.262732	−0.131366	+	0.991334i	$0.541936\pi$
$488$	5341.69	0.495506
$489$	0	0
$490$	2095.44	0.193188
$491$	13116.2	1.20555	0.602776	−	0.797911i	$-0.294061\pi$
$491$	13116.2	1.20555	0.602776	+	0.797911i	$0.294061\pi$
$492$	0	0
$493$	−1948.36	−0.177991
$494$	26480.2	2.41174
$495$	0	0
$496$	14303.8	1.29488
$497$	−13088.4	−1.18127
$498$	0	0
$499$	−16495.4	−1.47983	−0.739916	−	0.672699i	$-0.765135\pi$
$499$	−16495.4	−1.47983	−0.739916	+	0.672699i	$0.765135\pi$
$500$	−599.546	−0.0536250
$501$	0	0
$502$	10575.0	0.940209
$503$	16677.3	1.47834	0.739169	−	0.673520i	$-0.235219\pi$
$503$	16677.3	1.47834	0.739169	+	0.673520i	$0.235219\pi$
$504$	0	0
$505$	−7127.52	−0.628060
$506$	18378.2	1.61465
$507$	0	0
$508$	1912.52	0.167036
$509$	−9386.35	−0.817373	−0.408686	−	0.912675i	$-0.634013\pi$
$509$	−9386.35	−0.817373	−0.408686	+	0.912675i	$0.634013\pi$
$510$	0	0
$511$	6190.15	0.535883
$512$	7979.98	0.688806
$513$	0	0
$514$	−8077.75	−0.693180
$515$	7978.42	0.682662
$516$	0	0
$517$	8916.37	0.758495
$518$	−2572.02	−0.218162
$519$	0	0
$520$	3564.74	0.300624
$521$	16530.2	1.39002	0.695009	−	0.719001i	$-0.255400\pi$
$521$	16530.2	1.39002	0.695009	+	0.719001i	$0.255400\pi$
$522$	0	0
$523$	−18295.9	−1.52968	−0.764842	−	0.644217i	$-0.777183\pi$
$523$	−18295.9	−1.52968	−0.764842	+	0.644217i	$0.777183\pi$
$524$	1847.76	0.154045
$525$	0	0
$526$	−10404.9	−0.862500
$527$	−12108.5	−1.00086
$528$	0	0
$529$	−6905.20	−0.567535
$530$	−3859.16	−0.316285
$531$	0	0
$532$	8576.74	0.698964
$533$	−23140.0	−1.88050
$534$	0	0
$535$	−775.518	−0.0626703
$536$	6628.01	0.534117
$537$	0	0
$538$	1065.68	0.0853994
$539$	8297.63	0.663088
$540$	0	0
$541$	−949.695	−0.0754724	−0.0377362	−	0.999288i	$-0.512015\pi$
$541$	−949.695	−0.0754724	−0.0377362	+	0.999288i	$0.512015\pi$
$542$	25015.5	1.98249
$543$	0	0
$544$	−12914.7	−1.01786
$545$	−6061.16	−0.476388
$546$	0	0
$547$	18017.4	1.40835	0.704177	−	0.710024i	$-0.251316\pi$
$547$	18017.4	1.40835	0.704177	+	0.710024i	$0.251316\pi$
$548$	1687.73	0.131563
$549$	0	0
$550$	−6333.97	−0.491057
$551$	−3450.67	−0.266794
$552$	0	0
$553$	16834.9	1.29456
$554$	20396.6	1.56420
$555$	0	0
$556$	5460.58	0.416511
$557$	11387.6	0.866263	0.433131	−	0.901331i	$-0.357409\pi$
$557$	11387.6	0.866263	0.433131	+	0.901331i	$0.357409\pi$
$558$	0	0
$559$	−25448.9	−1.92553
$560$	5963.60	0.450014
$561$	0	0
$562$	4351.09	0.326583
$563$	277.822	0.0207971	0.0103986	−	0.999946i	$-0.496690\pi$
$563$	277.822	0.0207971	0.0103986	+	0.999946i	$0.496690\pi$
