LMFDB - Embedded newform 1088.4.a.t.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:	3.3.1524.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 1 $ x^3 - x^2 - 7*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 68)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.27307$ 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)

$f(q)$	$=$	$q-6.87987 q^{3} -19.0923 q^{5} -34.1567 q^{7} +20.3326 q^{9} +O(q^{10})$	$q-6.87987 q^{3} -19.0923 q^{5} -34.1567 q^{7} +20.3326 q^{9} -35.3047 q^{11} +16.3715 q^{13} +131.352 q^{15} +17.0000 q^{17} +91.4591 q^{19} +234.994 q^{21} +69.1353 q^{23} +239.516 q^{25} +45.8711 q^{27} +72.8010 q^{29} -314.737 q^{31} +242.892 q^{33} +652.130 q^{35} +103.729 q^{37} -112.634 q^{39} -34.8358 q^{41} -316.936 q^{43} -388.195 q^{45} -190.937 q^{47} +823.683 q^{49} -116.958 q^{51} -476.856 q^{53} +674.048 q^{55} -629.227 q^{57} +351.708 q^{59} +34.1756 q^{61} -694.494 q^{63} -312.569 q^{65} +114.361 q^{67} -475.642 q^{69} +611.520 q^{71} -11.4137 q^{73} -1647.84 q^{75} +1205.89 q^{77} -266.137 q^{79} -864.566 q^{81} -386.656 q^{83} -324.569 q^{85} -500.861 q^{87} +1654.30 q^{89} -559.197 q^{91} +2165.35 q^{93} -1746.16 q^{95} +1406.50 q^{97} -717.835 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} - 26 q^{5} + 8 q^{7} + 63 q^{9}+O(q^{10}) $ 3 * q - 4 * q^3 - 26 * q^5 + 8 * q^7 + 63 * q^9	$ 3 q - 4 q^{3} - 26 q^{5} + 8 q^{7} + 63 q^{9} - 60 q^{11} - 82 q^{13} + 40 q^{15} + 51 q^{17} + 124 q^{19} + 144 q^{21} - 120 q^{23} + 85 q^{25} + 296 q^{27} + 46 q^{29} - 272 q^{31} - 48 q^{33} + 640 q^{35} - 186 q^{37} - 728 q^{39} - 82 q^{41} - 300 q^{43} - 770 q^{45} - 224 q^{47} + 1275 q^{49} - 68 q^{51} - 202 q^{53} + 984 q^{55} + 624 q^{57} + 612 q^{59} - 786 q^{61} - 232 q^{63} + 444 q^{65} - 916 q^{67} + 240 q^{69} - 1272 q^{71} - 306 q^{73} - 1308 q^{75} + 1104 q^{77} + 1568 q^{79} - 1797 q^{81} - 948 q^{83} - 442 q^{85} - 2264 q^{87} + 478 q^{89} - 1856 q^{91} + 4688 q^{93} - 2920 q^{95} + 1318 q^{97} - 1980 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 - 26 * q^5 + 8 * q^7 + 63 * q^9 - 60 * q^11 - 82 * q^13 + 40 * q^15 + 51 * q^17 + 124 * q^19 + 144 * q^21 - 120 * q^23 + 85 * q^25 + 296 * q^27 + 46 * q^29 - 272 * q^31 - 48 * q^33 + 640 * q^35 - 186 * q^37 - 728 * q^39 - 82 * q^41 - 300 * q^43 - 770 * q^45 - 224 * q^47 + 1275 * q^49 - 68 * q^51 - 202 * q^53 + 984 * q^55 + 624 * q^57 + 612 * q^59 - 786 * q^61 - 232 * q^63 + 444 * q^65 - 916 * q^67 + 240 * q^69 - 1272 * q^71 - 306 * q^73 - 1308 * q^75 + 1104 * q^77 + 1568 * q^79 - 1797 * q^81 - 948 * q^83 - 442 * q^85 - 2264 * q^87 + 478 * q^89 - 1856 * q^91 + 4688 * q^93 - 2920 * q^95 + 1318 * q^97 - 1980 * 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
$3$	−6.87987	−1.32403	−0.662015	−	0.749490i	$-0.730299\pi$
$3$	−6.87987	−1.32403	−0.662015	+	0.749490i	$0.730299\pi$
$4$	0	0
$5$	−19.0923	−1.70767	−0.853833	−	0.520547i	$-0.825728\pi$
$5$	−19.0923	−1.70767	−0.853833	+	0.520547i	$0.825728\pi$
$6$	0	0
$7$	−34.1567	−1.84429	−0.922145	−	0.386844i	$-0.873565\pi$
$7$	−34.1567	−1.84429	−0.922145	+	0.386844i	$0.873565\pi$
$8$	0	0
$9$	20.3326	0.753058
$10$	0	0
$11$	−35.3047	−0.967707	−0.483853	−	0.875149i	$-0.660763\pi$
$11$	−35.3047	−0.967707	−0.483853	+	0.875149i	$0.660763\pi$
$12$	0	0
$13$	16.3715	0.349280	0.174640	−	0.984632i	$-0.444124\pi$
$13$	16.3715	0.349280	0.174640	+	0.984632i	$0.444124\pi$
$14$	0	0
$15$	131.352	2.26100
