LMFDB - Embedded newform 1088.4.a.be.1.2

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.550476.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 15x^{2} + 19x + 20 $ x^4 - x^3 - 15x^2 + 19x + 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.2
Root			$-3.84065$ 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)

$f(q)$	$=$	$q-5.49465 q^{3} +8.91638 q^{5} +17.5805 q^{7} +3.19121 q^{9} +O(q^{10})$	$q-5.49465 q^{3} +8.91638 q^{5} +17.5805 q^{7} +3.19121 q^{9} -66.4129 q^{11} -70.7484 q^{13} -48.9924 q^{15} -17.0000 q^{17} +100.127 q^{19} -96.5988 q^{21} +217.925 q^{23} -45.4982 q^{25} +130.821 q^{27} -23.0384 q^{29} -212.560 q^{31} +364.916 q^{33} +156.754 q^{35} -174.423 q^{37} +388.738 q^{39} +209.045 q^{41} +150.350 q^{43} +28.4541 q^{45} -252.201 q^{47} -33.9257 q^{49} +93.4091 q^{51} -60.6076 q^{53} -592.163 q^{55} -550.164 q^{57} +626.948 q^{59} +624.906 q^{61} +56.1031 q^{63} -630.820 q^{65} +114.386 q^{67} -1197.42 q^{69} +346.071 q^{71} -756.821 q^{73} +249.997 q^{75} -1167.57 q^{77} -146.530 q^{79} -804.979 q^{81} -328.488 q^{83} -151.578 q^{85} +126.588 q^{87} -583.027 q^{89} -1243.79 q^{91} +1167.95 q^{93} +892.772 q^{95} +1620.95 q^{97} -211.938 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 8 q^{5} - 22 q^{7} + 100 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 8 * q^5 - 22 * q^7 + 100 * q^9	$ 4 q + 2 q^{3} - 8 q^{5} - 22 q^{7} + 100 q^{9} - 70 q^{11} - 120 q^{13} + 140 q^{15} - 68 q^{17} + 44 q^{19} - 488 q^{21} + 158 q^{23} + 548 q^{25} + 392 q^{27} - 264 q^{29} + 122 q^{31} + 136 q^{33} + 44 q^{35} - 256 q^{37} - 528 q^{39} + 240 q^{41} + 1100 q^{43} + 880 q^{45} - 800 q^{47} + 12 q^{49} - 34 q^{51} - 432 q^{53} - 532 q^{55} + 472 q^{57} + 148 q^{59} + 728 q^{61} - 1450 q^{63} - 72 q^{65} + 1032 q^{67} + 1024 q^{69} + 798 q^{71} - 1544 q^{73} + 2974 q^{75} - 656 q^{77} + 758 q^{79} - 20 q^{81} - 244 q^{83} + 136 q^{85} + 1524 q^{87} + 1440 q^{89} + 1104 q^{91} + 2464 q^{93} + 7016 q^{95} - 1344 q^{97} + 86 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 8 * q^5 - 22 * q^7 + 100 * q^9 - 70 * q^11 - 120 * q^13 + 140 * q^15 - 68 * q^17 + 44 * q^19 - 488 * q^21 + 158 * q^23 + 548 * q^25 + 392 * q^27 - 264 * q^29 + 122 * q^31 + 136 * q^33 + 44 * q^35 - 256 * q^37 - 528 * q^39 + 240 * q^41 + 1100 * q^43 + 880 * q^45 - 800 * q^47 + 12 * q^49 - 34 * q^51 - 432 * q^53 - 532 * q^55 + 472 * q^57 + 148 * q^59 + 728 * q^61 - 1450 * q^63 - 72 * q^65 + 1032 * q^67 + 1024 * q^69 + 798 * q^71 - 1544 * q^73 + 2974 * q^75 - 656 * q^77 + 758 * q^79 - 20 * q^81 - 244 * q^83 + 136 * q^85 + 1524 * q^87 + 1440 * q^89 + 1104 * q^91 + 2464 * q^93 + 7016 * q^95 - 1344 * q^97 + 86 * 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
$3$	−5.49465	−1.05745	−0.528723	−	0.848794i	$-0.677329\pi$
$3$	−5.49465	−1.05745	−0.528723	+	0.848794i	$0.677329\pi$
$4$	0	0
$5$	8.91638	0.797505	0.398752	−	0.917059i	$-0.369443\pi$
$5$	8.91638	0.797505	0.398752	+	0.917059i	$0.369443\pi$
$6$	0	0
$7$	17.5805	0.949258	0.474629	−	0.880186i	$-0.342582\pi$
$7$	17.5805	0.949258	0.474629	+	0.880186i	$0.342582\pi$
$8$	0	0
$9$	3.19121	0.118193
$10$	0	0
$11$	−66.4129	−1.82039	−0.910193	−	0.414184i	$-0.864067\pi$
$11$	−66.4129	−1.82039	−0.910193	+	0.414184i	$0.864067\pi$
$12$	0	0
$13$	−70.7484	−1.50939	−0.754696	−	0.656075i	$-0.772215\pi$
$13$	−70.7484	−1.50939	−0.754696	+	0.656075i	$0.772215\pi$
$14$	0	0
$15$	−48.9924	−0.843319
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	100.127	1.20899	0.604494	−	0.796610i	$-0.293375\pi$
$19$	100.127	1.20899	0.604494	+	0.796610i	$0.293375\pi$
$20$	0	0
$21$	−96.5988	−1.00379
$22$	0	0
$23$	217.925	1.97567	0.987835	−	0.155504i	$-0.0497000\pi$
$23$	217.925	1.97567	0.987835	+	0.155504i	$0.0497000\pi$
$24$	0	0
$25$	−45.4982	−0.363986
$26$	0	0
$27$	130.821	0.932464
$28$	0	0
$29$	−23.0384	−0.147522	−0.0737609	−	0.997276i	$-0.523500\pi$
$29$	−23.0384	−0.147522	−0.0737609	+	0.997276i	$0.523500\pi$
$30$	0	0
$31$	−212.560	−1.23152	−0.615758	−	0.787935i	$-0.711150\pi$
$31$	−212.560	−1.23152	−0.615758	+	0.787935i	$0.711150\pi$
$32$	0	0
$33$	364.916	1.92496
$34$	0	0
$35$	156.754	0.757038
$36$	0	0
$37$	−174.423	−0.775000	−0.387500	−	0.921870i	$-0.626661\pi$
$37$	−174.423	−0.775000	−0.387500	+	0.921870i	$0.626661\pi$
$38$	0	0
$39$	388.738	1.59610
$40$	0	0
$41$	209.045	0.796277	0.398139	−	0.917325i	$-0.369656\pi$
$41$	209.045	0.796277	0.398139	+	0.917325i	$0.369656\pi$
$42$	0	0
$43$	150.350	0.533214	0.266607	−	0.963805i	$-0.414097\pi$
$43$	150.350	0.533214	0.266607	+	0.963805i	$0.414097\pi$
$44$	0	0
$45$	28.4541	0.0942595
$46$	0	0
$47$	−252.201	−0.782708	−0.391354	−	0.920240i	$-0.627993\pi$
$47$	−252.201	−0.782708	−0.391354	+	0.920240i	$0.627993\pi$
$48$	0	0
$49$	−33.9257	−0.0989086
$50$	0	0
$51$	93.4091	0.256468
$52$	0	0
$53$	−60.6076	−0.157077	−0.0785386	−	0.996911i	$-0.525025\pi$