$564$	0	0
$565$	−5760.04	−0.428897
$566$	14652.8	1.08817
$567$	0	0
$568$	9980.82	0.737299
$569$	−15475.5	−1.14019	−0.570093	−	0.821580i	$-0.693093\pi$
$569$	−15475.5	−1.14019	−0.570093	+	0.821580i	$0.693093\pi$
$570$	0	0
$571$	−3245.82	−0.237887	−0.118944	−	0.992901i	$-0.537951\pi$
$571$	−3245.82	−0.237887	−0.118944	+	0.992901i	$0.537951\pi$
$572$	−21133.8	−1.54484
$573$	0	0
$574$	−19995.8	−1.45402
$575$	−1813.46	−0.131524
$576$	0	0
$577$	−23087.9	−1.66579	−0.832895	−	0.553430i	$-0.813318\pi$
$577$	−23087.9	−1.66579	−0.832895	+	0.553430i	$0.813318\pi$
$578$	−1428.01	−0.102764
$579$	0	0
$580$	695.473	0.0497896
$581$	−16219.5	−1.15818
$582$	0	0
$583$	−15281.7	−1.08560
$584$	−4720.44	−0.334474
$585$	0	0
$586$	−20942.0	−1.47629
$587$	−13526.1	−0.951075	−0.475537	−	0.879696i	$-0.657746\pi$
$587$	−13526.1	−0.951075	−0.475537	+	0.879696i	$0.657746\pi$
$588$	0	0
$589$	−21444.9	−1.50021
$590$	6383.86	0.445457
$591$	0	0
$592$	3797.17	0.263619
$593$	−15964.6	−1.10554	−0.552770	−	0.833334i	$-0.686429\pi$
$593$	−15964.6	−1.10554	−0.552770	+	0.833334i	$0.686429\pi$
$594$	0	0
$595$	−5048.31	−0.347833
$596$	−5758.46	−0.395764
$597$	0	0
$598$	−16142.9	−1.10390
$599$	−22.9584	−0.00156603	−0.000783017	−	1.00000i	$-0.500249\pi$
$599$	−22.9584	−0.00156603	−0.000783017		1.00000i	$0.500249\pi$
$600$	0	0
$601$	−7471.32	−0.507090	−0.253545	−	0.967324i	$-0.581597\pi$
$601$	−7471.32	−0.507090	−0.253545	+	0.967324i	$0.581597\pi$
$602$	−21991.0	−1.48885
$603$	0	0
$604$	−7383.28	−0.497386
$605$	−18426.6	−1.23826
$606$	0	0
$607$	23036.9	1.54042	0.770212	−	0.637788i	$-0.220151\pi$
$607$	23036.9	1.54042	0.770212	+	0.637788i	$0.220151\pi$
$608$	−22872.8	−1.52568
$609$	0	0
$610$	8336.93	0.553365
$611$	−7831.91	−0.518569
$612$	0	0
$613$	−20040.0	−1.32040	−0.660202	−	0.751088i	$-0.729530\pi$
$613$	−20040.0	−1.32040	−0.660202	+	0.751088i	$0.729530\pi$
$614$	−9831.52	−0.646202
$615$	0	0
$616$	12197.9	0.797834
$617$	27464.4	1.79202	0.896010	−	0.444035i	$-0.146453\pi$
$617$	27464.4	1.79202	0.896010	+	0.444035i	$0.146453\pi$
$618$	0	0
$619$	3585.29	0.232803	0.116402	−	0.993202i	$-0.462864\pi$
$619$	3585.29	0.232803	0.116402	+	0.993202i	$0.462864\pi$
$620$	4322.17	0.279972
$621$	0	0
$622$	36805.9	2.37264
$623$	−5845.83	−0.375936
$624$	0	0
$625$	625.000	0.0400000
$626$	30552.6	1.95069
$627$	0	0
$628$	1070.23	0.0680048
$629$	−3214.38	−0.203761
$630$	0	0
$631$	−27135.3	−1.71195	−0.855973	−	0.517020i	$-0.827041\pi$
$631$	−27135.3	−1.71195	−0.855973	+	0.517020i	$0.827041\pi$