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	91.4591	1.10432	0.552162	−	0.833737i	$-0.313803\pi$
$19$	91.4591	1.10432	0.552162	+	0.833737i	$0.313803\pi$
$20$	0	0
$21$	234.994	2.44190
$22$	0	0
$23$	69.1353	0.626770	0.313385	−	0.949626i	$-0.398537\pi$
$23$	69.1353	0.626770	0.313385	+	0.949626i	$0.398537\pi$
$24$	0	0
$25$	239.516	1.91612
$26$	0	0
$27$	45.8711	0.326959
$28$	0	0
$29$	72.8010	0.466166	0.233083	−	0.972457i	$-0.425119\pi$
$29$	72.8010	0.466166	0.233083	+	0.972457i	$0.425119\pi$
$30$	0	0
$31$	−314.737	−1.82350	−0.911750	−	0.410746i	$-0.865268\pi$
$31$	−314.737	−1.82350	−0.911750	+	0.410746i	$0.865268\pi$
$32$	0	0
$33$	242.892	1.28127
$34$	0	0
$35$	652.130	3.14943
$36$	0	0
$37$	103.729	0.460888	0.230444	−	0.973086i	$-0.425982\pi$
$37$	103.729	0.460888	0.230444	+	0.973086i	$0.425982\pi$
$38$	0	0
$39$	−112.634	−0.462457
$40$	0	0
$41$	−34.8358	−0.132693	−0.0663467	−	0.997797i	$-0.521134\pi$
$41$	−34.8358	−0.132693	−0.0663467	+	0.997797i	$0.521134\pi$
$42$	0	0
$43$	−316.936	−1.12401	−0.562003	−	0.827135i	$-0.689969\pi$
$43$	−316.936	−1.12401	−0.562003	+	0.827135i	$0.689969\pi$
$44$	0	0
$45$	−388.195	−1.28597
$46$	0	0
$47$	−190.937	−0.592575	−0.296287	−	0.955099i	$-0.595749\pi$
$47$	−190.937	−0.592575	−0.296287	+	0.955099i	$0.595749\pi$
$48$	0	0
$49$	823.683	2.40141
$50$	0	0
$51$	−116.958	−0.321125
$52$	0	0
$53$	−476.856	−1.23587	−0.617936	−	0.786229i	$-0.712031\pi$
$53$	−476.856	−1.23587	−0.617936	+	0.786229i	$0.712031\pi$
$54$	0	0
$55$	674.048	1.65252
$56$	0	0
$57$	−629.227	−1.46216
$58$	0	0
$59$	351.708	0.776075	0.388038	−	0.921644i	$-0.373153\pi$
$59$	351.708	0.776075	0.388038	+	0.921644i	$0.373153\pi$
$60$	0	0
$61$	34.1756	0.0717333	0.0358667	−	0.999357i	$-0.488581\pi$
$61$	34.1756	0.0717333	0.0358667	+	0.999357i	$0.488581\pi$
$62$	0	0
$63$	−694.494	−1.38886
$64$	0	0
$65$	−312.569	−0.596453
$66$	0	0
$67$	114.361	0.208529	0.104265	−	0.994550i	$-0.466751\pi$
$67$	114.361	0.208529	0.104265	+	0.994550i	$0.466751\pi$
$68$	0	0
$69$	−475.642	−0.829863
$70$	0	0
$71$	611.520	1.02217	0.511085	−	0.859530i	$-0.329244\pi$
$71$	611.520	1.02217	0.511085	+	0.859530i	$0.329244\pi$
$72$	0	0
$73$	−11.4137	−0.0182997	−0.00914984	−	0.999958i	$-0.502913\pi$
$73$	−11.4137	−0.0182997	−0.00914984	+	0.999958i	$0.502913\pi$
$74$	0	0
$75$	−1647.84	−2.53701
$76$	0	0
$77$	1205.89	1.78473
$78$	0	0
$79$	−266.137	−0.379022	−0.189511	−	0.981879i	$-0.560690\pi$
$79$	−266.137	−0.379022	−0.189511	+	0.981879i	$0.560690\pi$
$80$	0	0
$81$	−864.566	−1.18596
$82$	0	0
$83$	−386.656	−0.511337	−0.255669	−	0.966764i	$-0.582296\pi$
$83$	−386.656	−0.511337	−0.255669	+	0.966764i	$0.582296\pi$
$84$	0	0
$85$	−324.569	−0.414170
$86$	0	0
$87$	−500.861	−0.617218
$88$	0	0
$89$	1654.30	1.97028	0.985140	−	0.171753i	$-0.0549430\pi$
$89$	1654.30	1.97028	0.985140	+	0.171753i	$0.0549430\pi$
$90$	0	0
$91$	−559.197	−0.644173
$92$	0	0
$93$	2165.35	2.41437
$94$	0	0
$95$	−1746.16	−1.88582
$96$	0	0
$97$	1406.50	1.47225	0.736126	−	0.676844i	$-0.236653\pi$
$97$	1406.50	1.47225	0.736126	+	0.676844i	$0.236653\pi$
$98$	0	0
$99$	−717.835	−0.728739
$100$	0	0
$101$	347.407	0.342261	0.171130	−	0.985248i	$-0.445258\pi$
$101$	347.407	0.342261	0.171130	+	0.985248i	$0.445258\pi$
$102$	0	0
$103$	−957.419	−0.915896	−0.457948	−	0.888979i	$-0.651415\pi$
$103$	−957.419	−0.915896	−0.457948	+	0.888979i	$0.651415\pi$
$104$	0	0
$105$	−4486.57	−4.16995
$106$	0	0
$107$	−1035.16	−0.935262	−0.467631	−	0.883924i	$-0.654892\pi$
