LMFDB - Embedded newform 1088.4.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,4,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, 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("1088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1088.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:	$64.1940780862$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1556.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 11 $ x^3 - x^2 - 9*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.27082$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.2284 q^{3} +3.08327 q^{5} -7.31168 q^{7} +77.6202 q^{9} +O(q^{10})$	$q-10.2284 q^{3} +3.08327 q^{5} -7.31168 q^{7} +77.6202 q^{9} -23.9381 q^{11} -35.5401 q^{13} -31.5370 q^{15} +17.0000 q^{17} +34.2840 q^{19} +74.7868 q^{21} -149.799 q^{23} -115.493 q^{25} -517.764 q^{27} +120.238 q^{29} +247.299 q^{31} +244.849 q^{33} -22.5439 q^{35} +448.644 q^{37} +363.518 q^{39} +303.980 q^{41} +194.024 q^{43} +239.324 q^{45} +21.0754 q^{47} -289.539 q^{49} -173.883 q^{51} +362.638 q^{53} -73.8079 q^{55} -350.671 q^{57} +364.334 q^{59} -478.279 q^{61} -567.534 q^{63} -109.580 q^{65} +5.17348 q^{67} +1532.20 q^{69} +335.956 q^{71} -1083.34 q^{73} +1181.31 q^{75} +175.028 q^{77} -561.669 q^{79} +3200.16 q^{81} -746.323 q^{83} +52.4157 q^{85} -1229.84 q^{87} +1113.50 q^{89} +259.858 q^{91} -2529.48 q^{93} +105.707 q^{95} +247.849 q^{97} -1858.08 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{3} - 2 q^{5} + 12 q^{7} + 63 q^{9}+O(q^{10}) $ 3 * q - 8 * q^3 - 2 * q^5 + 12 * q^7 + 63 * q^9	$ 3 q - 8 q^{3} - 2 q^{5} + 12 q^{7} + 63 q^{9} - 72 q^{11} - 50 q^{13} + 64 q^{15} + 51 q^{17} - 124 q^{19} + 32 q^{21} + 60 q^{23} - 75 q^{25} - 512 q^{27} + 54 q^{29} + 300 q^{31} - 48 q^{33} - 248 q^{35} + 542 q^{37} + 320 q^{39} + 30 q^{41} - 52 q^{43} + 502 q^{45} + 16 q^{47} - 677 q^{49} - 136 q^{51} + 518 q^{53} - 608 q^{55} - 896 q^{57} - 132 q^{59} + 614 q^{61} - 852 q^{63} - 148 q^{65} + 332 q^{67} + 1280 q^{69} - 268 q^{71} - 578 q^{73} + 712 q^{75} + 128 q^{77} - 668 q^{79} + 2427 q^{81} - 876 q^{83} - 34 q^{85} - 2400 q^{87} + 1790 q^{89} + 168 q^{91} - 2592 q^{93} - 504 q^{95} + 838 q^{97} - 2040 q^{99}+O(q^{100}) $ 3 * q - 8 * q^3 - 2 * q^5 + 12 * q^7 + 63 * q^9 - 72 * q^11 - 50 * q^13 + 64 * q^15 + 51 * q^17 - 124 * q^19 + 32 * q^21 + 60 * q^23 - 75 * q^25 - 512 * q^27 + 54 * q^29 + 300 * q^31 - 48 * q^33 - 248 * q^35 + 542 * q^37 + 320 * q^39 + 30 * q^41 - 52 * q^43 + 502 * q^45 + 16 * q^47 - 677 * q^49 - 136 * q^51 + 518 * q^53 - 608 * q^55 - 896 * q^57 - 132 * q^59 + 614 * q^61 - 852 * q^63 - 148 * q^65 + 332 * q^67 + 1280 * q^69 - 268 * q^71 - 578 * q^73 + 712 * q^75 + 128 * q^77 - 668 * q^79 + 2427 * q^81 - 876 * q^83 - 34 * q^85 - 2400 * q^87 + 1790 * q^89 + 168 * q^91 - 2592 * q^93 - 504 * q^95 + 838 * q^97 - 2040 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−10.2284	−1.96846	−0.984229	−	0.176901i	$-0.943393\pi$
$3$	−10.2284	−1.96846	−0.984229	+	0.176901i	$0.943393\pi$
$4$	0	0
$5$	3.08327	0.275776	0.137888	−	0.990448i	$-0.455969\pi$
$5$	3.08327	0.275776	0.137888	+	0.990448i	$0.455969\pi$
$6$	0	0
$7$	−7.31168	−0.394793	−0.197397	−	0.980324i	$-0.563249\pi$
$7$	−7.31168	−0.394793	−0.197397	+	0.980324i	$0.563249\pi$
$8$	0	0
$9$	77.6202	2.87482
$10$	0	0
$11$	−23.9381	−0.656147	−0.328074	−	0.944652i	$-0.606399\pi$
$11$	−23.9381	−0.656147	−0.328074	+	0.944652i	$0.606399\pi$
$12$	0	0
$13$	−35.5401	−0.758234	−0.379117	−	0.925349i	$-0.623772\pi$
$13$	−35.5401	−0.758234	−0.379117	+	0.925349i	$0.623772\pi$
$14$	0	0
$15$	−31.5370	−0.542854
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	34.2840	0.413963	0.206982	−	0.978345i	$-0.433636\pi$
$19$	34.2840	0.413963	0.206982	+	0.978345i	$0.433636\pi$
$20$	0	0
$21$	74.7868	0.777134
$22$	0	0
$23$	−149.799	−1.35805	−0.679027	−	0.734114i	$-0.737598\pi$
$23$	−149.799	−1.35805	−0.679027	+	0.734114i	$0.737598\pi$
$24$	0	0
$25$	−115.493	−0.923947
$26$	0	0
$27$	−517.764	−3.69051
$28$	0	0
$29$	120.238	0.769919	0.384960	−	0.922933i	$-0.374215\pi$
$29$	120.238	0.769919	0.384960	+	0.922933i	$0.374215\pi$
$30$	0	0
$31$	247.299	1.43278	0.716391	−	0.697699i	$-0.245793\pi$
$31$	247.299	1.43278	0.716391	+	0.697699i	$0.245793\pi$
$32$	0	0
$33$	244.849	1.29160
$34$	0	0
$35$	−22.5439	−0.108875
$36$	0	0
$37$	448.644	1.99342	0.996712	−	0.0810259i	$-0.0258197\pi$
$37$	448.644	1.99342	0.996712	+	0.0810259i	$0.0258197\pi$
$38$	0	0
$39$	363.518	1.49255
$40$	0	0
$41$	303.980	1.15789	0.578947	−	0.815365i	$-0.303464\pi$
$41$	303.980	1.15789	0.578947	+	0.815365i	$0.303464\pi$
$42$	0	0
$43$	194.024	0.688103	0.344051	−	0.938951i	$-0.388201\pi$
$43$	194.024	0.688103	0.344051	+	0.938951i	$0.388201\pi$
$44$	0	0
$45$	239.324	0.792809
$46$	0	0
$47$	21.0754	0.0654076	0.0327038	−	0.999465i	$-0.489588\pi$
$47$	21.0754	0.0654076	0.0327038	+	0.999465i	$0.489588\pi$