$632$	−12837.8	−0.808009
$633$	0	0
$634$	469.160	0.0293892
$635$	−1993.72	−0.124596
$636$	0	0
$637$	−7288.43	−0.453341
$638$	7347.40	0.455935
$639$	0	0
$640$	−6746.34	−0.416676
$641$	−23443.5	−1.44456	−0.722278	−	0.691603i	$-0.756905\pi$
$641$	−23443.5	−1.44456	−0.722278	+	0.691603i	$0.756905\pi$
$642$	0	0
$643$	−9339.29	−0.572792	−0.286396	−	0.958111i	$-0.592457\pi$
$643$	−9339.29	−0.572792	−0.286396	+	0.958111i	$0.592457\pi$
$644$	−5228.59	−0.319931
$645$	0	0
$646$	28596.9	1.74169
$647$	−1103.56	−0.0670562	−0.0335281	−	0.999438i	$-0.510674\pi$
$647$	−1103.56	−0.0670562	−0.0335281	+	0.999438i	$0.510674\pi$
$648$	0	0
$649$	25279.2	1.52896
$650$	5563.60	0.335726
$651$	0	0
$652$	−14939.3	−0.897344
$653$	3127.51	0.187425	0.0937127	−	0.995599i	$-0.470126\pi$
$653$	3127.51	0.187425	0.0937127	+	0.995599i	$0.470126\pi$
$654$	0	0
$655$	−1926.21	−0.114906
$656$	29520.6	1.75699
$657$	0	0
$658$	−6767.75	−0.400964
$659$	−16004.5	−0.946050	−0.473025	−	0.881049i	$-0.656838\pi$
$659$	−16004.5	−0.946050	−0.473025	+	0.881049i	$0.656838\pi$
$660$	0	0
$661$	10807.3	0.635939	0.317970	−	0.948101i	$-0.396999\pi$
$661$	10807.3	0.635939	0.317970	+	0.948101i	$0.396999\pi$
$662$	−26023.4	−1.52784
$663$	0	0
$664$	12368.6	0.722882
$665$	−8940.87	−0.521372
$666$	0	0
$667$	2103.61	0.122117
$668$	8253.47	0.478049
$669$	0	0
$670$	10344.5	0.596483
$671$	33013.0	1.89933
$672$	0	0
$673$	23835.6	1.36522	0.682612	−	0.730781i	$-0.260844\pi$
$673$	23835.6	1.36522	0.682612	+	0.730781i	$0.260844\pi$
$674$	16682.5	0.953392
$675$	0	0
$676$	8025.75	0.456631
$677$	−5081.10	−0.288453	−0.144227	−	0.989545i	$-0.546069\pi$
$677$	−5081.10	−0.288453	−0.144227	+	0.989545i	$0.546069\pi$
$678$	0	0
$679$	−22975.4	−1.29855
$680$	3849.70	0.217102
$681$	0	0
$682$	45662.0	2.56377
$683$	−7815.19	−0.437833	−0.218917	−	0.975744i	$-0.570252\pi$
$683$	−7815.19	−0.437833	−0.218917	+	0.975744i	$0.570252\pi$
$684$	0	0
$685$	−1759.38	−0.0981352
$686$	−24737.3	−1.37679
$687$	0	0
$688$	32466.1	1.79907
$689$	13423.1	0.742204
$690$	0	0
$691$	8815.96	0.485347	0.242674	−	0.970108i	$-0.421976\pi$
$691$	8815.96	0.485347	0.242674	+	0.970108i	$0.421976\pi$
$692$	−11696.9	−0.642557
$693$	0	0
$694$	−21644.9	−1.18391
$695$	−5692.41	−0.310684
$696$	0	0
$697$	−24989.8	−1.35804
$698$	45873.9	2.48761
$699$	0	0
$700$	1802.01	0.0972994
$701$	4100.40	0.220927	0.110464	−	0.993880i	$-0.464766\pi$
$701$	4100.40	0.220927	0.110464	+	0.993880i	$0.464766\pi$
$702$	0	0
$703$	−5692.87	−0.305420