$53$	−60.6076	−0.157077	−0.0785386	+	0.996911i	$0.525025\pi$
$54$	0	0
$55$	−592.163	−1.45177
$56$	0	0
$57$	−550.164	−1.27844
$58$	0	0
$59$	626.948	1.38342	0.691709	−	0.722176i	$-0.256858\pi$
$59$	626.948	1.38342	0.691709	+	0.722176i	$0.256858\pi$
$60$	0	0
$61$	624.906	1.31166	0.655828	−	0.754911i	$-0.272320\pi$
$61$	624.906	1.31166	0.655828	+	0.754911i	$0.272320\pi$
$62$	0	0
$63$	56.1031	0.112196
$64$	0	0
$65$	−630.820	−1.20375
$66$	0	0
$67$	114.386	0.208575	0.104287	−	0.994547i	$-0.466744\pi$
$67$	114.386	0.208575	0.104287	+	0.994547i	$0.466744\pi$
$68$	0	0
$69$	−1197.42	−2.08917
$70$	0	0
$71$	346.071	0.578466	0.289233	−	0.957259i	$-0.406600\pi$
$71$	346.071	0.578466	0.289233	+	0.957259i	$0.406600\pi$
$72$	0	0
$73$	−756.821	−1.21341	−0.606707	−	0.794925i	$-0.707510\pi$
$73$	−756.821	−1.21341	−0.606707	+	0.794925i	$0.707510\pi$
$74$	0	0
$75$	249.997	0.384895
$76$	0	0
$77$	−1167.57	−1.72802
$78$	0	0
$79$	−146.530	−0.208682	−0.104341	−	0.994542i	$-0.533273\pi$
$79$	−146.530	−0.208682	−0.104341	+	0.994542i	$0.533273\pi$
$80$	0	0
$81$	−804.979	−1.10422
$82$	0	0
$83$	−328.488	−0.434412	−0.217206	−	0.976126i	$-0.569694\pi$
$83$	−328.488	−0.434412	−0.217206	+	0.976126i	$0.569694\pi$
$84$	0	0
$85$	−151.578	−0.193423
$86$	0	0
$87$	126.588	0.155996
$88$	0	0
$89$	−583.027	−0.694390	−0.347195	−	0.937793i	$-0.612866\pi$
$89$	−583.027	−0.694390	−0.347195	+	0.937793i	$0.612866\pi$
$90$	0	0
$91$	−1243.79	−1.43280
$92$	0	0
$93$	1167.95	1.30226
$94$	0	0
$95$	892.772	0.964173
$96$	0	0
$97$	1620.95	1.69672	0.848362	−	0.529416i	$-0.177589\pi$
$97$	1620.95	1.69672	0.848362	+	0.529416i	$0.177589\pi$
$98$	0	0
$99$	−211.938	−0.215157
$100$	0	0
$101$	897.227	0.883935	0.441968	−	0.897031i	$-0.354281\pi$
$101$	897.227	0.883935	0.441968	+	0.897031i	$0.354281\pi$
$102$	0	0
$103$	1327.36	1.26979	0.634896	−	0.772597i	$-0.281043\pi$
$103$	1327.36	1.26979	0.634896	+	0.772597i	$0.281043\pi$
$104$	0	0
$105$	−861.311	−0.800527
$106$	0	0
$107$	−201.601	−0.182145	−0.0910727	−	0.995844i	$-0.529030\pi$
$107$	−201.601	−0.182145	−0.0910727	+	0.995844i	$0.529030\pi$
$108$	0	0
$109$	734.270	0.645232	0.322616	−	0.946530i	$-0.395438\pi$
$109$	734.270	0.645232	0.322616	+	0.946530i	$0.395438\pi$
$110$	0	0
$111$	958.395	0.819521
$112$	0	0
$113$	1679.39	1.39808	0.699042	−	0.715080i	$-0.253610\pi$
$113$	1679.39	1.39808	0.699042	+	0.715080i	$0.253610\pi$
$114$	0	0
$115$	1943.10	1.57561
$116$	0	0
$117$	−225.773	−0.178400
$118$	0	0
$119$	−298.869	−0.230229
$120$	0	0
$121$	3079.68	2.31381
$122$	0	0
$123$	−1148.63	−0.842021
$124$	0	0
$125$	−1520.23	−1.08779
$126$	0	0
$127$	145.675	0.101784	0.0508922	−	0.998704i	$-0.483794\pi$
$127$	145.675	0.101784	0.0508922	+	0.998704i	$0.483794\pi$
$128$	0	0
$129$	−826.123	−0.563846
$130$	0	0
$131$	1844.28	1.23004	0.615020	−	0.788512i	$-0.289148\pi$
$131$	1844.28	1.23004	0.615020	+	0.788512i	$0.289148\pi$
$132$	0	0
$133$	1760.29	1.14764
$134$	0	0
$135$	1166.45	0.743644
$136$	0	0
$137$	−1814.20	−1.13137	−0.565684	−	0.824622i	$-0.691388\pi$
$137$	−1814.20	−1.13137	−0.565684	+	0.824622i	$0.691388\pi$
$138$	0	0
$139$	552.260	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$139$	552.260	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$140$	0	0
$141$	1385.76	0.827672
$142$	0	0
$143$	4698.61	2.74768
$144$	0	0
$145$	−205.419	−0.117649
$146$	0	0
$147$	186.410	0.104591
$148$	0	0
$149$	788.674	0.433629	0.216814	−	0.976213i	$-0.430433\pi$
$149$	788.674	0.433629	0.216814	+	0.976213i	$0.430433\pi$
$150$	0	0
$151$	−1991.69	−1.07339	−0.536694	−	0.843777i	$-0.680327\pi$
$151$	−1991.69	−1.07339	−0.536694	+	0.843777i	$0.680327\pi$
$152$	0	0
$153$	−54.2506	−0.0286660
$154$	0	0
$155$	−1895.27	−0.982140
$156$	0	0
$157$	681.129	0.346242	0.173121	−	0.984901i	$-0.444615\pi$
$157$	681.129	0.346242	0.173121	+	0.984901i	$0.444615\pi$
$158$	0	0
$159$	333.018	0.166101
$160$	0	0
$161$	3831.23	1.87542
$162$	0	0
$163$	3272.98	1.57276	0.786380	−	0.617743i	$-0.211953\pi$
$163$	3272.98	1.57276	0.786380	+	0.617743i	$0.211953\pi$
$164$	0	0
$165$	3253.73	1.53517
$166$	0	0
$167$	697.294	0.323103	0.161552	−	0.986864i	$-0.448350\pi$
$167$	697.294	0.323103	0.161552	+	0.986864i	$0.448350\pi$
$168$	0	0
$169$	2808.34	1.27826
$170$	0	0
$171$	319.527	0.142894
$172$	0	0
$173$	−3173.78	−1.39479	−0.697394	−	0.716688i	$-0.745657\pi$
$173$	−3173.78	−1.39479	−0.697394	+	0.716688i	$0.745657\pi$
$174$	0	0
$175$	−799.882	−0.345517
$176$	0	0
$177$	−3444.86	−1.46289
$178$	0	0
$179$	−859.994	−0.359101	−0.179550	−	0.983749i	$-0.557464\pi$
$179$	−859.994	−0.359101	−0.179550	+	0.983749i	$0.557464\pi$
$180$	0	0
$181$	1947.39	0.799717	0.399858	−	0.916577i	$-0.369059\pi$
$181$	1947.39	0.799717	0.399858	+	0.916577i	$0.369059\pi$
$182$	0	0
$183$	−3433.64	−1.38701
$184$	0	0
$185$	−1555.22	−0.618066
$186$	0	0
$187$	1129.02	0.441509
$188$	0	0
$189$	2299.90	0.885149