$107$	−1035.16	−0.935262	−0.467631	+	0.883924i	$0.654892\pi$
$108$	0	0
$109$	89.0964	0.0782925	0.0391463	−	0.999233i	$-0.487536\pi$
$109$	89.0964	0.0782925	0.0391463	+	0.999233i	$0.487536\pi$
$110$	0	0
$111$	−713.639	−0.610230
$112$	0	0
$113$	981.016	0.816692	0.408346	−	0.912827i	$-0.366106\pi$
$113$	981.016	0.816692	0.408346	+	0.912827i	$0.366106\pi$
$114$	0	0
$115$	−1319.95	−1.07031
$116$	0	0
$117$	332.874	0.263028
$118$	0	0
$119$	−580.665	−0.447306
$120$	0	0
$121$	−84.5770	−0.0635439
$122$	0	0
$123$	239.665	0.175690
$124$	0	0
$125$	−2186.37	−1.56444
$126$	0	0
$127$	102.803	0.0718289	0.0359144	−	0.999355i	$-0.488566\pi$
$127$	102.803	0.0718289	0.0359144	+	0.999355i	$0.488566\pi$
$128$	0	0
$129$	2180.48	1.48822
$130$	0	0
$131$	284.690	0.189874	0.0949368	−	0.995483i	$-0.469735\pi$
$131$	284.690	0.189874	0.0949368	+	0.995483i	$0.469735\pi$
$132$	0	0
$133$	−3123.95	−2.03670
$134$	0	0
$135$	−875.784	−0.558337
$136$	0	0
$137$	646.494	0.403166	0.201583	−	0.979471i	$-0.435391\pi$
$137$	646.494	0.403166	0.201583	+	0.979471i	$0.435391\pi$
$138$	0	0
$139$	−902.289	−0.550584	−0.275292	−	0.961361i	$-0.588774\pi$
$139$	−902.289	−0.550584	−0.275292	+	0.961361i	$0.588774\pi$
$140$	0	0
$141$	1313.62	0.784587
$142$	0	0
$143$	−577.991	−0.338000
$144$	0	0
$145$	−1389.94	−0.796055
$146$	0	0
$147$	−5666.83	−3.17954
$148$	0	0
$149$	2592.83	1.42559	0.712794	−	0.701373i	$-0.247429\pi$
$149$	2592.83	1.42559	0.712794	+	0.701373i	$0.247429\pi$
$150$	0	0
$151$	3182.05	1.71491	0.857455	−	0.514560i	$-0.172045\pi$
$151$	3182.05	1.71491	0.857455	+	0.514560i	$0.172045\pi$
$152$	0	0
$153$	345.654	0.182643
$154$	0	0
$155$	6009.06	3.11393
$156$	0	0
$157$	1465.86	0.745148	0.372574	−	0.928002i	$-0.378475\pi$
$157$	1465.86	0.745148	0.372574	+	0.928002i	$0.378475\pi$
$158$	0	0
$159$	3280.70	1.63633
$160$	0	0
$161$	−2361.44	−1.15595
$162$	0	0
$163$	−1541.32	−0.740647	−0.370323	−	0.928903i	$-0.620753\pi$
$163$	−1541.32	−0.740647	−0.370323	+	0.928903i	$0.620753\pi$
$164$	0	0
$165$	−4637.36	−2.18799
$166$	0	0
$167$	2870.97	1.33031	0.665157	−	0.746703i	$-0.268364\pi$
$167$	2870.97	1.33031	0.665157	+	0.746703i	$0.268364\pi$
$168$	0	0
$169$	−1928.97	−0.878004
$170$	0	0
$171$	1859.60	0.831620
$172$	0	0
$173$	−778.716	−0.342224	−0.171112	−	0.985252i	$-0.554736\pi$
$173$	−778.716	−0.342224	−0.171112	+	0.985252i	$0.554736\pi$
$174$	0	0
$175$	−8181.07	−3.53389
$176$	0	0
$177$	−2419.70	−1.02755
$178$	0	0
$179$	−3179.68	−1.32771	−0.663856	−	0.747860i	$-0.731081\pi$
$179$	−3179.68	−1.32771	−0.663856	+	0.747860i	$0.731081\pi$
$180$	0	0
$181$	147.672	0.0606429	0.0303214	−	0.999540i	$-0.490347\pi$
$181$	147.672	0.0606429	0.0303214	+	0.999540i	$0.490347\pi$
$182$	0	0
$183$	−235.123	−0.0949771
$184$	0	0
$185$	−1980.42	−0.787044
$186$	0	0
$187$	−600.180	−0.234703
$188$	0	0
$189$	−1566.81	−0.603008
$190$	0	0
$191$	−23.3680	−0.00885260	−0.00442630	−	0.999990i	$-0.501409\pi$
$191$	−23.3680	−0.00885260	−0.00442630	+	0.999990i	$0.501409\pi$
$192$	0	0
$193$	394.629	0.147181	0.0735907	−	0.997289i	$-0.476554\pi$
$193$	394.629	0.147181	0.0735907	+	0.997289i	$0.476554\pi$
$194$	0	0
$195$	2150.43	0.789722
$196$	0	0
$197$	3656.93	1.32257	0.661283	−	0.750136i	$-0.270012\pi$
$197$	3656.93	1.32257	0.661283	+	0.750136i	$0.270012\pi$
$198$	0	0
$199$	−2612.26	−0.930545	−0.465272	−	0.885168i	$-0.654044\pi$
$199$	−2612.26	−0.930545	−0.465272	+	0.885168i	$0.654044\pi$
$200$	0	0
$201$	−786.791	−0.276099
$202$	0	0
$203$	−2486.64	−0.859745
$204$	0	0