$704$	3733.14	0.199855
$705$	0	0
$706$	−44767.0	−2.38644
$707$	21422.7	1.13958
$708$	0	0
$709$	−17803.3	−0.943042	−0.471521	−	0.881855i	$-0.656295\pi$
$709$	−17803.3	−0.943042	−0.471521	+	0.881855i	$0.656295\pi$
$710$	15577.4	0.823391
$711$	0	0
$712$	4457.87	0.234643
$713$	13073.3	0.686677
$714$	0	0
$715$	22031.0	1.15233
$716$	10782.2	0.562780
$717$	0	0
$718$	−12143.5	−0.631187
$719$	4213.37	0.218543	0.109271	−	0.994012i	$-0.465148\pi$
$719$	4213.37	0.218543	0.109271	+	0.994012i	$0.465148\pi$
$720$	0	0
$721$	−23980.1	−1.23865
$722$	26110.9	1.34591
$723$	0	0
$724$	−11305.6	−0.580347
$725$	−725.000	−0.0371391
$726$	0	0
$727$	29559.3	1.50797	0.753984	−	0.656893i	$-0.228130\pi$
$727$	29559.3	1.50797	0.753984	+	0.656893i	$0.228130\pi$
$728$	−10714.3	−0.545464
$729$	0	0
$730$	−7367.32	−0.373530
$731$	−27483.2	−1.39057
$732$	0	0
$733$	23312.9	1.17474	0.587369	−	0.809319i	$-0.300164\pi$
$733$	23312.9	1.17474	0.587369	+	0.809319i	$0.300164\pi$
$734$	−35743.1	−1.79742
$735$	0	0
$736$	13943.8	0.698337
$737$	40962.8	2.04733
$738$	0	0
$739$	27985.2	1.39303	0.696517	−	0.717540i	$-0.254732\pi$
$739$	27985.2	1.39303	0.696517	+	0.717540i	$0.254732\pi$
$740$	1147.38	0.0569982
$741$	0	0
$742$	11599.2	0.573881
$743$	2542.48	0.125538	0.0627690	−	0.998028i	$-0.480007\pi$
$743$	2542.48	0.125538	0.0627690	+	0.998028i	$0.480007\pi$
$744$	0	0
$745$	6002.94	0.295209
$746$	25825.3	1.26747
$747$	0	0
$748$	−22823.2	−1.11564
$749$	2330.92	0.113712
$750$	0	0
$751$	33600.2	1.63261	0.816305	−	0.577622i	$-0.196019\pi$
$751$	33600.2	1.63261	0.816305	+	0.577622i	$0.196019\pi$
$752$	9991.46	0.484510
$753$	0	0
$754$	−6453.77	−0.311714
$755$	7696.74	0.371011
$756$	0	0
$757$	−34022.4	−1.63351	−0.816754	−	0.576986i	$-0.804229\pi$
$757$	−34022.4	−1.63351	−0.816754	+	0.576986i	$0.804229\pi$
$758$	38626.0	1.85087
$759$	0	0
$760$	6818.06	0.325417
$761$	−15032.8	−0.716084	−0.358042	−	0.933705i	$-0.616556\pi$
$761$	−15032.8	−0.716084	−0.358042	+	0.933705i	$0.616556\pi$
$762$	0	0
$763$	18217.6	0.864378
$764$	−9221.28	−0.436668
$765$	0	0
$766$	12150.8	0.573144
$767$	−22204.6	−1.04532
$768$	0	0
$769$	−24042.9	−1.12745	−0.563725	−	0.825963i	$-0.690632\pi$
$769$	−24042.9	−1.12745	−0.563725	+	0.825963i	$0.690632\pi$
$770$	19037.5	0.890994
$771$	0	0
$772$	8828.59	0.411590
$773$	−17441.1	−0.811530	−0.405765	−	0.913977i	$-0.632995\pi$
$773$	−17441.1	−0.811530	−0.405765	+	0.913977i	$0.632995\pi$
$774$	0	0
$775$	−4505.67	−0.208837
$776$	17520.4	0.810496
$777$	0	0
$778$	17625.1	0.812200