$190$	0	0
$191$	4626.29	1.75260	0.876300	−	0.481766i	$-0.160005\pi$
$191$	4626.29	1.75260	0.876300	+	0.481766i	$0.160005\pi$
$192$	0	0
$193$	3169.27	1.18202	0.591008	−	0.806666i	$-0.298730\pi$
$193$	3169.27	1.18202	0.591008	+	0.806666i	$0.298730\pi$
$194$	0	0
$195$	3466.14	1.27290
$196$	0	0
$197$	−4537.05	−1.64087	−0.820435	−	0.571740i	$-0.806268\pi$
$197$	−4537.05	−1.64087	−0.820435	+	0.571740i	$0.806268\pi$
$198$	0	0
$199$	35.3976	0.0126094	0.00630469	−	0.999980i	$-0.497993\pi$
$199$	35.3976	0.0126094	0.00630469	+	0.999980i	$0.497993\pi$
$200$	0	0
$201$	−628.513	−0.220557
$202$	0	0
$203$	−405.028	−0.140036
$204$	0	0
$205$	1863.93	0.635035
$206$	0	0
$207$	695.444	0.233511
$208$	0	0
$209$	−6649.74	−2.20082
$210$	0	0
$211$	2055.80	0.670746	0.335373	−	0.942085i	$-0.391138\pi$
$211$	2055.80	0.670746	0.335373	+	0.942085i	$0.391138\pi$
$212$	0	0
$213$	−1901.54	−0.611697
$214$	0	0
$215$	1340.58	0.425241
$216$	0	0
$217$	−3736.92	−1.16903
$218$	0	0
$219$	4158.47	1.28312
$220$	0	0
$221$	1202.72	0.366081
$222$	0	0
$223$	−2948.14	−0.885302	−0.442651	−	0.896694i	$-0.645962\pi$
$223$	−2948.14	−0.885302	−0.442651	+	0.896694i	$0.645962\pi$
$224$	0	0
$225$	−145.194	−0.0430206
$226$	0	0
$227$	4596.69	1.34402	0.672011	−	0.740541i	$-0.265431\pi$
$227$	4596.69	1.34402	0.672011	+	0.740541i	$0.265431\pi$
$228$	0	0
$229$	54.4924	0.0157247	0.00786236	−	0.999969i	$-0.497497\pi$
$229$	54.4924	0.0157247	0.00786236	+	0.999969i	$0.497497\pi$
$230$	0	0
$231$	6415.41	1.82729
$232$	0	0
$233$	1502.17	0.422361	0.211181	−	0.977447i	$-0.432269\pi$
$233$	1502.17	0.422361	0.211181	+	0.977447i	$0.432269\pi$
$234$	0	0
$235$	−2248.72	−0.624214
$236$	0	0
$237$	805.130	0.220670
$238$	0	0
$239$	39.7347	0.0107541	0.00537704	−	0.999986i	$-0.498288\pi$
$239$	39.7347	0.0107541	0.00537704	+	0.999986i	$0.498288\pi$
$240$	0	0
$241$	2420.53	0.646972	0.323486	−	0.946233i	$-0.395145\pi$
$241$	2420.53	0.646972	0.323486	+	0.946233i	$0.395145\pi$
$242$	0	0
$243$	890.912	0.235194
$244$	0	0
$245$	−302.494	−0.0788801
$246$	0	0
$247$	−7083.84	−1.82484
$248$	0	0
$249$	1804.93	0.459367
$250$	0	0
$251$	1769.52	0.444984	0.222492	−	0.974935i	$-0.428581\pi$
$251$	1769.52	0.444984	0.222492	+	0.974935i	$0.428581\pi$
$252$	0	0
$253$	−14473.0	−3.59648
$254$	0	0
$255$	832.871	0.204535
$256$	0	0
$257$	5771.80	1.40091	0.700457	−	0.713694i	$-0.252979\pi$
$257$	5771.80	1.40091	0.700457	+	0.713694i	$0.252979\pi$
$258$	0	0
$259$	−3066.45	−0.735675
$260$	0	0
$261$	−73.5206	−0.0174360
$262$	0	0
$263$	−5175.24	−1.21338	−0.606690	−	0.794939i	$-0.707503\pi$
$263$	−5175.24	−1.21338	−0.606690	+	0.794939i	$0.707503\pi$
$264$	0	0
$265$	−540.400	−0.125270
$266$	0	0
$267$	3203.53	0.734281
$268$	0	0
$269$	−336.748	−0.0763266	−0.0381633	−	0.999272i	$-0.512151\pi$
$269$	−336.748	−0.0763266	−0.0381633	+	0.999272i	$0.512151\pi$
$270$	0	0
$271$	6323.01	1.41733	0.708664	−	0.705547i	$-0.249298\pi$
$271$	6323.01	1.41733	0.708664	+	0.705547i	$0.249298\pi$
$272$	0	0
$273$	6834.21	1.51511
$274$	0	0
$275$	3021.67	0.662595
$276$	0	0
$277$	3969.95	0.861124	0.430562	−	0.902561i	$-0.358315\pi$
$277$	3969.95	0.861124	0.430562	+	0.902561i	$0.358315\pi$
$278$	0	0
$279$	−678.326	−0.145557
$280$	0	0
$281$	2917.44	0.619360	0.309680	−	0.950841i	$-0.399778\pi$
$281$	2917.44	0.619360	0.309680	+	0.950841i	$0.399778\pi$
$282$	0	0
$283$	7294.66	1.53224	0.766118	−	0.642700i	$-0.222186\pi$
$283$	7294.66	1.53224	0.766118	+	0.642700i	$0.222186\pi$
$284$	0	0
$285$	−4905.47	−1.01956
$286$	0	0
$287$	3675.12	0.755873
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−8906.54	−1.79419
$292$	0	0
$293$	−4296.30	−0.856629	−0.428315	−	0.903630i	$-0.640892\pi$
$293$	−4296.30	−0.856629	−0.428315	+	0.903630i	$0.640892\pi$
$294$	0	0
$295$	5590.11	1.10328
$296$	0	0
$297$	−8688.21	−1.69744
$298$	0	0
$299$	−15417.8	−2.98206
$300$	0	0
$301$	2643.24	0.506158
$302$	0	0
$303$	−4929.95	−0.934714
$304$	0	0
$305$	5571.89	1.04605
$306$	0	0
$307$	−1443.87	−0.268423	−0.134211	−	0.990953i	$-0.542850\pi$
$307$	−1443.87	−0.268423	−0.134211	+	0.990953i	$0.542850\pi$
$308$	0	0
$309$	−7293.38	−1.34274
$310$	0	0
$311$	−5690.86	−1.03762	−0.518809	−	0.854890i	$-0.673624\pi$
$311$	−5690.86	−1.03762	−0.518809	+	0.854890i	$0.673624\pi$
$312$	0	0
$313$	−2494.40	−0.450453	−0.225226	−	0.974306i	$-0.572312\pi$
$313$	−2494.40	−0.450453	−0.225226	+	0.974306i	$0.572312\pi$
$314$	0	0
$315$	500.237	0.0894767
$316$	0	0
$317$	8024.35	1.42174	0.710871	−	0.703322i	$-0.248301\pi$
$317$	8024.35	1.42174	0.710871	+	0.703322i	$0.248301\pi$
$318$	0	0
$319$	1530.05	0.268547
$320$	0	0
$321$	1107.73	0.192609
$322$	0	0
$323$	−1702.16	−0.293223
$324$	0	0
$325$	3218.93	0.549397
$326$	0	0
$327$	−4034.56	−0.682298
$328$	0	0
$329$	−4433.82	−0.742993
$330$	0	0
$331$	−568.277	−0.0943665	−0.0471833	−	0.998886i	$-0.515024\pi$
$331$	−568.277	−0.0943665	−0.0471833	+	0.998886i	$0.515024\pi$