$48$	0	0
$49$	−289.539	−0.844138
$50$	0	0
$51$	−173.883	−0.477421
$52$	0	0
$53$	362.638	0.939852	0.469926	−	0.882706i	$-0.344281\pi$
$53$	362.638	0.939852	0.469926	+	0.882706i	$0.344281\pi$
$54$	0	0
$55$	−73.8079	−0.180950
$56$	0	0
$57$	−350.671	−0.814869
$58$	0	0
$59$	364.334	0.803936	0.401968	−	0.915654i	$-0.368326\pi$
$59$	364.334	0.803936	0.401968	+	0.915654i	$0.368326\pi$
$60$	0	0
$61$	−478.279	−1.00389	−0.501945	−	0.864899i	$-0.667382\pi$
$61$	−478.279	−1.00389	−0.501945	+	0.864899i	$0.667382\pi$
$62$	0	0
$63$	−567.534	−1.13496
$64$	0	0
$65$	−109.580	−0.209103
$66$	0	0
$67$	5.17348	0.00943345	0.00471673	−	0.999989i	$-0.498499\pi$
$67$	5.17348	0.00943345	0.00471673	+	0.999989i	$0.498499\pi$
$68$	0	0
$69$	1532.20	2.67327
$70$	0	0
$71$	335.956	0.561559	0.280779	−	0.959772i	$-0.409407\pi$
$71$	335.956	0.561559	0.280779	+	0.959772i	$0.409407\pi$
$72$	0	0
$73$	−1083.34	−1.73693	−0.868464	−	0.495752i	$-0.834892\pi$
$73$	−1083.34	−1.73693	−0.868464	+	0.495752i	$0.834892\pi$
$74$	0	0
$75$	1181.31	1.81875
$76$	0	0
$77$	175.028	0.259043
$78$	0	0
$79$	−561.669	−0.799908	−0.399954	−	0.916535i	$-0.630974\pi$
$79$	−561.669	−0.799908	−0.399954	+	0.916535i	$0.630974\pi$
$80$	0	0
$81$	3200.16	4.38979
$82$	0	0
$83$	−746.323	−0.986983	−0.493491	−	0.869751i	$-0.664280\pi$
$83$	−746.323	−0.986983	−0.493491	+	0.869751i	$0.664280\pi$
$84$	0	0
$85$	52.4157	0.0668856
$86$	0	0
$87$	−1229.84	−1.51555
$88$	0	0
$89$	1113.50	1.32619	0.663095	−	0.748535i	$-0.269243\pi$
$89$	1113.50	1.32619	0.663095	+	0.748535i	$0.269243\pi$
$90$	0	0
$91$	259.858	0.299346
$92$	0	0
$93$	−2529.48	−2.82037
$94$	0	0
$95$	105.707	0.114161
$96$	0	0
$97$	247.849	0.259436	0.129718	−	0.991551i	$-0.458593\pi$
$97$	247.849	0.259436	0.129718	+	0.991551i	$0.458593\pi$
$98$	0	0
$99$	−1858.08	−1.88631
$100$	0	0
$101$	182.948	0.180237	0.0901187	−	0.995931i	$-0.471275\pi$
$101$	182.948	0.180237	0.0901187	+	0.995931i	$0.471275\pi$
$102$	0	0
$103$	1392.85	1.33245	0.666223	−	0.745753i	$-0.267910\pi$
$103$	1392.85	1.33245	0.666223	+	0.745753i	$0.267910\pi$
$104$	0	0
$105$	230.588	0.214315
$106$	0	0
$107$	−1244.81	−1.12468	−0.562340	−	0.826906i	$-0.690099\pi$
$107$	−1244.81	−1.12468	−0.562340	+	0.826906i	$0.690099\pi$
$108$	0	0
$109$	1504.10	1.32171	0.660855	−	0.750513i	$-0.270194\pi$
$109$	1504.10	1.32171	0.660855	+	0.750513i	$0.270194\pi$
$110$	0	0
$111$	−4588.92	−3.92397
$112$	0	0
$113$	−1980.20	−1.64851	−0.824254	−	0.566220i	$-0.808405\pi$
$113$	−1980.20	−1.64851	−0.824254	+	0.566220i	$0.808405\pi$
$114$	0	0
$115$	−461.871	−0.374519
$116$	0	0
$117$	−2758.63	−2.17979
$118$	0	0
$119$	−124.299	−0.0957515
$120$	0	0
$121$	−757.965	−0.569471
$122$	0	0
$123$	−3109.23	−2.27927
$124$	0	0
$125$	−741.507	−0.530579
$126$	0	0
$127$	1052.54	0.735419	0.367710	−	0.929941i	$-0.380142\pi$
$127$	1052.54	0.735419	0.367710	+	0.929941i	$0.380142\pi$
$128$	0	0
$129$	−1984.56	−1.35450
$130$	0	0
$131$	−1083.37	−0.722556	−0.361278	−	0.932458i	$-0.617659\pi$
$131$	−1083.37	−0.722556	−0.361278	+	0.932458i	$0.617659\pi$
$132$	0	0
$133$	−250.674	−0.163430
$134$	0	0
$135$	−1596.41	−1.01776
$136$	0	0
$137$	−261.505	−0.163079	−0.0815397	−	0.996670i	$-0.525984\pi$
$137$	−261.505	−0.163079	−0.0815397	+	0.996670i	$0.525984\pi$
$138$	0	0
$139$	1112.50	0.678855	0.339428	−	0.940632i	$-0.389767\pi$
$139$	1112.50	0.678855	0.339428	+	0.940632i	$0.389767\pi$
$140$	0	0
$141$	−215.567	−0.128752
$142$	0	0
$143$	850.764	0.497513
$144$	0	0
$145$	370.727	0.212326
$146$	0	0
$147$	2961.53	1.66165
$148$	0	0
$149$	−2401.62	−1.32046	−0.660230	−	0.751063i	$-0.729541\pi$
$149$	−2401.62	−1.32046	−0.660230	+	0.751063i	$0.729541\pi$
$150$	0	0
$151$	509.585	0.274632	0.137316	−	0.990527i	$-0.456152\pi$
$151$	509.585	0.274632	0.137316	+	0.990527i	$0.456152\pi$
$152$	0	0
$153$	1319.54	0.697247
$154$	0	0
$155$	762.491	0.395128
$156$	0	0
$157$	2180.35	1.10835	0.554174	−	0.832401i	$-0.313034\pi$
$157$	2180.35	1.10835	0.554174	+	0.832401i	$0.313034\pi$
$158$	0	0
$159$	−3709.21	−1.85006
$160$	0	0
$161$	1095.28	0.536151
$162$	0	0
$163$	−1936.03	−0.930315	−0.465157	−	0.885228i	$-0.654002\pi$
$163$	−1936.03	−0.930315	−0.465157	+	0.885228i	$0.654002\pi$
$164$	0	0
$165$	754.937	0.356192
$166$	0	0
$167$	2696.48	1.24946	0.624730	−	0.780841i	$-0.285209\pi$
$167$	2696.48	1.24946	0.624730	+	0.780841i	$0.285209\pi$
$168$	0	0
$169$	−933.903	−0.425081
$170$	0	0
$171$	2661.14	1.19007
$172$	0	0
$173$	−3628.25	−1.59451	−0.797257	−	0.603640i	$-0.793716\pi$
$173$	−3628.25	−1.59451	−0.797257	+	0.603640i	$0.793716\pi$
$174$	0	0
$175$	844.451	0.364768
$176$	0	0
$177$	−3726.55	−1.58251
$178$	0	0
$179$	−1171.36	−0.489113	−0.244557	−	0.969635i	$-0.578642\pi$
$179$	−1171.36	−0.489113	−0.244557	+	0.969635i	$0.578642\pi$
$180$	0	0
$181$	−3244.93	−1.33256	−0.666280	−	0.745701i	$-0.732115\pi$
$181$	−3244.93	−1.33256	−0.666280	+	0.745701i	$0.732115\pi$
$182$	0	0