$779$	−44258.5	−2.03559
$780$	0	0
$781$	61684.0	2.82616
$782$	−17433.4	−0.797208
$783$	0	0
$784$	9298.12	0.423566
$785$	−1115.67	−0.0507262
$786$	0	0
$787$	−38211.9	−1.73076	−0.865379	−	0.501119i	$-0.832922\pi$
$787$	−38211.9	−1.73076	−0.865379	+	0.501119i	$0.832922\pi$
$788$	13011.6	0.588220
$789$	0	0
$790$	−20036.4	−0.902357
$791$	17312.5	0.778208
$792$	0	0
$793$	−28997.8	−1.29854
$794$	−27821.8	−1.24353
$795$	0	0
$796$	1077.41	0.0479748
$797$	−18149.9	−0.806655	−0.403327	−	0.915056i	$-0.632146\pi$
$797$	−18149.9	−0.806655	−0.403327	+	0.915056i	$0.632146\pi$
$798$	0	0
$799$	−8457.99	−0.374496
$800$	−4805.68	−0.212383
$801$	0	0
$802$	−21671.5	−0.954172
$803$	−29173.5	−1.28208
$804$	0	0
$805$	5450.57	0.238643
$806$	−40108.3	−1.75280
$807$	0	0
$808$	−16336.3	−0.711275
$809$	339.522	0.0147552	0.00737759	−	0.999973i	$-0.497652\pi$
$809$	339.522	0.0147552	0.00737759	+	0.999973i	$0.497652\pi$
$810$	0	0
$811$	26295.4	1.13854	0.569270	−	0.822151i	$-0.307226\pi$
$811$	26295.4	1.13854	0.569270	+	0.822151i	$0.307226\pi$
$812$	−2090.33	−0.0903403
$813$	0	0
$814$	12121.7	0.521946
$815$	15573.6	0.669347
$816$	0	0
$817$	−48674.5	−2.08434
$818$	−53937.8	−2.30549
$819$	0	0
$820$	8920.18	0.379886
$821$	46289.6	1.96775	0.983873	−	0.178867i	$-0.0572433\pi$
$821$	46289.6	1.96775	0.983873	+	0.178867i	$0.0572433\pi$
$822$	0	0
$823$	−28099.3	−1.19013	−0.595067	−	0.803676i	$-0.702874\pi$
$823$	−28099.3	−1.19013	−0.595067	+	0.803676i	$0.702874\pi$
$824$	18286.6	0.773111
$825$	0	0
$826$	−19187.5	−0.808255
$827$	38732.3	1.62860	0.814301	−	0.580443i	$-0.197121\pi$
$827$	38732.3	1.62860	0.814301	+	0.580443i	$0.197121\pi$
$828$	0	0
$829$	−9334.33	−0.391067	−0.195534	−	0.980697i	$-0.562644\pi$
$829$	−9334.33	−0.391067	−0.195534	+	0.980697i	$0.562644\pi$
$830$	19304.0	0.807290
$831$	0	0
$832$	−3279.09	−0.136637
$833$	−7871.06	−0.327390
$834$	0	0
$835$	−8603.88	−0.356586
$836$	−40421.3	−1.67225
$837$	0	0
$838$	−20383.7	−0.840266
$839$	40742.1	1.67649	0.838245	−	0.545294i	$-0.183582\pi$
$839$	40742.1	1.67649	0.838245	+	0.545294i	$0.183582\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	3029.31	0.123987
$843$	0	0
$844$	−19953.4	−0.813771
$845$	−8366.49	−0.340611
$846$	0	0
$847$	55383.5	2.24675
$848$	−17124.3	−0.693457
$849$	0	0
$850$	6008.34	0.242452
$851$	3470.51	0.139797
$852$	0	0
$853$	−29088.9	−1.16762	−0.583812	−	0.811889i	$-0.698440\pi$
$853$	−29088.9	−1.16762	−0.583812	+	0.811889i	$0.698440\pi$
$854$	−25057.7	−1.00405
$855$	0	0