$332$	0	0
$333$	−556.622	−0.0915996
$334$	0	0
$335$	1019.91	0.166339
$336$	0	0
$337$	−7384.12	−1.19359	−0.596793	−	0.802395i	$-0.703559\pi$
$337$	−7384.12	−1.19359	−0.596793	+	0.802395i	$0.703559\pi$
$338$	0	0
$339$	−9227.66	−1.47840
$340$	0	0
$341$	14116.8	2.24183
$342$	0	0
$343$	−6626.55	−1.04315
$344$	0	0
$345$	−10676.6	−1.66612
$346$	0	0
$347$	8038.59	1.24361	0.621807	−	0.783171i	$-0.286399\pi$
$347$	8038.59	1.24361	0.621807	+	0.783171i	$0.286399\pi$
$348$	0	0
$349$	2838.19	0.435315	0.217658	−	0.976025i	$-0.430158\pi$
$349$	2838.19	0.435315	0.217658	+	0.976025i	$0.430158\pi$
$350$	0	0
$351$	−9255.38	−1.40745
$352$	0	0
$353$	3194.83	0.481709	0.240855	−	0.970561i	$-0.422572\pi$
$353$	3194.83	0.481709	0.240855	+	0.970561i	$0.422572\pi$
$354$	0	0
$355$	3085.70	0.461330
$356$	0	0
$357$	1642.18	0.243455
$358$	0	0
$359$	3010.02	0.442515	0.221258	−	0.975215i	$-0.428984\pi$
$359$	3010.02	0.442515	0.221258	+	0.975215i	$0.428984\pi$
$360$	0	0
$361$	3166.46	0.461650
$362$	0	0
$363$	−16921.8	−2.44673
$364$	0	0
$365$	−6748.10	−0.967704
$366$	0	0
$367$	8472.22	1.20503	0.602515	−	0.798107i	$-0.294165\pi$
$367$	8472.22	1.20503	0.602515	+	0.798107i	$0.294165\pi$
$368$	0	0
$369$	667.107	0.0941144
$370$	0	0
$371$	−1065.51	−0.149107
$372$	0	0
$373$	−2770.27	−0.384556	−0.192278	−	0.981341i	$-0.561588\pi$
$373$	−2770.27	−0.384556	−0.192278	+	0.981341i	$0.561588\pi$
$374$	0	0
$375$	8353.12	1.15027
$376$	0	0
$377$	1629.93	0.222668
$378$	0	0
$379$	−2912.82	−0.394780	−0.197390	−	0.980325i	$-0.563247\pi$
$379$	−2912.82	−0.394780	−0.197390	+	0.980325i	$0.563247\pi$
$380$	0	0
$381$	−800.436	−0.107631
$382$	0	0
$383$	2599.44	0.346802	0.173401	−	0.984851i	$-0.444524\pi$
$383$	2599.44	0.346802	0.173401	+	0.984851i	$0.444524\pi$
$384$	0	0
$385$	−10410.5	−1.37810
$386$	0	0
$387$	479.800	0.0630222
$388$	0	0
$389$	9402.65	1.22554	0.612768	−	0.790263i	$-0.290056\pi$
$389$	9402.65	1.22554	0.612768	+	0.790263i	$0.290056\pi$
$390$	0	0
$391$	−3704.72	−0.479171
$392$	0	0
$393$	−10133.7	−1.30070
$394$	0	0
$395$	−1306.51	−0.166425
$396$	0	0
$397$	−9090.93	−1.14927	−0.574636	−	0.818409i	$-0.694856\pi$
$397$	−9090.93	−1.14927	−0.574636	+	0.818409i	$0.694856\pi$
$398$	0	0
$399$	−9672.17	−1.21357
$400$	0	0
$401$	−11718.2	−1.45930	−0.729649	−	0.683822i	$-0.760317\pi$
$401$	−11718.2	−1.45930	−0.729649	+	0.683822i	$0.760317\pi$
$402$	0	0
$403$	15038.3	1.85884
$404$	0	0
$405$	−7177.50	−0.880624
$406$	0	0
$407$	11584.0	1.41080
$408$	0	0
$409$	12694.6	1.53473	0.767367	−	0.641208i	$-0.221566\pi$
$409$	12694.6	1.53473	0.767367	+	0.641208i	$0.221566\pi$
$410$	0	0
$411$	9968.40	1.19636
$412$	0	0
$413$	11022.1	1.31322
$414$	0	0
$415$	−2928.92	−0.346446
$416$	0	0
$417$	−3034.48	−0.356353
$418$	0	0
$419$	−10688.0	−1.24616	−0.623079	−	0.782159i	$-0.714119\pi$
$419$	−10688.0	−1.24616	−0.623079	+	0.782159i	$0.714119\pi$
$420$	0	0
$421$	−3078.97	−0.356437	−0.178218	−	0.983991i	$-0.557033\pi$
$421$	−3078.97	−0.356437	−0.178218	+	0.983991i	$0.557033\pi$
$422$	0	0
$423$	−804.827	−0.0925107
$424$	0	0
$425$	773.470	0.0882795
$426$	0	0
$427$	10986.2	1.24510
$428$	0	0
$429$	−25817.2	−2.90552
$430$	0	0
$431$	−11143.4	−1.24537	−0.622687	−	0.782471i	$-0.713959\pi$
$431$	−11143.4	−1.24537	−0.622687	+	0.782471i	$0.713959\pi$
$432$	0	0
$433$	−683.362	−0.0758436	−0.0379218	−	0.999281i	$-0.512074\pi$
$433$	−683.362	−0.0758436	−0.0379218	+	0.999281i	$0.512074\pi$
$434$	0	0
$435$	1128.71	0.124408
$436$	0	0
$437$	21820.2	2.38856
$438$	0	0
$439$	1464.68	0.159238	0.0796189	−	0.996825i	$-0.474630\pi$
$439$	1464.68	0.159238	0.0796189	+	0.996825i	$0.474630\pi$
$440$	0	0
$441$	−108.264	−0.0116903
$442$	0	0
$443$	614.092	0.0658609	0.0329304	−	0.999458i	$-0.489516\pi$
$443$	614.092	0.0658609	0.0329304	+	0.999458i	$0.489516\pi$
$444$	0	0
$445$	−5198.49	−0.553780
$446$	0	0
$447$	−4333.49	−0.458539
$448$	0	0
$449$	−8814.90	−0.926505	−0.463252	−	0.886226i	$-0.653318\pi$
$449$	−8814.90	−0.926505	−0.463252	+	0.886226i	$0.653318\pi$
$450$	0	0
$451$	−13883.3	−1.44953
$452$	0	0
$453$	10943.7	1.13505
$454$	0	0
$455$	−11090.1	−1.14267
$456$	0	0
$457$	16470.3	1.68588	0.842942	−	0.538004i	$-0.180821\pi$
$457$	16470.3	1.68588	0.842942	+	0.538004i	$0.180821\pi$
$458$	0	0
$459$	−2223.96	−0.226156
$460$	0	0
$461$	3207.17	0.324019	0.162010	−	0.986789i	$-0.448202\pi$
$461$	3207.17	0.324019	0.162010	+	0.986789i	$0.448202\pi$
$462$	0	0
$463$	−358.409	−0.0359755	−0.0179878	−	0.999838i	$-0.505726\pi$
$463$	−358.409	−0.0359755	−0.0179878	+	0.999838i	$0.505726\pi$
$464$	0	0
$465$	10413.8	1.03856
$466$	0	0
$467$	1165.88	0.115525	0.0577627	−	0.998330i	$-0.481603\pi$
$467$	1165.88	0.115525	0.0577627	+	0.998330i	$0.481603\pi$
$468$	0	0
$469$	2010.97	0.197991
$470$	0	0
$471$	−3742.57	−0.366133
$472$	0	0
$473$	−9985.21	−0.970656
$474$	0	0
$475$	−4555.61	−0.440054
$476$	0	0
$477$	−193.412	−0.0185654