$183$	4892.03	1.97612
$184$	0	0
$185$	1383.29	0.549739
$186$	0	0
$187$	−406.948	−0.159139
$188$	0	0
$189$	3785.73	1.45699
$190$	0	0
$191$	−2008.08	−0.760732	−0.380366	−	0.924836i	$-0.624202\pi$
$191$	−2008.08	−0.760732	−0.380366	+	0.924836i	$0.624202\pi$
$192$	0	0
$193$	−2599.92	−0.969672	−0.484836	−	0.874605i	$-0.661121\pi$
$193$	−2599.92	−0.969672	−0.484836	+	0.874605i	$0.661121\pi$
$194$	0	0
$195$	1120.83	0.411611
$196$	0	0
$197$	−1674.07	−0.605446	−0.302723	−	0.953079i	$-0.597896\pi$
$197$	−1674.07	−0.605446	−0.302723	+	0.953079i	$0.597896\pi$
$198$	0	0
$199$	−3378.35	−1.20344	−0.601721	−	0.798706i	$-0.705518\pi$
$199$	−3378.35	−1.20344	−0.601721	+	0.798706i	$0.705518\pi$
$200$	0	0
$201$	−52.9165	−0.0185694
$202$	0	0
$203$	−879.142	−0.303959
$204$	0	0
$205$	937.253	0.319320
$206$	0	0
$207$	−11627.4	−3.90416
$208$	0	0
$209$	−820.696	−0.271621
$210$	0	0
$211$	−4113.98	−1.34226	−0.671132	−	0.741338i	$-0.734192\pi$
$211$	−4113.98	−1.34226	−0.671132	+	0.741338i	$0.734192\pi$
$212$	0	0
$213$	−3436.30	−1.10540
$214$	0	0
$215$	598.230	0.189762
$216$	0	0
$217$	−1808.17	−0.565653
$218$	0	0
$219$	11080.9	3.41907
$220$	0	0
$221$	−604.181	−0.183899
$222$	0	0
$223$	−379.552	−0.113976	−0.0569881	−	0.998375i	$-0.518150\pi$
$223$	−379.552	−0.113976	−0.0569881	+	0.998375i	$0.518150\pi$
$224$	0	0
$225$	−8964.63	−2.65619
$226$	0	0
$227$	−5532.04	−1.61751	−0.808754	−	0.588148i	$-0.799857\pi$
$227$	−5532.04	−1.61751	−0.808754	+	0.588148i	$0.799857\pi$
$228$	0	0
$229$	1903.85	0.549389	0.274695	−	0.961532i	$-0.411423\pi$
$229$	1903.85	0.549389	0.274695	+	0.961532i	$0.411423\pi$
$230$	0	0
$231$	−1790.26	−0.509915
$232$	0	0
$233$	872.319	0.245268	0.122634	−	0.992452i	$-0.460866\pi$
$233$	872.319	0.245268	0.122634	+	0.992452i	$0.460866\pi$
$234$	0	0
$235$	64.9811	0.0180379
$236$	0	0
$237$	5744.98	1.57458
$238$	0	0
$239$	5714.72	1.54667	0.773335	−	0.633997i	$-0.218587\pi$
$239$	5714.72	1.54667	0.773335	+	0.633997i	$0.218587\pi$
$240$	0	0
$241$	588.062	0.157180	0.0785900	−	0.996907i	$-0.474958\pi$
$241$	588.062	0.157180	0.0785900	+	0.996907i	$0.474958\pi$
$242$	0	0
$243$	−18752.8	−4.95060
$244$	0	0
$245$	−892.729	−0.232793
$246$	0	0
$247$	−1218.46	−0.313881
$248$	0	0
$249$	7633.69	1.94283
$250$	0	0
$251$	1396.01	0.351058	0.175529	−	0.984474i	$-0.443836\pi$
$251$	1396.01	0.351058	0.175529	+	0.984474i	$0.443836\pi$
$252$	0	0
$253$	3585.91	0.891083
$254$	0	0
$255$	−536.129	−0.131661
$256$	0	0
$257$	751.154	0.182318	0.0911590	−	0.995836i	$-0.470943\pi$
$257$	751.154	0.182318	0.0911590	+	0.995836i	$0.470943\pi$
$258$	0	0
$259$	−3280.34	−0.786991
$260$	0	0
$261$	9332.91	2.21338
$262$	0	0
$263$	−3274.86	−0.767819	−0.383909	−	0.923371i	$-0.625423\pi$
$263$	−3274.86	−0.767819	−0.383909	+	0.923371i	$0.625423\pi$
$264$	0	0
$265$	1118.11	0.259189
$266$	0	0
$267$	−11389.3	−2.61055
$268$	0	0
$269$	2182.81	0.494752	0.247376	−	0.968920i	$-0.420432\pi$
$269$	2182.81	0.494752	0.247376	+	0.968920i	$0.420432\pi$
$270$	0	0
$271$	2557.21	0.573209	0.286604	−	0.958049i	$-0.407473\pi$
$271$	2557.21	0.573209	0.286604	+	0.958049i	$0.407473\pi$
$272$	0	0
$273$	−2657.93	−0.589250
$274$	0	0
$275$	2764.70	0.606246
$276$	0	0
$277$	−1196.93	−0.259627	−0.129813	−	0.991538i	$-0.541438\pi$
$277$	−1196.93	−0.259627	−0.129813	+	0.991538i	$0.541438\pi$
$278$	0	0
$279$	19195.4	4.11900
$280$	0	0
$281$	3298.30	0.700213	0.350106	−	0.936710i	$-0.386145\pi$
$281$	3298.30	0.700213	0.350106	+	0.936710i	$0.386145\pi$
$282$	0	0
$283$	−1253.93	−0.263387	−0.131694	−	0.991290i	$-0.542041\pi$
$283$	−1253.93	−0.263387	−0.131694	+	0.991290i	$0.542041\pi$
$284$	0	0
$285$	−1081.21	−0.224722
$286$	0	0
$287$	−2222.60	−0.457129
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−2535.10	−0.510688
$292$	0	0
$293$	−574.805	−0.114609	−0.0573046	−	0.998357i	$-0.518251\pi$
$293$	−574.805	−0.114609	−0.0573046	+	0.998357i	$0.518251\pi$
$294$	0	0
$295$	1123.34	0.221707
$296$	0	0
$297$	12394.3	2.42152
$298$	0	0
$299$	5323.86	1.02972
$300$	0	0
$301$	−1418.64	−0.271658
$302$	0	0
$303$	−1871.26	−0.354789
$304$	0	0
$305$	−1474.66	−0.276849
$306$	0	0
$307$	8704.04	1.61813	0.809065	−	0.587719i	$-0.199974\pi$
$307$	8704.04	1.61813	0.809065	+	0.587719i	$0.199974\pi$
$308$	0	0
$309$	−14246.7	−2.62286
$310$	0	0
$311$	−2244.00	−0.409150	−0.204575	−	0.978851i	$-0.565581\pi$
$311$	−2244.00	−0.409150	−0.204575	+	0.978851i	$0.565581\pi$
$312$	0	0
$313$	−5932.65	−1.07135	−0.535676	−	0.844424i	$-0.679943\pi$
$313$	−5932.65	−1.07135	−0.535676	+	0.844424i	$0.679943\pi$
$314$	0	0
$315$	−1749.86	−0.312996
$316$	0	0
$317$	−6242.81	−1.10609	−0.553046	−	0.833150i	$-0.686535\pi$
$317$	−6242.81	−1.10609	−0.553046	+	0.833150i	$0.686535\pi$
$318$	0	0
$319$	−2878.28	−0.505180
$320$	0	0
$321$	12732.5	2.21388
$322$	0	0
$323$	582.829	0.100401
$324$	0	0
$325$	4104.65	0.700569
$326$	0	0
$327$	−15384.5	−2.60173
$328$	0	0
$329$	−154.096	−0.0258225