$856$	−1777.49	−0.0709737
$857$	−31564.0	−1.25812	−0.629059	−	0.777357i	$-0.716560\pi$
$857$	−31564.0	−1.25812	−0.629059	+	0.777357i	$0.716560\pi$
$858$	0	0
$859$	21654.5	0.860120	0.430060	−	0.902800i	$-0.358492\pi$
$859$	21654.5	0.860120	0.430060	+	0.902800i	$0.358492\pi$
$860$	9810.22	0.388983
$861$	0	0
$862$	26894.4	1.06267
$863$	−2304.51	−0.0908996	−0.0454498	−	0.998967i	$-0.514472\pi$
$863$	−2304.51	−0.0908996	−0.0454498	+	0.998967i	$0.514472\pi$
$864$	0	0
$865$	12193.5	0.479297
$866$	45380.5	1.78071
$867$	0	0
$868$	−12990.8	−0.507992
$869$	−79341.1	−3.09720
$870$	0	0
$871$	−35980.7	−1.39972
$872$	−13892.2	−0.539507
$873$	0	0
$874$	−30875.6	−1.19495
$875$	−1878.52	−0.0725777
$876$	0	0
$877$	42141.0	1.62258	0.811289	−	0.584645i	$-0.198766\pi$
$877$	42141.0	1.62258	0.811289	+	0.584645i	$0.198766\pi$
$878$	37669.4	1.44793
$879$	0	0
$880$	−28105.8	−1.07664
$881$	−3482.39	−0.133172	−0.0665861	−	0.997781i	$-0.521211\pi$
$881$	−3482.39	−0.133172	−0.0665861	+	0.997781i	$0.521211\pi$
$882$	0	0
$883$	28559.6	1.08846	0.544229	−	0.838937i	$-0.316822\pi$
$883$	28559.6	1.08846	0.544229	+	0.838937i	$0.316822\pi$
$884$	20047.3	0.762742
$885$	0	0
$886$	18598.5	0.705224
$887$	42485.5	1.60826	0.804128	−	0.594456i	$-0.202633\pi$
$887$	42485.5	1.60826	0.804128	+	0.594456i	$0.202633\pi$
$888$	0	0
$889$	5992.37	0.226072
$890$	6957.52	0.262041
$891$	0	0
$892$	−8376.43	−0.314421
$893$	−14979.6	−0.561337
$894$	0	0
$895$	−11240.0	−0.419789
$896$	20277.0	0.756034
$897$	0	0
$898$	42342.3	1.57347
$899$	5226.58	0.193900
$900$	0	0
$901$	14496.1	0.535999
$902$	94238.2	3.47870
$903$	0	0
$904$	−13202.1	−0.485724
$905$	11785.6	0.432893
$906$	0	0
$907$	−22359.4	−0.818559	−0.409280	−	0.912409i	$-0.634220\pi$
$907$	−22359.4	−0.818559	−0.409280	+	0.912409i	$0.634220\pi$
$908$	−621.166	−0.0227028
$909$	0	0
$910$	−16722.1	−0.609156
$911$	14833.6	0.539471	0.269735	−	0.962934i	$-0.413064\pi$
$911$	14833.6	0.539471	0.269735	+	0.962934i	$0.413064\pi$
$912$	0	0
$913$	76440.9	2.77089
$914$	1446.80	0.0523588
$915$	0	0
$916$	−3271.92	−0.118021
$917$	5789.46	0.208489
$918$	0	0
$919$	−41161.8	−1.47748	−0.738740	−	0.673991i	$-0.764579\pi$
$919$	−41161.8	−1.47748	−0.738740	+	0.673991i	$0.764579\pi$
$920$	−4156.46	−0.148950
$921$	0	0
$922$	−25300.1	−0.903702
$923$	−54181.7	−1.93219
$924$	0	0
$925$	−1196.10	−0.0425161
$926$	−23252.6	−0.825192
$927$	0	0
$928$	5574.59	0.197193
$929$	−29321.8	−1.03554	−0.517771	−	0.855519i	$-0.673238\pi$