$205$	665.095	0.226596
$206$	0	0
$207$	1405.70	0.471994
$208$	0	0
$209$	−3228.94	−1.06866
$210$	0	0
$211$	4008.42	1.30782	0.653912	−	0.756570i	$-0.273127\pi$
$211$	4008.42	1.30782	0.653912	+	0.756570i	$0.273127\pi$
$212$	0	0
$213$	−4207.18	−1.35339
$214$	0	0
$215$	6051.03	1.91943
$216$	0	0
$217$	10750.4	3.36306
$218$	0	0
$219$	78.5249	0.0242293
$220$	0	0
$221$	278.315	0.0847127
$222$	0	0
$223$	−3531.76	−1.06056	−0.530278	−	0.847824i	$-0.677912\pi$
$223$	−3531.76	−1.06056	−0.530278	+	0.847824i	$0.677912\pi$
$224$	0	0
$225$	4869.96	1.44295
$226$	0	0
$227$	−336.895	−0.0985046	−0.0492523	−	0.998786i	$-0.515684\pi$
$227$	−336.895	−0.0985046	−0.0492523	+	0.998786i	$0.515684\pi$
$228$	0	0
$229$	5253.56	1.51600	0.758002	−	0.652252i	$-0.226176\pi$
$229$	5253.56	1.51600	0.758002	+	0.652252i	$0.226176\pi$
$230$	0	0
$231$	−8296.39	−2.36304
$232$	0	0
$233$	−3486.05	−0.980167	−0.490084	−	0.871675i	$-0.663034\pi$
$233$	−3486.05	−0.980167	−0.490084	+	0.871675i	$0.663034\pi$
$234$	0	0
$235$	3645.42	1.01192
$236$	0	0
$237$	1830.98	0.501836
$238$	0	0
$239$	−2435.01	−0.659028	−0.329514	−	0.944151i	$-0.606885\pi$
$239$	−2435.01	−0.659028	−0.329514	+	0.944151i	$0.606885\pi$
$240$	0	0
$241$	−6151.11	−1.64410	−0.822049	−	0.569416i	$-0.807169\pi$
$241$	−6151.11	−1.64410	−0.822049	+	0.569416i	$0.807169\pi$
$242$	0	0
$243$	4709.58	1.24329
$244$	0	0
$245$	−15726.0	−4.10080
$246$	0	0
$247$	1497.32	0.385718
$248$	0	0
$249$	2660.14	0.677026
$250$	0	0
$251$	1609.83	0.404827	0.202413	−	0.979300i	$-0.435122\pi$
$251$	1609.83	0.404827	0.202413	+	0.979300i	$0.435122\pi$
$252$	0	0
$253$	−2440.80	−0.606530
$254$	0	0
$255$	2232.99	0.548374
$256$	0	0
$257$	−2328.06	−0.565060	−0.282530	−	0.959259i	$-0.591174\pi$
$257$	−2328.06	−0.565060	−0.282530	+	0.959259i	$0.591174\pi$
$258$	0	0
$259$	−3543.03	−0.850012
$260$	0	0
$261$	1480.23	0.351050
$262$	0	0
$263$	3138.99	0.735964	0.367982	−	0.929833i	$-0.380049\pi$
$263$	3138.99	0.735964	0.367982	+	0.929833i	$0.380049\pi$
$264$	0	0
$265$	9104.27	2.11046
$266$	0	0
$267$	−11381.3	−2.60871
$268$	0	0
$269$	−4887.94	−1.10789	−0.553946	−	0.832552i	$-0.686879\pi$
$269$	−4887.94	−1.10789	−0.553946	+	0.832552i	$0.686879\pi$
$270$	0	0
$271$	−2746.40	−0.615617	−0.307808	−	0.951448i	$-0.599596\pi$
$271$	−2746.40	−0.615617	−0.307808	+	0.951448i	$0.599596\pi$
$272$	0	0
$273$	3847.20	0.852905
$274$	0	0
$275$	−8456.03	−1.85425
$276$	0	0
$277$	−8743.46	−1.89655	−0.948274	−	0.317454i	$-0.897172\pi$
$277$	−8743.46	−1.89655	−0.948274	+	0.317454i	$0.897172\pi$
$278$	0	0
$279$	−6399.41	−1.37320
$280$	0	0
$281$	−6214.16	−1.31924	−0.659618	−	0.751601i	$-0.729282\pi$
$281$	−6214.16	−1.31924	−0.659618	+	0.751601i	$0.729282\pi$
$282$	0	0
$283$	6138.63	1.28941	0.644706	−	0.764430i	$-0.276980\pi$
$283$	6138.63	1.28941	0.644706	+	0.764430i	$0.276980\pi$
$284$	0	0
$285$	12013.4	2.49688
$286$	0	0
$287$	1189.88	0.244725
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−9676.53	−1.94931
$292$	0	0
$293$	2033.08	0.405371	0.202685	−	0.979244i	$-0.435033\pi$
$293$	2033.08	0.405371	0.202685	+	0.979244i	$0.435033\pi$
$294$	0	0
$295$	−6714.90	−1.32528
$296$	0	0
$297$	−1619.47	−0.316401
$298$	0	0
$299$	1131.85	0.218918
$300$	0	0
$301$	10825.5	2.07299
$302$	0	0
$303$	−2390.12	−0.453164
$304$	0	0
$305$	−652.490	−0.122497
$306$	0	0
$307$	3139.90	0.583725	0.291862	−	0.956460i	$-0.405725\pi$
$307$	3139.90	0.583725	0.291862	+	0.956460i	$0.405725\pi$
$308$	0	0
$309$	6586.91	1.21267
$310$	0	0