$330$	0	0
$331$	998.323	0.165779	0.0828894	−	0.996559i	$-0.473585\pi$
$331$	998.323	0.165779	0.0828894	+	0.996559i	$0.473585\pi$
$332$	0	0
$333$	34823.9	5.73074
$334$	0	0
$335$	15.9513	0.00260152
$336$	0	0
$337$	−2738.89	−0.442721	−0.221360	−	0.975192i	$-0.571050\pi$
$337$	−2738.89	−0.442721	−0.221360	+	0.975192i	$0.571050\pi$
$338$	0	0
$339$	20254.3	3.24502
$340$	0	0
$341$	−5919.88	−0.940117
$342$	0	0
$343$	4624.92	0.728054
$344$	0	0
$345$	4724.20	0.737225
$346$	0	0
$347$	12215.5	1.88980	0.944902	−	0.327353i	$-0.106157\pi$
$347$	12215.5	1.88980	0.944902	+	0.327353i	$0.106157\pi$
$348$	0	0
$349$	−3054.27	−0.468457	−0.234228	−	0.972182i	$-0.575256\pi$
$349$	−3054.27	−0.468457	−0.234228	+	0.972182i	$0.575256\pi$
$350$	0	0
$351$	18401.4	2.79827
$352$	0	0
$353$	1591.87	0.240019	0.120010	−	0.992773i	$-0.461707\pi$
$353$	1591.87	0.240019	0.120010	+	0.992773i	$0.461707\pi$
$354$	0	0
$355$	1035.85	0.154865
$356$	0	0
$357$	1271.38	0.188483
$358$	0	0
$359$	3597.68	0.528909	0.264455	−	0.964398i	$-0.414808\pi$
$359$	3597.68	0.528909	0.264455	+	0.964398i	$0.414808\pi$
$360$	0	0
$361$	−5683.60	−0.828635
$362$	0	0
$363$	7752.77	1.12098
$364$	0	0
$365$	−3340.24	−0.479004
$366$	0	0
$367$	−1741.83	−0.247745	−0.123873	−	0.992298i	$-0.539531\pi$
$367$	−1741.83	−0.247745	−0.123873	+	0.992298i	$0.539531\pi$
$368$	0	0
$369$	23595.0	3.32874
$370$	0	0
$371$	−2651.49	−0.371047
$372$	0	0
$373$	5647.06	0.783897	0.391948	−	0.919987i	$-0.371801\pi$
$373$	5647.06	0.783897	0.391948	+	0.919987i	$0.371801\pi$
$374$	0	0
$375$	7584.43	1.04442
$376$	0	0
$377$	−4273.27	−0.583779
$378$	0	0
$379$	−3298.42	−0.447041	−0.223520	−	0.974699i	$-0.571755\pi$
$379$	−3298.42	−0.447041	−0.223520	+	0.974699i	$0.571755\pi$
$380$	0	0
$381$	−10765.8	−1.44764
$382$	0	0
$383$	−8530.71	−1.13812	−0.569059	−	0.822297i	$-0.692692\pi$
$383$	−8530.71	−1.13812	−0.569059	+	0.822297i	$0.692692\pi$
$384$	0	0
$385$	539.659	0.0714379
$386$	0	0
$387$	15060.2	1.97817
$388$	0	0
$389$	9806.36	1.27816	0.639078	−	0.769142i	$-0.279316\pi$
$389$	9806.36	1.27816	0.639078	+	0.769142i	$0.279316\pi$
$390$	0	0
$391$	−2546.58	−0.329376
$392$	0	0
$393$	11081.2	1.42232
$394$	0	0
$395$	−1731.78	−0.220596
$396$	0	0
$397$	−13667.4	−1.72783	−0.863916	−	0.503635i	$-0.831995\pi$
$397$	−13667.4	−1.72783	−0.863916	+	0.503635i	$0.831995\pi$
$398$	0	0
$399$	2563.99	0.321705
$400$	0	0
$401$	−10156.2	−1.26478	−0.632389	−	0.774651i	$-0.717926\pi$
$401$	−10156.2	−1.26478	−0.632389	+	0.774651i	$0.717926\pi$
$402$	0	0
$403$	−8789.03	−1.08638
$404$	0	0
$405$	9866.96	1.21060
$406$	0	0
$407$	−10739.7	−1.30798
$408$	0	0
$409$	1605.25	0.194070	0.0970350	−	0.995281i	$-0.469064\pi$
$409$	1605.25	0.194070	0.0970350	+	0.995281i	$0.469064\pi$
$410$	0	0
$411$	2674.78	0.321015
$412$	0	0
$413$	−2663.89	−0.317389
$414$	0	0
$415$	−2301.12	−0.272187
$416$	0	0
$417$	−11379.1	−1.33630
$418$	0	0
$419$	5445.18	0.634880	0.317440	−	0.948278i	$-0.397177\pi$
$419$	5445.18	0.634880	0.317440	+	0.948278i	$0.397177\pi$
$420$	0	0
$421$	−1227.22	−0.142069	−0.0710344	−	0.997474i	$-0.522630\pi$
$421$	−1227.22	−0.142069	−0.0710344	+	0.997474i	$0.522630\pi$
$422$	0	0
$423$	1635.88	0.188035
$424$	0	0
$425$	−1963.39	−0.224090
$426$	0	0
$427$	3497.02	0.396329
$428$	0	0
$429$	−8701.95	−0.979334
$430$	0	0
$431$	−15343.7	−1.71480	−0.857400	−	0.514650i	$-0.827922\pi$
$431$	−15343.7	−1.71480	−0.857400	+	0.514650i	$0.827922\pi$
$432$	0	0
$433$	−459.294	−0.0509752	−0.0254876	−	0.999675i	$-0.508114\pi$
$433$	−459.294	−0.0509752	−0.0254876	+	0.999675i	$0.508114\pi$
$434$	0	0
$435$	−3791.95	−0.417954
$436$	0	0
$437$	−5135.71	−0.562184
$438$	0	0
$439$	11719.7	1.27415	0.637074	−	0.770802i	$-0.280144\pi$
$439$	11719.7	1.27415	0.637074	+	0.770802i	$0.280144\pi$
$440$	0	0
$441$	−22474.1	−2.42675
$442$	0	0
$443$	−6265.67	−0.671989	−0.335995	−	0.941864i	$-0.609072\pi$
$443$	−6265.67	−0.671989	−0.335995	+	0.941864i	$0.609072\pi$
$444$	0	0
$445$	3433.23	0.365732
$446$	0	0
$447$	24564.8	2.59927
$448$	0	0
$449$	7087.24	0.744917	0.372458	−	0.928049i	$-0.378515\pi$
$449$	7087.24	0.744917	0.372458	+	0.928049i	$0.378515\pi$
$450$	0	0
$451$	−7276.71	−0.759750
$452$	0	0
$453$	−5212.24	−0.540602
$454$	0	0
$455$	801.212	0.0825525
$456$	0	0
$457$	−5562.62	−0.569383	−0.284692	−	0.958619i	$-0.591891\pi$
$457$	−5562.62	−0.569383	−0.284692	+	0.958619i	$0.591891\pi$
$458$	0	0
$459$	−8801.99	−0.895080
$460$	0	0
$461$	216.328	0.0218555	0.0109278	−	0.999940i	$-0.496522\pi$
$461$	216.328	0.0218555	0.0109278	+	0.999940i	$0.496522\pi$
$462$	0	0
$463$	−11177.9	−1.12199	−0.560997	−	0.827818i	$-0.689582\pi$
$463$	−11177.9	−1.12199	−0.560997	+	0.827818i	$0.689582\pi$
$464$	0	0
$465$	−7799.07	−0.777792
$466$	0	0
$467$	4355.01	0.431533	0.215767	−	0.976445i	$-0.430775\pi$
$467$	4355.01	0.431533	0.215767	+	0.976445i	$0.430775\pi$
$468$	0	0
$469$	−37.8268	−0.00372427
$470$	0	0
$471$	−22301.5	−2.18174