$929$	−29321.8	−1.03554	−0.517771	+	0.855519i	$0.673238\pi$
$930$	0	0
$931$	−13940.1	−0.490729
$932$	−8303.02	−0.291818
$933$	0	0
$934$	31660.4	1.10917
$935$	23792.2	0.832178
$936$	0	0
$937$	−42922.5	−1.49650	−0.748249	−	0.663418i	$-0.769105\pi$
$937$	−42922.5	−1.49650	−0.748249	+	0.663418i	$0.769105\pi$
$938$	−31091.8	−1.08228
$939$	0	0
$940$	3019.10	0.104758
$941$	13239.3	0.458650	0.229325	−	0.973350i	$-0.426348\pi$
$941$	13239.3	0.458650	0.229325	+	0.973350i	$0.426348\pi$
$942$	0	0
$943$	26981.0	0.931732
$944$	28327.2	0.976665
$945$	0	0
$946$	103641.	3.56201
$947$	19266.8	0.661126	0.330563	−	0.943784i	$-0.392761\pi$
$947$	19266.8	0.661126	0.330563	+	0.943784i	$0.392761\pi$
$948$	0	0
$949$	25625.3	0.876534
$950$	10641.1	0.363415
$951$	0	0
$952$	−11570.8	−0.393919
$953$	14816.2	0.503615	0.251807	−	0.967777i	$-0.418975\pi$
$953$	14816.2	0.503615	0.251807	+	0.967777i	$0.418975\pi$
$954$	0	0
$955$	9612.78	0.325720
$956$	−15868.2	−0.536835
$957$	0	0
$958$	30702.8	1.03545
$959$	5288.05	0.178061
$960$	0	0
$961$	2690.68	0.0903185
$962$	−10647.4	−0.356845
$963$	0	0
$964$	17201.6	0.574717
$965$	−9203.41	−0.307014
$966$	0	0
$967$	1342.65	0.0446502	0.0223251	−	0.999751i	$-0.492893\pi$
$967$	1342.65	0.0446502	0.0223251	+	0.999751i	$0.492893\pi$
$968$	−42233.9	−1.40232
$969$	0	0
$970$	27344.6	0.905135
$971$	11211.1	0.370526	0.185263	−	0.982689i	$-0.440686\pi$
$971$	11211.1	0.370526	0.185263	+	0.982689i	$0.440686\pi$
$972$	0	0
$973$	17109.3	0.563718
$974$	−10100.7	−0.332285
$975$	0	0
$976$	36993.6	1.21325
$977$	−30870.2	−1.01088	−0.505438	−	0.862863i	$-0.668669\pi$
$977$	−30870.2	−1.01088	−0.505438	+	0.862863i	$0.668669\pi$
$978$	0	0
$979$	27550.8	0.899414
$980$	2809.60	0.0915809
$981$	0	0
$982$	46919.3	1.52470
$983$	−28549.4	−0.926332	−0.463166	−	0.886272i	$-0.653287\pi$
$983$	−28549.4	−0.926332	−0.463166	+	0.886272i	$0.653287\pi$
$984$	0	0
$985$	−13564.0	−0.438765
$986$	−6969.68	−0.225111
$987$	0	0
$988$	35505.0	1.14328
$989$	29673.1	0.954046
$990$	0	0
$991$	−20806.0	−0.666927	−0.333464	−	0.942763i	$-0.608217\pi$
$991$	−20806.0	−0.666927	−0.333464	+	0.942763i	$0.608217\pi$
$992$	34644.5	1.10883
$993$	0	0
$994$	−46819.7	−1.49399
$995$	−1123.16	−0.0357854
$996$	0	0
$997$	57999.9	1.84240	0.921201	−	0.389088i	$-0.127210\pi$
$997$	57999.9	1.84240	0.921201	+	0.389088i	$0.127210\pi$
$998$	−59007.4	−1.87159
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.n.1.6		7
3.2	odd	2		435.4.a.i.1.2	✓	7
15.14	odd	2		2175.4.a.n.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.i.1.2	✓	7	3.2	odd	2