$478$	0	0
$479$	7603.13	0.725253	0.362626	−	0.931935i	$-0.381880\pi$
$479$	7603.13	0.725253	0.362626	+	0.931935i	$0.381880\pi$
$480$	0	0
$481$	12340.2	1.16978
$482$	0	0
$483$	−21051.3	−1.98316
$484$	0	0
$485$	14453.0	1.35315
$486$	0	0
$487$	273.581	0.0254561	0.0127281	−	0.999919i	$-0.495948\pi$
$487$	273.581	0.0254561	0.0127281	+	0.999919i	$0.495948\pi$
$488$	0	0
$489$	−17983.9	−1.66311
$490$	0	0
$491$	−10321.5	−0.948680	−0.474340	−	0.880342i	$-0.657313\pi$
$491$	−10321.5	−0.948680	−0.474340	+	0.880342i	$0.657313\pi$
$492$	0	0
$493$	391.654	0.0357793
$494$	0	0
$495$	−1889.72	−0.171589
$496$	0	0
$497$	6084.11	0.549114
$498$	0	0
$499$	−164.611	−0.0147676	−0.00738379	−	0.999973i	$-0.502350\pi$
$499$	−164.611	−0.0147676	−0.00738379	+	0.999973i	$0.502350\pi$
$500$	0	0
$501$	−3831.39	−0.341664
$502$	0	0
$503$	2478.45	0.219699	0.109849	−	0.993948i	$-0.464963\pi$
$503$	2478.45	0.219699	0.109849	+	0.993948i	$0.464963\pi$
$504$	0	0
$505$	8000.02	0.704943
$506$	0	0
$507$	−15430.9	−1.35169
$508$	0	0
$509$	−16976.8	−1.47836	−0.739179	−	0.673509i	$-0.764786\pi$
$509$	−16976.8	−1.47836	−0.739179	+	0.673509i	$0.764786\pi$
$510$	0	0
$511$	−13305.3	−1.15184
$512$	0	0
$513$	13098.7	1.12734
$514$	0	0
$515$	11835.2	1.01267
$516$	0	0
$517$	16749.4	1.42483
$518$	0	0
$519$	17438.8	1.47491
$520$	0	0
$521$	10468.6	0.880300	0.440150	−	0.897924i	$-0.354925\pi$
$521$	10468.6	0.880300	0.440150	+	0.897924i	$0.354925\pi$
$522$	0	0
$523$	15665.7	1.30978	0.654890	−	0.755724i	$-0.272715\pi$
$523$	15665.7	1.30978	0.654890	+	0.755724i	$0.272715\pi$
$524$	0	0
$525$	4395.07	0.365365
$526$	0	0
$527$	3613.53	0.298686
$528$	0	0
$529$	35324.1	2.90327
$530$	0	0
$531$	2000.72	0.163510
$532$	0	0
$533$	−14789.6	−1.20189
$534$	0	0
$535$	−1797.55	−0.145262
$536$	0	0
$537$	4725.37	0.379730
$538$	0	0
$539$	2253.10	0.180052
$540$	0	0
$541$	−1809.97	−0.143839	−0.0719194	−	0.997410i	$-0.522912\pi$
$541$	−1809.97	−0.143839	−0.0719194	+	0.997410i	$0.522912\pi$
$542$	0	0
$543$	−10700.3	−0.845658
$544$	0	0
$545$	6547.03	0.514576
$546$	0	0
$547$	−2511.47	−0.196312	−0.0981559	−	0.995171i	$-0.531294\pi$
$547$	−2511.47	−0.196312	−0.0981559	+	0.995171i	$0.531294\pi$
$548$	0	0
$549$	1994.21	0.155029
$550$	0	0
$551$	−2306.78	−0.178352
$552$	0	0
$553$	−2576.07	−0.198093
$554$	0	0
$555$	8545.41	0.653572
$556$	0	0
$557$	7702.34	0.585922	0.292961	−	0.956124i	$-0.405359\pi$
$557$	7702.34	0.585922	0.292961	+	0.956124i	$0.405359\pi$
$558$	0	0
$559$	−10637.1	−0.804829
$560$	0	0
$561$	−6203.57	−0.466872
$562$	0	0
$563$	−7351.64	−0.550328	−0.275164	−	0.961397i	$-0.588732\pi$
$563$	−7351.64	−0.550328	−0.275164	+	0.961397i	$0.588732\pi$
$564$	0	0
$565$	14974.1	1.11498
$566$	0	0
$567$	−14151.9	−1.04819
$568$	0	0
$569$	4307.15	0.317337	0.158669	−	0.987332i	$-0.449280\pi$
$569$	4307.15	0.317337	0.158669	+	0.987332i	$0.449280\pi$
$570$	0	0
$571$	−24101.6	−1.76641	−0.883205	−	0.468988i	$-0.844619\pi$
$571$	−24101.6	−1.76641	−0.883205	+	0.468988i	$0.844619\pi$
$572$	0	0
$573$	−25419.9	−1.85328
$574$	0	0
$575$	−9915.18	−0.719116
$576$	0	0
$577$	−24395.7	−1.76015	−0.880075	−	0.474835i	$-0.842508\pi$
$577$	−24395.7	−1.76015	−0.880075	+	0.474835i	$0.842508\pi$
$578$	0	0
$579$	−17414.0	−1.24992
$580$	0	0
$581$	−5774.98	−0.412369
$582$	0	0
$583$	4025.13	0.285941
$584$	0	0
$585$	−2013.08	−0.142275
$586$	0	0
$587$	−11785.0	−0.828650	−0.414325	−	0.910129i	$-0.635982\pi$
$587$	−11785.0	−0.828650	−0.414325	+	0.910129i	$0.635982\pi$
$588$	0	0
$589$	−21283.1	−1.48889
$590$	0	0
$591$	24929.5	1.73513
$592$	0	0
$593$	12910.4	0.894040	0.447020	−	0.894524i	$-0.352485\pi$
$593$	12910.4	0.894040	0.447020	+	0.894524i	$0.352485\pi$
$594$	0	0
$595$	−2664.83	−0.183609
$596$	0	0
$597$	−194.497	−0.0133337
$598$	0	0
$599$	−21704.4	−1.48050	−0.740249	−	0.672333i	$-0.765292\pi$
$599$	−21704.4	−1.48050	−0.740249	+	0.672333i	$0.765292\pi$
$600$	0	0
$601$	5383.92	0.365416	0.182708	−	0.983167i	$-0.441514\pi$
$601$	5383.92	0.365416	0.182708	+	0.983167i	$0.441514\pi$
$602$	0	0
$603$	365.031	0.0246521
$604$	0	0
$605$	27459.6	1.84527
$606$	0	0
$607$	−10689.2	−0.714762	−0.357381	−	0.933959i	$-0.616330\pi$
$607$	−10689.2	−0.714762	−0.357381	+	0.933959i	$0.616330\pi$
$608$	0	0
$609$	2225.49	0.148081
$610$	0	0
$611$	17842.8	1.18141
$612$	0	0
$613$	−23689.4	−1.56086	−0.780429	−	0.625244i	$-0.784999\pi$
$613$	−23689.4	−1.56086	−0.780429	+	0.625244i	$0.784999\pi$
$614$	0	0
$615$	−10241.6	−0.671516
$616$	0	0
$617$	−13910.1	−0.907617	−0.453809	−	0.891099i	$-0.649935\pi$
$617$	−13910.1	−0.907617	−0.453809	+	0.891099i	$0.649935\pi$
$618$	0	0
$619$	−10555.2	−0.685377	−0.342689	−	0.939449i	$-0.611338\pi$
$619$	−10555.2	−0.685377	−0.342689	+	0.939449i	$0.611338\pi$
$620$	0	0
$621$	28509.1	1.84224
$622$	0	0
$623$	−10249.9	−0.659156
$624$	0	0
$625$	−7867.63	−0.503529
$626$	0	0
$627$	36538.0	2.32725
$628$	0	0
$629$	2965.19	0.187965