$472$	0	0
$473$	−4644.58	−0.451497
$474$	0	0
$475$	−3959.58	−0.382480
$476$	0	0
$477$	28148.0	2.70191
$478$	0	0
$479$	148.437	0.0141592	0.00707961	−	0.999975i	$-0.497746\pi$
$479$	148.437	0.0141592	0.00707961	+	0.999975i	$0.497746\pi$
$480$	0	0
$481$	−15944.9	−1.51148
$482$	0	0
$483$	−11203.0	−1.05539
$484$	0	0
$485$	764.187	0.0715463
$486$	0	0
$487$	5946.01	0.553263	0.276632	−	0.960976i	$-0.410782\pi$
$487$	5946.01	0.553263	0.276632	+	0.960976i	$0.410782\pi$
$488$	0	0
$489$	19802.5	1.83128
$490$	0	0
$491$	9945.17	0.914092	0.457046	−	0.889443i	$-0.348907\pi$
$491$	9945.17	0.914092	0.457046	+	0.889443i	$0.348907\pi$
$492$	0	0
$493$	2044.05	0.186733
$494$	0	0
$495$	−5728.98	−0.520199
$496$	0	0
$497$	−2456.40	−0.221700
$498$	0	0
$499$	−4505.40	−0.404187	−0.202094	−	0.979366i	$-0.564775\pi$
$499$	−4505.40	−0.404187	−0.202094	+	0.979366i	$0.564775\pi$
$500$	0	0
$501$	−27580.7	−2.45951
$502$	0	0
$503$	10100.8	0.895374	0.447687	−	0.894190i	$-0.352248\pi$
$503$	10100.8	0.895374	0.447687	+	0.894190i	$0.352248\pi$
$504$	0	0
$505$	564.078	0.0497052
$506$	0	0
$507$	9552.33	0.836753
$508$	0	0
$509$	−19366.6	−1.68646	−0.843229	−	0.537554i	$-0.819348\pi$
$509$	−19366.6	−1.68646	−0.843229	+	0.537554i	$0.819348\pi$
$510$	0	0
$511$	7921.06	0.685728
$512$	0	0
$513$	−17751.1	−1.52773
$514$	0	0
$515$	4294.55	0.367457
$516$	0	0
$517$	−504.505	−0.0429170
$518$	0	0
$519$	37111.2	3.13873
$520$	0	0
$521$	−4394.01	−0.369491	−0.184746	−	0.982786i	$-0.559146\pi$
$521$	−4394.01	−0.369491	−0.184746	+	0.982786i	$0.559146\pi$
$522$	0	0
$523$	−8065.99	−0.674381	−0.337191	−	0.941436i	$-0.609477\pi$
$523$	−8065.99	−0.674381	−0.337191	+	0.941436i	$0.609477\pi$
$524$	0	0
$525$	−8637.38	−0.718031
$526$	0	0
$527$	4204.09	0.347501
$528$	0	0
$529$	10272.7	0.844309
$530$	0	0
$531$	28279.7	2.31117
$532$	0	0
$533$	−10803.5	−0.877955
$534$	0	0
$535$	−3838.10	−0.310160
$536$	0	0
$537$	11981.1	0.962799
$538$	0	0
$539$	6931.04	0.553879
$540$	0	0
$541$	−23946.0	−1.90299	−0.951496	−	0.307661i	$-0.900454\pi$
$541$	−23946.0	−1.90299	−0.951496	+	0.307661i	$0.900454\pi$
$542$	0	0
$543$	33190.4	2.62309
$544$	0	0
$545$	4637.55	0.364497
$546$	0	0
$547$	−640.284	−0.0500486	−0.0250243	−	0.999687i	$-0.507966\pi$
$547$	−640.284	−0.0500486	−0.0250243	+	0.999687i	$0.507966\pi$
$548$	0	0
$549$	−37124.1	−2.88601
$550$	0	0
$551$	4122.25	0.318718
$552$	0	0
$553$	4106.74	0.315798
$554$	0	0
$555$	−14148.9	−1.08214
$556$	0	0
$557$	5381.50	0.409374	0.204687	−	0.978827i	$-0.434382\pi$
$557$	5381.50	0.409374	0.204687	+	0.978827i	$0.434382\pi$
$558$	0	0
$559$	−6895.63	−0.521743
$560$	0	0
$561$	4162.43	0.313259
$562$	0	0
$563$	17871.9	1.33785	0.668925	−	0.743330i	$-0.266755\pi$
$563$	17871.9	1.33785	0.668925	+	0.743330i	$0.266755\pi$
$564$	0	0
$565$	−6105.50	−0.454620
$566$	0	0
$567$	−23398.5	−1.73306
$568$	0	0
$569$	−22286.3	−1.64198	−0.820992	−	0.570939i	$-0.806579\pi$
$569$	−22286.3	−1.64198	−0.820992	+	0.570939i	$0.806579\pi$
$570$	0	0
$571$	−17760.8	−1.30169	−0.650845	−	0.759210i	$-0.725585\pi$
$571$	−17760.8	−1.30169	−0.650845	+	0.759210i	$0.725585\pi$
$572$	0	0
$573$	20539.5	1.49747
$574$	0	0
$575$	17300.8	1.25477
$576$	0	0
$577$	−14447.9	−1.04242	−0.521208	−	0.853430i	$-0.674518\pi$
$577$	−14447.9	−1.04242	−0.521208	+	0.853430i	$0.674518\pi$
$578$	0	0
$579$	26593.1	1.90876
$580$	0	0
$581$	5456.87	0.389654
$582$	0	0
$583$	−8680.88	−0.616681
$584$	0	0
$585$	−8505.61	−0.601135
$586$	0	0
$587$	−8160.77	−0.573818	−0.286909	−	0.957958i	$-0.592628\pi$
$587$	−8160.77	−0.573818	−0.286909	+	0.957958i	$0.592628\pi$
$588$	0	0
$589$	8478.42	0.593119
$590$	0	0
$591$	17123.1	1.19179
$592$	0	0
$593$	−10912.0	−0.755650	−0.377825	−	0.925877i	$-0.623328\pi$
$593$	−10912.0	−0.755650	−0.377825	+	0.925877i	$0.623328\pi$
$594$	0	0
$595$	−383.246	−0.0264060
$596$	0	0
$597$	34555.2	2.36893
$598$	0	0
$599$	13560.1	0.924962	0.462481	−	0.886629i	$-0.346959\pi$
$599$	13560.1	0.924962	0.462481	+	0.886629i	$0.346959\pi$
$600$	0	0
$601$	−1724.43	−0.117040	−0.0585199	−	0.998286i	$-0.518638\pi$
$601$	−1724.43	−0.117040	−0.0585199	+	0.998286i	$0.518638\pi$
$602$	0	0
$603$	401.567	0.0271195
$604$	0	0
$605$	−2337.01	−0.157047
$606$	0	0
$607$	22311.2	1.49190	0.745949	−	0.666003i	$-0.231996\pi$
$607$	22311.2	1.49190	0.745949	+	0.666003i	$0.231996\pi$
$608$	0	0
$609$	8992.22	0.598330
$610$	0	0
$611$	−749.020	−0.0495943
$612$	0	0
$613$	−16905.6	−1.11388	−0.556941	−	0.830552i	$-0.688025\pi$
$613$	−16905.6	−1.11388	−0.556941	+	0.830552i	$0.688025\pi$
$614$	0	0
$615$	−9586.61	−0.628568
$616$	0	0
$617$	881.345	0.0575067	0.0287533	−	0.999587i	$-0.490846\pi$
$617$	881.345	0.0575067	0.0287533	+	0.999587i	$0.490846\pi$
$618$	0	0
$619$	7055.06	0.458104	0.229052	−	0.973414i	$-0.426437\pi$
$619$	7055.06	0.458104	0.229052	+	0.973414i	$0.426437\pi$
$620$	0	0
$621$	77560.5	5.01191
$622$	0	0
$623$	−8141.56	−0.523571
$624$	0	0
$625$	12150.4	0.777626