$311$	9907.83	1.80650	0.903250	−	0.429114i	$-0.141174\pi$
$311$	9907.83	1.80650	0.903250	+	0.429114i	$0.141174\pi$
$312$	0	0
$313$	1445.19	0.260981	0.130491	−	0.991450i	$-0.458345\pi$
$313$	1445.19	0.260981	0.130491	+	0.991450i	$0.458345\pi$
$314$	0	0
$315$	13259.5	2.37171
$316$	0	0
$317$	−1315.32	−0.233046	−0.116523	−	0.993188i	$-0.537175\pi$
$317$	−1315.32	−0.233046	−0.116523	+	0.993188i	$0.537175\pi$
$318$	0	0
$319$	−2570.22	−0.451112
$320$	0	0
$321$	7121.79	1.23832
$322$	0	0
$323$	1554.81	0.267838
$324$	0	0
$325$	3921.23	0.669263
$326$	0	0
$327$	−612.971	−0.103662
$328$	0	0
$329$	6521.78	1.09288
$330$	0	0
$331$	−1679.47	−0.278888	−0.139444	−	0.990230i	$-0.544532\pi$
$331$	−1679.47	−0.278888	−0.139444	+	0.990230i	$0.544532\pi$
$332$	0	0
$333$	2109.07	0.347076
$334$	0	0
$335$	−2183.42	−0.356099
$336$	0	0
$337$	6427.98	1.03903	0.519517	−	0.854460i	$-0.326112\pi$
$337$	6427.98	1.03903	0.519517	+	0.854460i	$0.326112\pi$
$338$	0	0
$339$	−6749.26	−1.08133
$340$	0	0
$341$	11111.7	1.76461
$342$	0	0
$343$	−16418.6	−2.58460
$344$	0	0
$345$	9081.09	1.41713
$346$	0	0
$347$	355.706	0.0550296	0.0275148	−	0.999621i	$-0.491241\pi$
$347$	355.706	0.0550296	0.0275148	+	0.999621i	$0.491241\pi$
$348$	0	0
$349$	4790.26	0.734719	0.367359	−	0.930079i	$-0.380262\pi$
$349$	4790.26	0.734719	0.367359	+	0.930079i	$0.380262\pi$
$350$	0	0
$351$	750.978	0.114200
$352$	0	0
$353$	−9135.45	−1.37742	−0.688712	−	0.725035i	$-0.741824\pi$
$353$	−9135.45	−1.37742	−0.688712	+	0.725035i	$0.741824\pi$
$354$	0	0
$355$	−11675.3	−1.74553
$356$	0	0
$357$	3994.89	0.592247
$358$	0	0
$359$	10649.0	1.56555	0.782773	−	0.622308i	$-0.213805\pi$
$359$	10649.0	1.56555	0.782773	+	0.622308i	$0.213805\pi$
$360$	0	0
$361$	1505.77	0.219532
$362$	0	0
$363$	581.878	0.0841341
$364$	0	0
$365$	217.914	0.0312497
$366$	0	0
$367$	6913.73	0.983362	0.491681	−	0.870775i	$-0.336383\pi$
$367$	6913.73	0.983362	0.491681	+	0.870775i	$0.336383\pi$
$368$	0	0
$369$	−708.300	−0.0999258
$370$	0	0
$371$	16287.8	2.27931
$372$	0	0
$373$	−6993.44	−0.970795	−0.485398	−	0.874294i	$-0.661325\pi$
$373$	−6993.44	−0.970795	−0.485398	+	0.874294i	$0.661325\pi$
$374$	0	0
$375$	15041.9	2.07136
$376$	0	0
$377$	1191.86	0.162822
$378$	0	0
$379$	602.412	0.0816460	0.0408230	−	0.999166i	$-0.487002\pi$
$379$	602.412	0.0816460	0.0408230	+	0.999166i	$0.487002\pi$
$380$	0	0
$381$	−707.269	−0.0951037
$382$	0	0
$383$	8931.74	1.19162	0.595810	−	0.803125i	$-0.296831\pi$
$383$	8931.74	1.19162	0.595810	+	0.803125i	$0.296831\pi$
$384$	0	0
$385$	−23023.3	−3.04773
$386$	0	0
$387$	−6444.12	−0.846441
$388$	0	0
$389$	12111.7	1.57864	0.789318	−	0.613985i	$-0.210434\pi$
$389$	12111.7	1.57864	0.789318	+	0.613985i	$0.210434\pi$
$390$	0	0
$391$	1175.30	0.152014
$392$	0	0
$393$	−1958.63	−0.251399
$394$	0	0
$395$	5081.16	0.647243
$396$	0	0
$397$	−9016.37	−1.13985	−0.569923	−	0.821698i	$-0.693027\pi$
$397$	−9016.37	−1.13985	−0.569923	+	0.821698i	$0.693027\pi$
$398$	0	0
$399$	21492.3	2.69665
$400$	0	0
$401$	4046.55	0.503928	0.251964	−	0.967737i	$-0.418924\pi$
$401$	4046.55	0.503928	0.251964	+	0.967737i	$0.418924\pi$
$402$	0	0
$403$	−5152.72	−0.636911
$404$	0	0
$405$	16506.5	2.02523
$406$	0	0
$407$	−3662.11	−0.446005
$408$	0	0
$409$	−3490.95	−0.422045	−0.211023	−	0.977481i	$-0.567679\pi$
$409$	−3490.95	−0.422045	−0.211023	+	0.977481i	$0.567679\pi$
$410$	0	0
$411$	−4447.79	−0.533804
$412$	0	0
$413$	−12013.2	−1.43131
$414$	0	0
$415$	7382.15	0.873194
$416$	0	0