1305.4.a.n.1.6		7	1.1	even	1	trivial
2175.4.a.n.1.6		7	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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q+3.57720 q^{2} +4.79637 q^{4} -5.00000 q^{5} +15.0281 q^{7} -11.4600 q^{8} -17.8860 q^{10} -70.8260 q^{11} +62.2117 q^{13} +53.7587 q^{14} -79.3658 q^{16} +67.1848 q^{17} +118.988 q^{19} -23.9818 q^{20} -253.359 q^{22} -72.5383 q^{23} +25.0000 q^{25} +222.544 q^{26} +72.0804 q^{28} -29.0000 q^{29} -180.227 q^{31} -192.227 q^{32} +240.334 q^{34} -75.1407 q^{35} -47.8439 q^{37} +425.646 q^{38} +57.3002 q^{40} -371.956 q^{41} -409.069 q^{43} -339.707 q^{44} -259.484 q^{46} -125.891 q^{47} -117.155 q^{49} +89.4300 q^{50} +298.390 q^{52} +215.764 q^{53} +354.130 q^{55} -172.223 q^{56} -103.739 q^{58} -356.919 q^{59} -466.115 q^{61} -644.707 q^{62} -52.7086 q^{64} -311.059 q^{65} -578.359 q^{67} +322.243 q^{68} -268.793 q^{70} -870.924 q^{71} +411.904 q^{73} -171.147 q^{74} +570.712 q^{76} -1064.38 q^{77} +1120.23 q^{79} +396.829 q^{80} -1330.56 q^{82} -1079.28 q^{83} -335.924 q^{85} -1463.32 q^{86} +811.668 q^{88} -388.993 q^{89} +934.926 q^{91} -347.920 q^{92} -450.339 q^{94} -594.942 q^{95} -1528.82 q^{97} -419.088 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 22 q^{4} - 35 q^{5} - 50 q^{7} + 33 q^{8} - 10 q^{10} - 76 q^{11} + 30 q^{13} - 89 q^{14} + 138 q^{16} + 140 q^{17} + 90 q^{19} - 110 q^{20} + 61 q^{22} - 34 q^{23} + 175 q^{25} + 241 q^{26}+ \cdots + 761 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 22 * q^4 - 35 * q^5 - 50 * q^7 + 33 * q^8 - 10 * q^10 - 76 * q^11 + 30 * q^13 - 89 * q^14 + 138 * q^16 + 140 * q^17 + 90 * q^19 - 110 * q^20 + 61 * q^22 - 34 * q^23 + 175 * q^25 + 241 * q^26 - 57 * q^28 - 203 * q^29 + 524 * q^31 + 6 * q^32 + 255 * q^34 + 250 * q^35 - 28 * q^37 - 222 * q^38 - 165 * q^40 - 1532 * q^41 - 464 * q^43 - 1475 * q^44 + 72 * q^46 + 360 * q^47 + 569 * q^49 + 50 * q^50 - 205 * q^52 - 282 * q^53 + 380 * q^55 - 1102 * q^56 - 58 * q^58 - 766 * q^59 + 1200 * q^61 - 2856 * q^62 + 701 * q^64 - 150 * q^65 + 1546 * q^67 - 1801 * q^68 + 445 * q^70 - 1802 * q^71 - 220 * q^73 - 1594 * q^74 + 1960 * q^76 - 3222 * q^77 + 1298 * q^79 - 690 * q^80 + 856 * q^82 - 1652 * q^83 - 700 * q^85 - 7628 * q^86 + 550 * q^88 - 2846 * q^89 - 816 * q^91 - 472 * q^92 + 745 * q^94 - 450 * q^95 + 1110 * q^97 + 761 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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