$626$	0	0
$627$	8394.41	0.534674
$628$	0	0
$629$	7626.96	0.483476
$630$	0	0
$631$	−14724.9	−0.928981	−0.464491	−	0.885578i	$-0.653763\pi$
$631$	−14724.9	−0.928981	−0.464491	+	0.885578i	$0.653763\pi$
$632$	0	0
$633$	42079.4	2.64219
$634$	0	0
$635$	3245.28	0.202811
$636$	0	0
$637$	10290.3	0.640054
$638$	0	0
$639$	26077.0	1.61438
$640$	0	0
$641$	4518.80	0.278443	0.139222	−	0.990261i	$-0.455540\pi$
$641$	4518.80	0.278443	0.139222	+	0.990261i	$0.455540\pi$
$642$	0	0
$643$	−5293.30	−0.324646	−0.162323	−	0.986738i	$-0.551899\pi$
$643$	−5293.30	−0.324646	−0.162323	+	0.986738i	$0.551899\pi$
$644$	0	0
$645$	−6118.93	−0.373539
$646$	0	0
$647$	−28408.1	−1.72618	−0.863088	−	0.505053i	$-0.831473\pi$
$647$	−28408.1	−1.72618	−0.863088	+	0.505053i	$0.831473\pi$
$648$	0	0
$649$	−8721.48	−0.527501
$650$	0	0
$651$	18494.7	1.11346
$652$	0	0
$653$	18708.3	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$653$	18708.3	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$654$	0	0
$655$	−3340.34	−0.199264
$656$	0	0
$657$	−84089.4	−4.99336
$658$	0	0
$659$	−24835.9	−1.46809	−0.734044	−	0.679102i	$-0.762369\pi$
$659$	−24835.9	−1.46809	−0.734044	+	0.679102i	$0.762369\pi$
$660$	0	0
$661$	−21416.6	−1.26023	−0.630114	−	0.776503i	$-0.716992\pi$
$661$	−21416.6	−1.26023	−0.630114	+	0.776503i	$0.716992\pi$
$662$	0	0
$663$	6179.81	0.361997
$664$	0	0
$665$	−772.896	−0.0450701
$666$	0	0
$667$	−18011.5	−1.04559
$668$	0	0
$669$	3882.21	0.224357
$670$	0	0
$671$	11449.1	0.658700
$672$	0	0
$673$	−10872.9	−0.622765	−0.311382	−	0.950285i	$-0.600792\pi$
$673$	−10872.9	−0.622765	−0.311382	+	0.950285i	$0.600792\pi$
$674$	0	0
$675$	59798.4	3.40984
$676$	0	0
$677$	−11072.3	−0.628571	−0.314286	−	0.949328i	$-0.601765\pi$
$677$	−11072.3	−0.628571	−0.314286	+	0.949328i	$0.601765\pi$
$678$	0	0
$679$	−1812.19	−0.102424
$680$	0	0
$681$	56583.9	3.18399
$682$	0	0
$683$	−8696.44	−0.487204	−0.243602	−	0.969875i	$-0.578329\pi$
$683$	−8696.44	−0.487204	−0.243602	+	0.969875i	$0.578329\pi$
$684$	0	0
$685$	−806.292	−0.0449735
$686$	0	0
$687$	−19473.4	−1.08145
$688$	0	0
$689$	−12888.2	−0.712628
$690$	0	0
$691$	−11936.1	−0.657120	−0.328560	−	0.944483i	$-0.606563\pi$
$691$	−11936.1	−0.657120	−0.328560	+	0.944483i	$0.606563\pi$
$692$	0	0
$693$	13585.7	0.744702
$694$	0	0
$695$	3430.14	0.187212
$696$	0	0
$697$	5167.66	0.280831
$698$	0	0
$699$	−8922.43	−0.482800
$700$	0	0
$701$	10833.2	0.583685	0.291842	−	0.956466i	$-0.405732\pi$
$701$	10833.2	0.583685	0.291842	+	0.956466i	$0.405732\pi$
$702$	0	0
$703$	15381.3	0.825204
$704$	0	0
$705$	−664.653	−0.0355068
$706$	0	0
$707$	−1337.65	−0.0711565
$708$	0	0
$709$	−9205.66	−0.487624	−0.243812	−	0.969822i	$-0.578398\pi$
$709$	−9205.66	−0.487624	−0.243812	+	0.969822i	$0.578398\pi$
$710$	0	0
$711$	−43596.9	−2.29959
$712$	0	0
$713$	−37045.1	−1.94579
$714$	0	0
$715$	2623.14	0.137202
$716$	0	0
$717$	−58452.4	−3.04455
$718$	0	0
$719$	−21716.0	−1.12638	−0.563191	−	0.826327i	$-0.690426\pi$
$719$	−21716.0	−1.12638	−0.563191	+	0.826327i	$0.690426\pi$
$720$	0	0
$721$	−10184.1	−0.526041
$722$	0	0
$723$	−6014.93	−0.309402
$724$	0	0
$725$	−13886.7	−0.711365
$726$	0	0
$727$	32011.8	1.63308	0.816541	−	0.577287i	$-0.195889\pi$
$727$	32011.8	1.63308	0.816541	+	0.577287i	$0.195889\pi$
$728$	0	0
$729$	105408.	5.35526
$730$	0	0
$731$	3298.41	0.166889
$732$	0	0
$733$	16136.0	0.813094	0.406547	−	0.913630i	$-0.366733\pi$
$733$	16136.0	0.813094	0.406547	+	0.913630i	$0.366733\pi$
$734$	0	0
$735$	9131.19	0.458244
$736$	0	0
$737$	−123.844	−0.00618974
$738$	0	0
$739$	−34532.2	−1.71893	−0.859464	−	0.511197i	$-0.829202\pi$
$739$	−34532.2	−1.71893	−0.859464	+	0.511197i	$0.829202\pi$
$740$	0	0
$741$	12462.9	0.617861
$742$	0	0
$743$	17223.1	0.850409	0.425204	−	0.905097i	$-0.360202\pi$
$743$	17223.1	0.850409	0.425204	+	0.905097i	$0.360202\pi$
$744$	0	0
$745$	−7404.86	−0.364152
$746$	0	0
$747$	−57929.8	−2.83740
$748$	0	0
$749$	9101.68	0.444016
$750$	0	0
$751$	−2343.37	−0.113863	−0.0569314	−	0.998378i	$-0.518132\pi$
$751$	−2343.37	−0.113863	−0.0569314	+	0.998378i	$0.518132\pi$
$752$	0	0
$753$	−14279.0	−0.691043
$754$	0	0
$755$	1571.19	0.0757371
$756$	0	0
$757$	18908.8	0.907862	0.453931	−	0.891037i	$-0.350021\pi$
$757$	18908.8	0.907862	0.453931	+	0.891037i	$0.350021\pi$
$758$	0	0
$759$	−36678.1	−1.75406
$760$	0	0
$761$	12379.8	0.589707	0.294854	−	0.955542i	$-0.404729\pi$
$761$	12379.8	0.589707	0.294854	+	0.955542i	$0.404729\pi$
$762$	0	0
$763$	−10997.5	−0.521803
$764$	0	0
$765$	4068.52	0.192284
$766$	0	0
$767$	−12948.5	−0.609572
$768$	0	0
$769$	−20772.2	−0.974076	−0.487038	−	0.873381i	$-0.661923\pi$
$769$	−20772.2	−0.974076	−0.487038	+	0.873381i	$0.661923\pi$
$770$	0	0
$771$	−7683.11	−0.358885
$772$	0	0
$773$	−11247.5	−0.523341	−0.261671	−	0.965157i	$-0.584273\pi$
$773$	−11247.5	−0.523341	−0.261671	+	0.965157i	$0.584273\pi$
$774$	0	0
$775$	−28561.4	−1.32382
$776$	0	0
$777$	33552.7	1.54916