$417$	6207.62	0.728990
$418$	0	0
$419$	6940.21	0.809192	0.404596	−	0.914496i	$-0.367412\pi$
$419$	6940.21	0.809192	0.404596	+	0.914496i	$0.367412\pi$
$420$	0	0
$421$	867.961	0.100479	0.0502397	−	0.998737i	$-0.484001\pi$
$421$	867.961	0.100479	0.0502397	+	0.998737i	$0.484001\pi$
$422$	0	0
$423$	−3882.23	−0.446243
$424$	0	0
$425$	4071.77	0.464729
$426$	0	0
$427$	−1167.33	−0.132297
$428$	0	0
$429$	3976.50	0.447523
$430$	0	0
$431$	−2713.40	−0.303248	−0.151624	−	0.988438i	$-0.548450\pi$
$431$	−2713.40	−0.303248	−0.151624	+	0.988438i	$0.548450\pi$
$432$	0	0
$433$	499.370	0.0554231	0.0277115	−	0.999616i	$-0.491178\pi$
$433$	499.370	0.0554231	0.0277115	+	0.999616i	$0.491178\pi$
$434$	0	0
$435$	9562.58	1.05400
$436$	0	0
$437$	6323.06	0.692158
$438$	0	0
$439$	13153.8	1.43006	0.715029	−	0.699095i	$-0.246414\pi$
$439$	13153.8	1.43006	0.715029	+	0.699095i	$0.246414\pi$
$440$	0	0
$441$	16747.6	1.80840
$442$	0	0
$443$	−3950.78	−0.423718	−0.211859	−	0.977300i	$-0.567952\pi$
$443$	−3950.78	−0.423718	−0.211859	+	0.977300i	$0.567952\pi$
$444$	0	0
$445$	−31584.3	−3.36458
$446$	0	0
$447$	−17838.3	−1.88752
$448$	0	0
$449$	−9373.70	−0.985239	−0.492619	−	0.870245i	$-0.663961\pi$
$449$	−9373.70	−0.985239	−0.492619	+	0.870245i	$0.663961\pi$
$450$	0	0
$451$	1229.87	0.128408
$452$	0	0
$453$	−21892.1	−2.27059
$454$	0	0
$455$	10676.3	1.10003
$456$	0	0
$457$	−14528.6	−1.48713	−0.743566	−	0.668663i	$-0.766867\pi$
$457$	−14528.6	−1.48713	−0.743566	+	0.668663i	$0.766867\pi$
$458$	0	0
$459$	779.809	0.0792992
$460$	0	0
$461$	−694.967	−0.0702122	−0.0351061	−	0.999384i	$-0.511177\pi$
$461$	−694.967	−0.0702122	−0.0351061	+	0.999384i	$0.511177\pi$
$462$	0	0
$463$	2953.87	0.296497	0.148248	−	0.988950i	$-0.452636\pi$
$463$	2953.87	0.296497	0.148248	+	0.988950i	$0.452636\pi$
$464$	0	0
$465$	−41341.5	−4.12294
$466$	0	0
$467$	17509.4	1.73498	0.867491	−	0.497453i	$-0.165731\pi$
$467$	17509.4	1.73498	0.867491	+	0.497453i	$0.165731\pi$
$468$	0	0
$469$	−3906.21	−0.384589
$470$	0	0
$471$	−10084.9	−0.986599
$472$	0	0
$473$	11189.3	1.08771
$474$	0	0
$475$	21905.9	2.11602
$476$	0	0
$477$	−9695.70	−0.930683
$478$	0	0
$479$	14221.5	1.35657	0.678284	−	0.734800i	$-0.262724\pi$
$479$	14221.5	1.35657	0.678284	+	0.734800i	$0.262724\pi$
$480$	0	0
$481$	1698.19	0.160979
$482$	0	0
$483$	16246.4	1.53051
$484$	0	0
$485$	−26853.3	−2.51412
$486$	0	0
$487$	−14872.8	−1.38388	−0.691940	−	0.721955i	$-0.743244\pi$
$487$	−14872.8	−1.38388	−0.691940	+	0.721955i	$0.743244\pi$
$488$	0	0
$489$	10604.1	0.980639
$490$	0	0
$491$	−8765.65	−0.805679	−0.402839	−	0.915271i	$-0.631977\pi$
$491$	−8765.65	−0.805679	−0.402839	+	0.915271i	$0.631977\pi$
$492$	0	0
$493$	1237.62	0.113062
$494$	0	0
$495$	13705.1	1.24444
$496$	0	0
$497$	−20887.5	−1.88518
$498$	0	0
$499$	8061.92	0.723249	0.361624	−	0.932324i	$-0.382222\pi$
$499$	8061.92	0.723249	0.361624	+	0.932324i	$0.382222\pi$
$500$	0	0
$501$	−19751.9	−1.76138
$502$	0	0
$503$	−15748.0	−1.39596	−0.697980	−	0.716118i	$-0.745917\pi$
$503$	−15748.0	−1.39596	−0.697980	+	0.716118i	$0.745917\pi$
$504$	0	0
$505$	−6632.80	−0.584467
$506$	0	0
$507$	13271.1	1.16250
$508$	0	0
$509$	11294.7	0.983557	0.491779	−	0.870720i	$-0.336347\pi$
$509$	11294.7	0.983557	0.491779	+	0.870720i	$0.336347\pi$
$510$	0	0
$511$	389.856	0.0337499
$512$	0	0
$513$	4195.33	0.361069
$514$	0	0
$515$	18279.3	1.56404
$516$	0	0
$517$	6740.97	0.573438
$518$	0	0
$519$	5357.46	0.453115
$520$	0	0
$521$	−12847.7	−1.08036	−0.540180	−	0.841550i	$-0.681644\pi$