$630$	0	0
$631$	13599.4	0.857979	0.428990	−	0.903309i	$-0.358870\pi$
$631$	13599.4	0.857979	0.428990	+	0.903309i	$0.358870\pi$
$632$	0	0
$633$	−11295.9	−0.709278
$634$	0	0
$635$	1298.90	0.0811735
$636$	0	0
$637$	2400.19	0.149292
$638$	0	0
$639$	1104.39	0.0683707
$640$	0	0
$641$	17474.2	1.07674	0.538370	−	0.842708i	$-0.319040\pi$
$641$	17474.2	1.07674	0.538370	+	0.842708i	$0.319040\pi$
$642$	0	0
$643$	26232.8	1.60890	0.804450	−	0.594021i	$-0.202460\pi$
$643$	26232.8	1.60890	0.804450	+	0.594021i	$0.202460\pi$
$644$	0	0
$645$	−7366.03	−0.449670
$646$	0	0
$647$	−21466.2	−1.30436	−0.652182	−	0.758063i	$-0.726146\pi$
$647$	−21466.2	−1.30436	−0.652182	+	0.758063i	$0.726146\pi$
$648$	0	0
$649$	−41637.5	−2.51836
$650$	0	0
$651$	20533.1	1.23618
$652$	0	0
$653$	−13855.3	−0.830323	−0.415161	−	0.909748i	$-0.636275\pi$
$653$	−13855.3	−0.830323	−0.415161	+	0.909748i	$0.636275\pi$
$654$	0	0
$655$	16444.3	0.980963
$656$	0	0
$657$	−2415.18	−0.143417
$658$	0	0
$659$	−1174.37	−0.0694186	−0.0347093	−	0.999397i	$-0.511051\pi$
$659$	−1174.37	−0.0694186	−0.0347093	+	0.999397i	$0.511051\pi$
$660$	0	0
$661$	5020.31	0.295412	0.147706	−	0.989031i	$-0.452811\pi$
$661$	5020.31	0.295412	0.147706	+	0.989031i	$0.452811\pi$
$662$	0	0
$663$	−6608.55	−0.387111
$664$	0	0
$665$	15695.4	0.915250
$666$	0	0
$667$	−5020.64	−0.291454
$668$	0	0
$669$	16199.0	0.936159
$670$	0	0
$671$	−41501.8	−2.38772
$672$	0	0
$673$	25279.2	1.44791	0.723955	−	0.689848i	$-0.242322\pi$
$673$	25279.2	1.44791	0.723955	+	0.689848i	$0.242322\pi$
$674$	0	0
$675$	−5952.12	−0.339404
$676$	0	0
$677$	15082.4	0.856225	0.428113	−	0.903725i	$-0.359179\pi$
$677$	15082.4	0.856225	0.428113	+	0.903725i	$0.359179\pi$
$678$	0	0
$679$	28497.1	1.61063
$680$	0	0
$681$	−25257.2	−1.42123
$682$	0	0
$683$	−19353.7	−1.08426	−0.542129	−	0.840295i	$-0.682382\pi$
$683$	−19353.7	−1.08426	−0.542129	+	0.840295i	$0.682382\pi$
$684$	0	0
$685$	−16176.1	−0.902272
$686$	0	0
$687$	−299.417	−0.0166281
$688$	0	0
$689$	4287.89	0.237091
$690$	0	0
$691$	−26175.3	−1.44103	−0.720517	−	0.693437i	$-0.756095\pi$
$691$	−26175.3	−1.44103	−0.720517	+	0.693437i	$0.756095\pi$
$692$	0	0
$693$	−3725.97	−0.204240
$694$	0	0
$695$	4924.16	0.268754
$696$	0	0
$697$	−3553.77	−0.193126
$698$	0	0
$699$	−8253.88	−0.446624
$700$	0	0
$701$	−20440.4	−1.10132	−0.550659	−	0.834730i	$-0.685623\pi$
$701$	−20440.4	−1.10132	−0.550659	+	0.834730i	$0.685623\pi$
$702$	0	0
$703$	−17464.5	−0.936965
$704$	0	0
$705$	12355.9	0.660073
$706$	0	0
$707$	15773.7	0.839083
$708$	0	0
$709$	−28524.0	−1.51092	−0.755460	−	0.655195i	$-0.772586\pi$
$709$	−28524.0	−1.51092	−0.755460	+	0.655195i	$0.772586\pi$
$710$	0	0
$711$	−467.608	−0.0246648
$712$	0	0
$713$	−46322.2	−2.43307
$714$	0	0
$715$	41894.6	2.19128
$716$	0	0
$717$	−218.329	−0.0113719
$718$	0	0
$719$	25448.8	1.32000	0.660000	−	0.751266i	$-0.270556\pi$
$719$	25448.8	1.32000	0.660000	+	0.751266i	$0.270556\pi$
$720$	0	0
$721$	23335.7	1.20536
$722$	0	0
$723$	−13300.0	−0.684138
$724$	0	0
$725$	1048.21	0.0536958
$726$	0	0
$727$	−25519.4	−1.30187	−0.650937	−	0.759132i	$-0.725624\pi$
$727$	−25519.4	−1.30187	−0.650937	+	0.759132i	$0.725624\pi$
$728$	0	0
$729$	16839.2	0.855519
$730$	0	0
$731$	−2555.96	−0.129323
$732$	0	0
$733$	16888.0	0.850986	0.425493	−	0.904962i	$-0.360101\pi$
$733$	16888.0	0.850986	0.425493	+	0.904962i	$0.360101\pi$
$734$	0	0
$735$	1662.10	0.0834115
$736$	0	0
$737$	−7596.72	−0.379686
$738$	0	0
$739$	−30597.5	−1.52307	−0.761534	−	0.648125i	$-0.775554\pi$
$739$	−30597.5	−1.52307	−0.761534	+	0.648125i	$0.775554\pi$
$740$	0	0
$741$	38923.3	1.92967
$742$	0	0
$743$	5339.39	0.263638	0.131819	−	0.991274i	$-0.457918\pi$
$743$	5339.39	0.263638	0.131819	+	0.991274i	$0.457918\pi$
$744$	0	0
$745$	7032.12	0.345821
$746$	0	0
$747$	−1048.27	−0.0513445
$748$	0	0
$749$	−3544.26	−0.172903
$750$	0	0
$751$	11116.1	0.540121	0.270061	−	0.962843i	$-0.412956\pi$
$751$	11116.1	0.540121	0.270061	+	0.962843i	$0.412956\pi$
$752$	0	0
$753$	−9722.89	−0.470547
$754$	0	0
$755$	−17758.7	−0.856033
$756$	0	0
$757$	23575.3	1.13191	0.565956	−	0.824435i	$-0.308507\pi$
$757$	23575.3	1.13191	0.565956	+	0.824435i	$0.308507\pi$
$758$	0	0
$759$	79524.2	3.80309
$760$	0	0
$761$	3251.49	0.154884	0.0774418	−	0.996997i	$-0.475325\pi$
$761$	3251.49	0.154884	0.0774418	+	0.996997i	$0.475325\pi$
$762$	0	0
$763$	12908.8	0.612492
$764$	0	0
$765$	−483.719	−0.0228613
$766$	0	0
$767$	−44355.6	−2.08812
$768$	0	0
$769$	−15247.9	−0.715023	−0.357511	−	0.933909i	$-0.616375\pi$
$769$	−15247.9	−0.715023	−0.357511	+	0.933909i	$0.616375\pi$
$770$	0	0
$771$	−31714.0	−1.48139
$772$	0	0
$773$	−12654.3	−0.588803	−0.294401	−	0.955682i	$-0.595120\pi$
$773$	−12654.3	−0.588803	−0.294401	+	0.955682i	$0.595120\pi$
$774$	0	0
$775$	9671.12	0.448254
$776$	0	0
$777$	16849.1	0.777937
$778$	0	0
$779$	20931.1	0.962689
$780$	0	0
$781$	−22983.6	−1.05303
$782$	0	0