$778$	0	0
$779$	10421.7	0.479326
$780$	0	0
$781$	−8042.17	−0.368465
$782$	0	0
$783$	−62255.0	−2.84139
$784$	0	0
$785$	6722.61	0.305656
$786$	0	0
$787$	38565.4	1.74677	0.873384	−	0.487032i	$-0.161920\pi$
$787$	38565.4	1.74677	0.873384	+	0.487032i	$0.161920\pi$
$788$	0	0
$789$	33496.6	1.51142
$790$	0	0
$791$	14478.6	0.650821
$792$	0	0
$793$	16998.1	0.761184
$794$	0	0
$795$	−11436.5	−0.510202
$796$	0	0
$797$	36628.7	1.62792	0.813961	−	0.580919i	$-0.197307\pi$
$797$	36628.7	1.62792	0.813961	+	0.580919i	$0.197307\pi$
$798$	0	0
$799$	358.281	0.0158637
$800$	0	0
$801$	86430.2	3.81256
$802$	0	0
$803$	25933.2	1.13968
$804$	0	0
$805$	3377.05	0.147858
$806$	0	0
$807$	−22326.6	−0.973897
$808$	0	0
$809$	36647.9	1.59267	0.796336	−	0.604855i	$-0.206769\pi$
$809$	36647.9	1.59267	0.796336	+	0.604855i	$0.206769\pi$
$810$	0	0
$811$	35905.8	1.55465	0.777327	−	0.629097i	$-0.216575\pi$
$811$	35905.8	1.55465	0.777327	+	0.629097i	$0.216575\pi$
$812$	0	0
$813$	−26156.2	−1.12834
$814$	0	0
$815$	−5969.30	−0.256559
$816$	0	0
$817$	6651.93	0.284849
$818$	0	0
$819$	20170.2	0.860567
$820$	0	0
$821$	13474.4	0.572789	0.286394	−	0.958112i	$-0.407543\pi$
$821$	13474.4	0.572789	0.286394	+	0.958112i	$0.407543\pi$
$822$	0	0
$823$	−35659.4	−1.51034	−0.755169	−	0.655531i	$-0.772445\pi$
$823$	−35659.4	−1.51034	−0.755169	+	0.655531i	$0.772445\pi$
$824$	0	0
$825$	−28278.4	−1.19337
$826$	0	0
$827$	20833.7	0.876009	0.438005	−	0.898973i	$-0.355685\pi$
$827$	20833.7	0.876009	0.438005	+	0.898973i	$0.355685\pi$
$828$	0	0
$829$	19919.3	0.834532	0.417266	−	0.908784i	$-0.362988\pi$
$829$	19919.3	0.834532	0.417266	+	0.908784i	$0.362988\pi$
$830$	0	0
$831$	12242.7	0.511064
$832$	0	0
$833$	−4922.17	−0.204734
$834$	0	0
$835$	8313.98	0.344571
$836$	0	0
$837$	−128043.	−5.28770
$838$	0	0
$839$	35246.0	1.45033	0.725166	−	0.688574i	$-0.241763\pi$
$839$	35246.0	1.45033	0.725166	+	0.688574i	$0.241763\pi$
$840$	0	0
$841$	−9931.80	−0.407225
$842$	0	0
$843$	−33736.3	−1.37834
$844$	0	0
$845$	−2879.48	−0.117227
$846$	0	0
$847$	5542.00	0.224823
$848$	0	0
$849$	12825.7	0.518466
$850$	0	0
$851$	−67206.4	−2.70718
$852$	0	0
$853$	−13250.7	−0.531884	−0.265942	−	0.963989i	$-0.585683\pi$
$853$	−13250.7	−0.531884	−0.265942	+	0.963989i	$0.585683\pi$
$854$	0	0
$855$	8205.01	0.328193
$856$	0	0
$857$	1907.93	0.0760485	0.0380242	−	0.999277i	$-0.487894\pi$
$857$	1907.93	0.0760485	0.0380242	+	0.999277i	$0.487894\pi$
$858$	0	0
$859$	15241.3	0.605386	0.302693	−	0.953088i	$-0.402114\pi$
$859$	15241.3	0.605386	0.302693	+	0.953088i	$0.402114\pi$
$860$	0	0
$861$	22733.7	0.899840
$862$	0	0
$863$	−8042.77	−0.317241	−0.158621	−	0.987340i	$-0.550705\pi$
$863$	−8042.77	−0.317241	−0.158621	+	0.987340i	$0.550705\pi$
$864$	0	0
$865$	−11186.9	−0.439729
$866$	0	0
$867$	−2956.01	−0.115792
$868$	0	0
$869$	13445.3	0.524857
$870$	0	0
$871$	−183.866	−0.00715277
$872$	0	0
$873$	19238.1	0.745832
$874$	0	0
$875$	5421.66	0.209469
$876$	0	0
$877$	16636.8	0.640576	0.320288	−	0.947320i	$-0.396220\pi$
$877$	16636.8	0.640576	0.320288	+	0.947320i	$0.396220\pi$
$878$	0	0
$879$	5879.34	0.225603
$880$	0	0
$881$	39048.2	1.49327	0.746633	−	0.665236i	$-0.231669\pi$
$881$	39048.2	1.49327	0.746633	+	0.665236i	$0.231669\pi$
$882$	0	0
$883$	−10251.5	−0.390703	−0.195351	−	0.980733i	$-0.562585\pi$
$883$	−10251.5	−0.390703	−0.195351	+	0.980733i	$0.562585\pi$
$884$	0	0
$885$	−11490.0	−0.436420
$886$	0	0
$887$	−450.274	−0.0170448	−0.00852239	−	0.999964i	$-0.502713\pi$
$887$	−450.274	−0.0170448	−0.00852239	+	0.999964i	$0.502713\pi$
$888$	0	0
$889$	−7695.87	−0.290339
$890$	0	0
$891$	−76605.8	−2.88035
$892$	0	0
$893$	722.549	0.0270763
$894$	0	0
$895$	−3611.61	−0.134886
$896$	0	0
$897$	−54454.6	−2.02696
$898$	0	0
$899$	29734.8	1.10313
$900$	0	0
$901$	6164.84	0.227948
$902$	0	0
$903$	14510.4	0.534748
$904$	0	0
$905$	−10005.0	−0.367489
$906$	0	0
$907$	−5679.97	−0.207939	−0.103969	−	0.994581i	$-0.533154\pi$
$907$	−5679.97	−0.207939	−0.103969	+	0.994581i	$0.533154\pi$
$908$	0	0
$909$	14200.4	0.518151
$910$	0	0
$911$	36865.9	1.34075	0.670374	−	0.742023i	$-0.266133\pi$
$911$	36865.9	1.34075	0.670374	+	0.742023i	$0.266133\pi$
$912$	0	0
$913$	17865.6	0.647606
$914$	0	0
$915$	15083.5	0.544966
$916$	0	0
$917$	7921.28	0.285260
$918$	0	0
$919$	−10878.9	−0.390490	−0.195245	−	0.980754i	$-0.562550\pi$
$919$	−10878.9	−0.390490	−0.195245	+	0.980754i	$0.562550\pi$
$920$	0	0
$921$	−89028.5	−3.18522
$922$	0	0
$923$	−11939.9	−0.425793
$924$	0	0
$925$	−51815.5	−1.84182
$926$	0	0
$927$	108114.	3.83055
$928$	0	0
$929$	−20336.1	−0.718197	−0.359099	−	0.933300i	$-0.616916\pi$
$929$	−20336.1	−0.718197	−0.359099	+	0.933300i	$0.616916\pi$
$930$	0	0
$931$	−9926.58	−0.349442
$932$	0	0
$933$	22952.5	0.805394
$934$	0	0
$935$	−1254.73	−0.0438868
$936$	0	0
$937$	3695.48	0.128843	0.0644215	−	0.997923i	$-0.479480\pi$
$937$	3695.48	0.128843	0.0644215	+	0.997923i	$0.479480\pi$