$521$	−12847.7	−1.08036	−0.540180	+	0.841550i	$0.681644\pi$
$522$	0	0
$523$	−22081.9	−1.84622	−0.923112	−	0.384531i	$-0.874363\pi$
$523$	−22081.9	−1.84622	−0.923112	+	0.384531i	$0.874363\pi$
$524$	0	0
$525$	56284.7	4.67898
$526$	0	0
$527$	−5350.53	−0.442264
$528$	0	0
$529$	−7387.30	−0.607159
$530$	0	0
$531$	7151.12	0.584429
$532$	0	0
$533$	−570.313	−0.0463471
$534$	0	0
$535$	19763.7	1.59712
$536$	0	0
$537$	21875.8	1.75793
$538$	0	0
$539$	−29079.9	−2.32386
$540$	0	0
$541$	17994.4	1.43002	0.715008	−	0.699116i	$-0.246423\pi$
$541$	17994.4	1.43002	0.715008	+	0.699116i	$0.246423\pi$
$542$	0	0
$543$	−1015.96	−0.0802930
$544$	0	0
$545$	−1701.05	−0.133698
$546$	0	0
$547$	−11528.6	−0.901147	−0.450573	−	0.892739i	$-0.648780\pi$
$547$	−11528.6	−0.901147	−0.450573	+	0.892739i	$0.648780\pi$
$548$	0	0
$549$	694.877	0.0540193
$550$	0	0
$551$	6658.32	0.514798
$552$	0	0
$553$	9090.36	0.699026
$554$	0	0
$555$	13625.0	1.04207
$556$	0	0
$557$	−9312.88	−0.708437	−0.354219	−	0.935163i	$-0.615253\pi$
$557$	−9312.88	−0.708437	−0.354219	+	0.935163i	$0.615253\pi$
$558$	0	0
$559$	−5188.71	−0.392592
$560$	0	0
$561$	4129.16	0.310754
$562$	0	0
$563$	−552.235	−0.0413391	−0.0206695	−	0.999786i	$-0.506580\pi$
$563$	−552.235	−0.0413391	−0.0206695	+	0.999786i	$0.506580\pi$
$564$	0	0
$565$	−18729.8	−1.39464
$566$	0	0
$567$	29530.8	2.18726
$568$	0	0
$569$	13836.8	1.01945	0.509727	−	0.860336i	$-0.329746\pi$
$569$	13836.8	1.01945	0.509727	+	0.860336i	$0.329746\pi$
$570$	0	0
$571$	−8568.10	−0.627958	−0.313979	−	0.949430i	$-0.601662\pi$
$571$	−8568.10	−0.627958	−0.313979	+	0.949430i	$0.601662\pi$
$572$	0	0
$573$	160.768	0.0117211
$574$	0	0
$575$	16559.0	1.20097
$576$	0	0
$577$	15080.1	1.08803	0.544014	−	0.839076i	$-0.316904\pi$
$577$	15080.1	1.08803	0.544014	+	0.839076i	$0.316904\pi$
$578$	0	0
$579$	−2714.99	−0.194873
$580$	0	0
$581$	13206.9	0.943055
$582$	0	0
$583$	16835.3	1.19596
$584$	0	0
$585$	−6355.33	−0.449164
$586$	0	0
$587$	19594.7	1.37779	0.688893	−	0.724863i	$-0.258097\pi$
$587$	19594.7	1.37779	0.688893	+	0.724863i	$0.258097\pi$
$588$	0	0
$589$	−28785.6	−2.01374
$590$	0	0
$591$	−25159.2	−1.75112
$592$	0	0
$593$	12775.1	0.884674	0.442337	−	0.896849i	$-0.354149\pi$
$593$	12775.1	0.884674	0.442337	+	0.896849i	$0.354149\pi$
$594$	0	0
$595$	11086.2	0.763850
$596$	0	0
$597$	17972.0	1.23207
$598$	0	0
$599$	−17535.3	−1.19612	−0.598058	−	0.801452i	$-0.704061\pi$
$599$	−17535.3	−1.19612	−0.598058	+	0.801452i	$0.704061\pi$
$600$	0	0
$601$	20352.9	1.38138	0.690691	−	0.723150i	$-0.257306\pi$
$601$	20352.9	1.38138	0.690691	+	0.723150i	$0.257306\pi$
$602$	0	0
$603$	2325.26	0.157035
$604$	0	0
$605$	1614.77	0.108512
$606$	0	0
$607$	748.483	0.0500494	0.0250247	−	0.999687i	$-0.492034\pi$
$607$	748.483	0.0500494	0.0250247	+	0.999687i	$0.492034\pi$
$608$	0	0
$609$	17107.8	1.13833
$610$	0	0
$611$	−3125.92	−0.206974
$612$	0	0
$613$	−28364.1	−1.86887	−0.934434	−	0.356136i	$-0.884094\pi$
$613$	−28364.1	−1.86887	−0.934434	+	0.356136i	$0.884094\pi$
$614$	0	0
$615$	−4575.76	−0.300020
$616$	0	0
$617$	−27240.5	−1.77741	−0.888703	−	0.458483i	$-0.848393\pi$
$617$	−27240.5	−1.77741	−0.888703	+	0.458483i	$0.848393\pi$
$618$	0	0
$619$	7222.04	0.468947	0.234474	−	0.972122i	$-0.424663\pi$
$619$	7222.04	0.468947	0.234474	+	0.972122i	$0.424663\pi$
$620$	0	0
$621$	3171.31	0.204928
$622$	0	0
$623$	−56505.3	−3.63377
$624$	0	0
$625$	11803.3	0.755409
$626$	0	0
$627$	22214.7	1.41494
$628$	0	0
$629$	1763.39	0.111782