$783$	−3013.91	−0.137559
$784$	0	0
$785$	6073.20	0.276130
$786$	0	0
$787$	−35609.4	−1.61288	−0.806440	−	0.591316i	$-0.798609\pi$
$787$	−35609.4	−1.61288	−0.806440	+	0.591316i	$0.798609\pi$
$788$	0	0
$789$	28436.1	1.28308
$790$	0	0
$791$	29524.5	1.32714
$792$	0	0
$793$	−44211.1	−1.97980
$794$	0	0
$795$	2969.31	0.132466
$796$	0	0
$797$	−24383.5	−1.08370	−0.541849	−	0.840476i	$-0.682276\pi$
$797$	−24383.5	−1.08370	−0.541849	+	0.840476i	$0.682276\pi$
$798$	0	0
$799$	4287.42	0.189835
$800$	0	0
$801$	−1860.56	−0.0820721
$802$	0	0
$803$	50262.7	2.20888
$804$	0	0
$805$	34160.7	1.49566
$806$	0	0
$807$	1850.31	0.0807113
$808$	0	0
$809$	−41571.6	−1.80665	−0.903325	−	0.428956i	$-0.858882\pi$
$809$	−41571.6	−1.80665	−0.903325	+	0.428956i	$0.858882\pi$
$810$	0	0
$811$	34587.9	1.49759	0.748795	−	0.662801i	$-0.230633\pi$
$811$	34587.9	1.49759	0.748795	+	0.662801i	$0.230633\pi$
$812$	0	0
$813$	−34742.8	−1.49875
$814$	0	0
$815$	29183.2	1.25428
$816$	0	0
$817$	15054.2	0.644649
$818$	0	0
$819$	−3969.21	−0.169347
$820$	0	0
$821$	−41937.4	−1.78273	−0.891367	−	0.453282i	$-0.850253\pi$
$821$	−41937.4	−1.78273	−0.891367	+	0.453282i	$0.850253\pi$
$822$	0	0
$823$	39926.7	1.69108	0.845538	−	0.533915i	$-0.179280\pi$
$823$	39926.7	1.69108	0.845538	+	0.533915i	$0.179280\pi$
$824$	0	0
$825$	−16603.0	−0.700658
$826$	0	0
$827$	−6121.83	−0.257409	−0.128704	−	0.991683i	$-0.541082\pi$
$827$	−6121.83	−0.257409	−0.128704	+	0.991683i	$0.541082\pi$
$828$	0	0
$829$	23065.3	0.966334	0.483167	−	0.875528i	$-0.339486\pi$
$829$	23065.3	0.966334	0.483167	+	0.875528i	$0.339486\pi$
$830$	0	0
$831$	−21813.5	−0.910592
$832$	0	0
$833$	576.736	0.0239889
$834$	0	0
$835$	6217.33	0.257676
$836$	0	0
$837$	−27807.4	−1.14834
$838$	0	0
$839$	2896.04	0.119169	0.0595844	−	0.998223i	$-0.481022\pi$
$839$	2896.04	0.119169	0.0595844	+	0.998223i	$0.481022\pi$
$840$	0	0
$841$	−23858.2	−0.978237
$842$	0	0
$843$	−16030.3	−0.654940
$844$	0	0
$845$	25040.2	1.01942
$846$	0	0
$847$	54142.3	2.19640
$848$	0	0
$849$	−40081.6	−1.62026
$850$	0	0
$851$	−38011.1	−1.53114
$852$	0	0
$853$	−43149.5	−1.73202	−0.866009	−	0.500028i	$-0.833323\pi$
$853$	−43149.5	−1.73202	−0.866009	+	0.500028i	$0.833323\pi$
$854$	0	0
$855$	2849.02	0.113959
$856$	0	0
$857$	12611.9	0.502699	0.251350	−	0.967896i	$-0.419126\pi$
$857$	12611.9	0.502699	0.251350	+	0.967896i	$0.419126\pi$
$858$	0	0
$859$	42433.8	1.68547	0.842737	−	0.538325i	$-0.180943\pi$
$859$	42433.8	1.68547	0.842737	+	0.538325i	$0.180943\pi$
$860$	0	0
$861$	−20193.5	−0.799295
$862$	0	0
$863$	31133.7	1.22805	0.614024	−	0.789287i	$-0.289550\pi$
$863$	31133.7	1.22805	0.614024	+	0.789287i	$0.289550\pi$
$864$	0	0
$865$	−28298.7	−1.11235
$866$	0	0
$867$	−1587.95	−0.0622027
$868$	0	0
$869$	9731.47	0.379882
$870$	0	0
$871$	−8092.65	−0.314821
$872$	0	0
$873$	5172.79	0.200541
$874$	0	0
$875$	−26726.4	−1.03259
$876$	0	0
$877$	20001.4	0.770123	0.385061	−	0.922891i	$-0.374180\pi$
$877$	20001.4	0.770123	0.385061	+	0.922891i	$0.374180\pi$
$878$	0	0
$879$	23606.7	0.905839
$880$	0	0
$881$	542.075	0.0207298	0.0103649	−	0.999946i	$-0.496701\pi$
$881$	542.075	0.0207298	0.0103649	+	0.999946i	$0.496701\pi$
$882$	0	0
$883$	20445.5	0.779212	0.389606	−	0.920982i	$-0.372611\pi$
$883$	20445.5	0.779212	0.389606	+	0.920982i	$0.372611\pi$
$884$	0	0
$885$	−30715.7	−1.16666
$886$	0	0
$887$	−8038.69	−0.304298	−0.152149	−	0.988358i	$-0.548619\pi$
$887$	−8038.69	−0.304298	−0.152149	+	0.988358i	$0.548619\pi$
$888$	0	0
$889$	2561.05	0.0966196
$890$	0	0
$891$	53461.0	2.01011
$892$	0	0
$893$	−25252.2	−0.946285
$894$	0	0
$895$	−7668.03	−0.286384
$896$	0	0
$897$	84715.6	3.15337
$898$	0	0
$899$	4897.06	0.181675
$900$	0	0
$901$	1030.33	0.0380968
$902$	0	0
$903$	−14523.7	−0.535235
$904$	0	0
$905$	17363.7	0.637778
$906$	0	0
$907$	32912.0	1.20488	0.602440	−	0.798164i	$-0.294195\pi$
$907$	32912.0	1.20488	0.602440	+	0.798164i	$0.294195\pi$
$908$	0	0
$909$	2863.24	0.104475
$910$	0	0
$911$	2693.87	0.0979712	0.0489856	−	0.998799i	$-0.484401\pi$
$911$	2693.87	0.0979712	0.0489856	+	0.998799i	$0.484401\pi$
$912$	0	0
$913$	21815.8	0.790798
$914$	0	0
$915$	−30615.6	−1.10614
$916$	0	0
$917$	32423.3	1.16763
$918$	0	0
$919$	−1977.03	−0.0709642	−0.0354821	−	0.999370i	$-0.511297\pi$
$919$	−1977.03	−0.0709642	−0.0354821	+	0.999370i	$0.511297\pi$
$920$	0	0
$921$	7933.54	0.283843
$922$	0	0
$923$	−24484.0	−0.873132
$924$	0	0
$925$	7935.95	0.282089
$926$	0	0
$927$	4235.89	0.150081
$928$	0	0
$929$	−27648.8	−0.976457	−0.488229	−	0.872716i	$-0.662357\pi$
$929$	−27648.8	−0.976457	−0.488229	+	0.872716i	$0.662357\pi$
$930$	0	0
$931$	−3396.88	−0.119579
$932$	0	0
$933$	31269.3	1.09722
$934$	0	0
$935$	10066.8	0.352105
$936$	0	0
$937$	26761.2	0.933030	0.466515	−	0.884513i	$-0.345509\pi$
$937$	26761.2	0.933030	0.466515	+	0.884513i	$0.345509\pi$
$938$	0	0
$939$	13705.9	0.476330
$940$	0	0