$938$	0	0
$939$	60681.5	2.10891
$940$	0	0
$941$	−2691.58	−0.0932446	−0.0466223	−	0.998913i	$-0.514846\pi$
$941$	−2691.58	−0.0932446	−0.0466223	+	0.998913i	$0.514846\pi$
$942$	0	0
$943$	−45535.8	−1.57248
$944$	0	0
$945$	11672.4	0.401803
$946$	0	0
$947$	27132.6	0.931035	0.465517	−	0.885039i	$-0.345868\pi$
$947$	27132.6	0.931035	0.465517	+	0.885039i	$0.345868\pi$
$948$	0	0
$949$	38502.1	1.31700
$950$	0	0
$951$	63854.0	2.17730
$952$	0	0
$953$	−5402.07	−0.183620	−0.0918102	−	0.995777i	$-0.529265\pi$
$953$	−5402.07	−0.183620	−0.0918102	+	0.995777i	$0.529265\pi$
$954$	0	0
$955$	−6191.47	−0.209792
$956$	0	0
$957$	29440.2	0.994426
$958$	0	0
$959$	1912.04	0.0643827
$960$	0	0
$961$	31365.9	1.05287
$962$	0	0
$963$	−96622.7	−3.23325
$964$	0	0
$965$	−8016.28	−0.267413
$966$	0	0
$967$	−57540.8	−1.91353	−0.956767	−	0.290854i	$-0.906061\pi$
$967$	−57540.8	−1.91353	−0.956767	+	0.290854i	$0.906061\pi$
$968$	0	0
$969$	−5961.41	−0.197635
$970$	0	0
$971$	−25631.7	−0.847126	−0.423563	−	0.905867i	$-0.639221\pi$
$971$	−25631.7	−0.847126	−0.423563	+	0.905867i	$0.639221\pi$
$972$	0	0
$973$	−8134.23	−0.268008
$974$	0	0
$975$	−41984.0	−1.37904
$976$	0	0
$977$	−15389.8	−0.503953	−0.251977	−	0.967733i	$-0.581081\pi$
$977$	−15389.8	−0.503953	−0.251977	+	0.967733i	$0.581081\pi$
$978$	0	0
$979$	−26655.2	−0.870176
$980$	0	0
$981$	116748.	3.79968
$982$	0	0
$983$	28812.6	0.934873	0.467437	−	0.884027i	$-0.345178\pi$
$983$	28812.6	0.934873	0.467437	+	0.884027i	$0.345178\pi$
$984$	0	0
$985$	−5161.63	−0.166968
$986$	0	0
$987$	1576.16	0.0508305
$988$	0	0
$989$	−29064.6	−0.934480
$990$	0	0
$991$	−27829.1	−0.892049	−0.446024	−	0.895021i	$-0.647161\pi$
$991$	−27829.1	−0.892049	−0.446024	+	0.895021i	$0.647161\pi$
$992$	0	0
$993$	−10211.2	−0.326328
$994$	0	0
$995$	−10416.4	−0.331881
$996$	0	0
$997$	−33204.6	−1.05476	−0.527382	−	0.849628i	$-0.676826\pi$
$997$	−33204.6	−1.05476	−0.527382	+	0.849628i	$0.676826\pi$
$998$	0	0
$999$	−232292.	−7.35675

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.s.1.1		3
4.3	odd	2		1088.4.a.z.1.3		3
8.3	odd	2		272.4.a.g.1.1		3
8.5	even	2		136.4.a.c.1.3	✓	3
24.5	odd	2		1224.4.a.f.1.2		3
24.11	even	2		2448.4.a.bf.1.2		3
136.101	even	2		2312.4.a.c.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.a.c.1.3	✓	3	8.5	even	2
272.4.a.g.1.1		3	8.3	odd	2
1088.4.a.s.1.1		3	1.1	even	1	trivial
1088.4.a.z.1.3		3	4.3	odd	2
1224.4.a.f.1.2		3	24.5	odd	2
2312.4.a.c.1.1		3	136.101	even	2
2448.4.a.bf.1.2		3	24.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q-10.2284 q^{3} +3.08327 q^{5} -7.31168 q^{7} +77.6202 q^{9} +O(q^{10})\)	\(q-10.2284 q^{3} +3.08327 q^{5} -7.31168 q^{7} +77.6202 q^{9} -23.9381 q^{11} -35.5401 q^{13} -31.5370 q^{15} +17.0000 q^{17} +34.2840 q^{19} +74.7868 q^{21} -149.799 q^{23} -115.493 q^{25} -517.764 q^{27} +120.238 q^{29} +247.299 q^{31} +244.849 q^{33} -22.5439 q^{35} +448.644 q^{37} +363.518 q^{39} +303.980 q^{41} +194.024 q^{43} +239.324 q^{45} +21.0754 q^{47} -289.539 q^{49} -173.883 q^{51} +362.638 q^{53} -73.8079 q^{55} -350.671 q^{57} +364.334 q^{59} -478.279 q^{61} -567.534 q^{63} -109.580 q^{65} +5.17348 q^{67} +1532.20 q^{69} +335.956 q^{71} -1083.34 q^{73} +1181.31 q^{75} +175.028 q^{77} -561.669 q^{79} +3200.16 q^{81} -746.323 q^{83} +52.4157 q^{85} -1229.84 q^{87} +1113.50 q^{89} +259.858 q^{91} -2529.48 q^{93} +105.707 q^{95} +247.849 q^{97} -1858.08 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{3} - 2 q^{5} + 12 q^{7} + 63 q^{9}+O(q^{10}) \) 3 * q - 8 * q^3 - 2 * q^5 + 12 * q^7 + 63 * q^9	\( 3 q - 8 q^{3} - 2 q^{5} + 12 q^{7} + 63 q^{9} - 72 q^{11} - 50 q^{13} + 64 q^{15} + 51 q^{17} - 124 q^{19} + 32 q^{21} + 60 q^{23} - 75 q^{25} - 512 q^{27} + 54 q^{29} + 300 q^{31} - 48 q^{33} - 248 q^{35} + 542 q^{37} + 320 q^{39} + 30 q^{41} - 52 q^{43} + 502 q^{45} + 16 q^{47} - 677 q^{49} - 136 q^{51} + 518 q^{53} - 608 q^{55} - 896 q^{57} - 132 q^{59} + 614 q^{61} - 852 q^{63} - 148 q^{65} + 332 q^{67} + 1280 q^{69} - 268 q^{71} - 578 q^{73} + 712 q^{75} + 128 q^{77} - 668 q^{79} + 2427 q^{81} - 876 q^{83} - 34 q^{85} - 2400 q^{87} + 1790 q^{89} + 168 q^{91} - 2592 q^{93} - 504 q^{95} + 838 q^{97} - 2040 q^{99}+O(q^{100}) \) 3 * q - 8 * q^3 - 2 * q^5 + 12 * q^7 + 63 * q^9 - 72 * q^11 - 50 * q^13 + 64 * q^15 + 51 * q^17 - 124 * q^19 + 32 * q^21 + 60 * q^23 - 75 * q^25 - 512 * q^27 + 54 * q^29 + 300 * q^31 - 48 * q^33 - 248 * q^35 + 542 * q^37 + 320 * q^39 + 30 * q^41 - 52 * q^43 + 502 * q^45 + 16 * q^47 - 677 * q^49 - 136 * q^51 + 518 * q^53 - 608 * q^55 - 896 * q^57 - 132 * q^59 + 614 * q^61 - 852 * q^63 - 148 * q^65 + 332 * q^67 + 1280 * q^69 - 268 * q^71 - 578 * q^73 + 712 * q^75 + 128 * q^77 - 668 * q^79 + 2427 * q^81 - 876 * q^83 - 34 * q^85 - 2400 * q^87 + 1790 * q^89 + 168 * q^91 - 2592 * q^93 - 504 * q^95 + 838 * q^97 - 2040 * q^99

Introduction

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