$941$	−34719.5	−1.20279	−0.601394	−	0.798953i	$-0.705388\pi$
$941$	−34719.5	−1.20279	−0.601394	+	0.798953i	$0.705388\pi$
$942$	0	0
$943$	45556.1	1.57318
$944$	0	0
$945$	20506.8	0.705911
$946$	0	0
$947$	24232.0	0.831505	0.415753	−	0.909478i	$-0.363518\pi$
$947$	24232.0	0.831505	0.415753	+	0.909478i	$0.363518\pi$
$948$	0	0
$949$	53543.9	1.83152
$950$	0	0
$951$	−44091.0	−1.50342
$952$	0	0
$953$	173.278	0.00588986	0.00294493	−	0.999996i	$-0.499063\pi$
$953$	173.278	0.00588986	0.00294493	+	0.999996i	$0.499063\pi$
$954$	0	0
$955$	41249.7	1.39771
$956$	0	0
$957$	−8407.10	−0.283974
$958$	0	0
$959$	−31894.5	−1.07396
$960$	0	0
$961$	15390.9	0.516631
$962$	0	0
$963$	−643.353	−0.0215283
$964$	0	0
$965$	28258.4	0.942663
$966$	0	0
$967$	12097.9	0.402317	0.201159	−	0.979559i	$-0.435529\pi$
$967$	12097.9	0.402317	0.201159	+	0.979559i	$0.435529\pi$
$968$	0	0
$969$	9352.79	0.310067
$970$	0	0
$971$	40540.3	1.33986	0.669928	−	0.742426i	$-0.266325\pi$
$971$	40540.3	1.33986	0.669928	+	0.742426i	$0.266325\pi$
$972$	0	0
$973$	9709.02	0.319894
$974$	0	0
$975$	−17686.9	−0.580958
$976$	0	0
$977$	7407.63	0.242570	0.121285	−	0.992618i	$-0.461298\pi$
$977$	7407.63	0.242570	0.121285	+	0.992618i	$0.461298\pi$
$978$	0	0
$979$	38720.5	1.26406
$980$	0	0
$981$	2343.21	0.0762619
$982$	0	0
$983$	−19681.4	−0.638596	−0.319298	−	0.947654i	$-0.603447\pi$
$983$	−19681.4	−0.638596	−0.319298	+	0.947654i	$0.603447\pi$
$984$	0	0
$985$	−40454.0	−1.30860
$986$	0	0
$987$	24362.3	0.785675
$988$	0	0
$989$	32765.0	1.05346
$990$	0	0
$991$	33928.1	1.08755	0.543774	−	0.839231i	$-0.316995\pi$
$991$	33928.1	1.08755	0.543774	+	0.839231i	$0.316995\pi$
$992$	0	0
$993$	3122.48	0.0997876
$994$	0	0
$995$	315.618	0.0100560
$996$	0	0
$997$	6091.11	0.193488	0.0967440	−	0.995309i	$-0.469157\pi$
$997$	6091.11	0.193488	0.0967440	+	0.995309i	$0.469157\pi$
$998$	0	0
$999$	−22818.2	−0.722659

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.be.1.2		4
4.3	odd	2		1088.4.a.bb.1.3		4
8.3	odd	2		272.4.a.k.1.2		4
8.5	even	2		136.4.a.d.1.3	✓	4
24.5	odd	2		1224.4.a.l.1.3		4
24.11	even	2		2448.4.a.bq.1.3		4
136.101	even	2		2312.4.a.e.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.a.d.1.3	✓	4	8.5	even	2
272.4.a.k.1.2		4	8.3	odd	2
1088.4.a.bb.1.3		4	4.3	odd	2
1088.4.a.be.1.2		4	1.1	even	1	trivial
1224.4.a.l.1.3		4	24.5	odd	2
2312.4.a.e.1.2		4	136.101	even	2
2448.4.a.bq.1.3		4	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)

\(f(q)\)	\(=\)	\(q-5.49465 q^{3} +8.91638 q^{5} +17.5805 q^{7} +3.19121 q^{9} +O(q^{10})\)	\(q-5.49465 q^{3} +8.91638 q^{5} +17.5805 q^{7} +3.19121 q^{9} -66.4129 q^{11} -70.7484 q^{13} -48.9924 q^{15} -17.0000 q^{17} +100.127 q^{19} -96.5988 q^{21} +217.925 q^{23} -45.4982 q^{25} +130.821 q^{27} -23.0384 q^{29} -212.560 q^{31} +364.916 q^{33} +156.754 q^{35} -174.423 q^{37} +388.738 q^{39} +209.045 q^{41} +150.350 q^{43} +28.4541 q^{45} -252.201 q^{47} -33.9257 q^{49} +93.4091 q^{51} -60.6076 q^{53} -592.163 q^{55} -550.164 q^{57} +626.948 q^{59} +624.906 q^{61} +56.1031 q^{63} -630.820 q^{65} +114.386 q^{67} -1197.42 q^{69} +346.071 q^{71} -756.821 q^{73} +249.997 q^{75} -1167.57 q^{77} -146.530 q^{79} -804.979 q^{81} -328.488 q^{83} -151.578 q^{85} +126.588 q^{87} -583.027 q^{89} -1243.79 q^{91} +1167.95 q^{93} +892.772 q^{95} +1620.95 q^{97} -211.938 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 8 q^{5} - 22 q^{7} + 100 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 8 * q^5 - 22 * q^7 + 100 * q^9	\( 4 q + 2 q^{3} - 8 q^{5} - 22 q^{7} + 100 q^{9} - 70 q^{11} - 120 q^{13} + 140 q^{15} - 68 q^{17} + 44 q^{19} - 488 q^{21} + 158 q^{23} + 548 q^{25} + 392 q^{27} - 264 q^{29} + 122 q^{31} + 136 q^{33} + 44 q^{35} - 256 q^{37} - 528 q^{39} + 240 q^{41} + 1100 q^{43} + 880 q^{45} - 800 q^{47} + 12 q^{49} - 34 q^{51} - 432 q^{53} - 532 q^{55} + 472 q^{57} + 148 q^{59} + 728 q^{61} - 1450 q^{63} - 72 q^{65} + 1032 q^{67} + 1024 q^{69} + 798 q^{71} - 1544 q^{73} + 2974 q^{75} - 656 q^{77} + 758 q^{79} - 20 q^{81} - 244 q^{83} + 136 q^{85} + 1524 q^{87} + 1440 q^{89} + 1104 q^{91} + 2464 q^{93} + 7016 q^{95} - 1344 q^{97} + 86 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 8 * q^5 - 22 * q^7 + 100 * q^9 - 70 * q^11 - 120 * q^13 + 140 * q^15 - 68 * q^17 + 44 * q^19 - 488 * q^21 + 158 * q^23 + 548 * q^25 + 392 * q^27 - 264 * q^29 + 122 * q^31 + 136 * q^33 + 44 * q^35 - 256 * q^37 - 528 * q^39 + 240 * q^41 + 1100 * q^43 + 880 * q^45 - 800 * q^47 + 12 * q^49 - 34 * q^51 - 432 * q^53 - 532 * q^55 + 472 * q^57 + 148 * q^59 + 728 * q^61 - 1450 * q^63 - 72 * q^65 + 1032 * q^67 + 1024 * q^69 + 798 * q^71 - 1544 * q^73 + 2974 * q^75 - 656 * q^77 + 758 * q^79 - 20 * q^81 - 244 * q^83 + 136 * q^85 + 1524 * q^87 + 1440 * q^89 + 1104 * q^91 + 2464 * q^93 + 7016 * q^95 - 1344 * q^97 + 86 * q^99

Introduction

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