$630$	0	0
$631$	−1099.75	−0.0693823	−0.0346911	−	0.999398i	$-0.511045\pi$
$631$	−1099.75	−0.0693823	−0.0346911	+	0.999398i	$0.511045\pi$
$632$	0	0
$633$	−27577.4	−1.73160
$634$	0	0
$635$	−1962.74	−0.122660
$636$	0	0
$637$	13484.9	0.838763
$638$	0	0
$639$	12433.8	0.769754
$640$	0	0
$641$	−1846.78	−0.113796	−0.0568982	−	0.998380i	$-0.518121\pi$
$641$	−1846.78	−0.113796	−0.0568982	+	0.998380i	$0.518121\pi$
$642$	0	0
$643$	−1995.66	−0.122397	−0.0611984	−	0.998126i	$-0.519492\pi$
$643$	−1995.66	−0.122397	−0.0611984	+	0.998126i	$0.519492\pi$
$644$	0	0
$645$	−41630.3	−2.54138
$646$	0	0
$647$	21623.2	1.31390	0.656952	−	0.753933i	$-0.271846\pi$
$647$	21623.2	1.31390	0.656952	+	0.753933i	$0.271846\pi$
$648$	0	0
$649$	−12416.9	−0.751013
$650$	0	0
$651$	−73961.3	−4.45280
$652$	0	0
$653$	9359.42	0.560892	0.280446	−	0.959870i	$-0.409518\pi$
$653$	9359.42	0.560892	0.280446	+	0.959870i	$0.409518\pi$
$654$	0	0
$655$	−5435.38	−0.324241
$656$	0	0
$657$	−232.070	−0.0137807
$658$	0	0
$659$	14648.4	0.865888	0.432944	−	0.901421i	$-0.357475\pi$
$659$	14648.4	0.865888	0.432944	+	0.901421i	$0.357475\pi$
$660$	0	0
$661$	−15894.8	−0.935306	−0.467653	−	0.883912i	$-0.654900\pi$
$661$	−15894.8	−0.935306	−0.467653	+	0.883912i	$0.654900\pi$
$662$	0	0
$663$	−1914.77	−0.112162
$664$	0	0
$665$	59643.3	3.47800
$666$	0	0
$667$	5033.12	0.292179
$668$	0	0
$669$	24298.0	1.40421
$670$	0	0
$671$	−1206.56	−0.0694168
$672$	0	0
$673$	1312.42	0.0751712	0.0375856	−	0.999293i	$-0.488033\pi$
$673$	1312.42	0.0751712	0.0375856	+	0.999293i	$0.488033\pi$
$674$	0	0
$675$	10986.8	0.626495
$676$	0	0
$677$	−18835.2	−1.06927	−0.534635	−	0.845083i	$-0.679551\pi$
$677$	−18835.2	−1.06927	−0.534635	+	0.845083i	$0.679551\pi$
$678$	0	0
$679$	−48041.5	−2.71526
$680$	0	0
$681$	2317.80	0.130423
$682$	0	0
$683$	−9093.24	−0.509434	−0.254717	−	0.967016i	$-0.581982\pi$
$683$	−9093.24	−0.509434	−0.254717	+	0.967016i	$0.581982\pi$
$684$	0	0
$685$	−12343.1	−0.688473
$686$	0	0
$687$	−36143.8	−2.00724
$688$	0	0
$689$	−7806.84	−0.431665
$690$	0	0
$691$	−10920.5	−0.601210	−0.300605	−	0.953749i	$-0.597189\pi$
$691$	−10920.5	−0.601210	−0.300605	+	0.953749i	$0.597189\pi$
$692$	0	0
$693$	24518.9	1.34401
$694$	0	0
$695$	17226.8	0.940213
$696$	0	0
$697$	−592.208	−0.0321829
$698$	0	0
$699$	23983.6	1.29777
$700$	0	0
$701$	14249.4	0.767750	0.383875	−	0.923385i	$-0.374589\pi$
$701$	14249.4	0.767750	0.383875	+	0.923385i	$0.374589\pi$
$702$	0	0
$703$	9486.92	0.508970
$704$	0	0
$705$	−25080.0	−1.33981
$706$	0	0
$707$	−11866.3	−0.631228
$708$	0	0
$709$	19183.7	1.01616	0.508082	−	0.861309i	$-0.330355\pi$
$709$	19183.7	1.01616	0.508082	+	0.861309i	$0.330355\pi$
$710$	0	0
$711$	−5411.24	−0.285425
$712$	0	0
$713$	−21759.5	−1.14292
$714$	0	0
$715$	11035.2	0.577191
$716$	0	0
$717$	16752.6	0.872574
$718$	0	0
$719$	−1872.24	−0.0971112	−0.0485556	−	0.998820i	$-0.515462\pi$
$719$	−1872.24	−0.0971112	−0.0485556	+	0.998820i	$0.515462\pi$
$720$	0	0
$721$	32702.3	1.68918
$722$	0	0
$723$	42318.8	2.17684
$724$	0	0
$725$	17437.0	0.893232
$726$	0	0
$727$	−33391.6	−1.70347	−0.851737	−	0.523969i	$-0.824451\pi$
$727$	−33391.6	−1.70347	−0.851737	+	0.523969i	$0.824451\pi$
$728$	0	0
$729$	−9057.99	−0.460194
$730$	0	0
$731$	−5387.91	−0.272611
$732$	0	0
$733$	30489.9	1.53639	0.768193	−	0.640218i	$-0.221156\pi$
$733$	30489.9	1.53639	0.768193	+	0.640218i	$0.221156\pi$
$734$	0	0
$735$	108193.	5.42959
$736$	0	0
$737$	−4037.50	−0.201795
$738$	0	0
$739$	−12342.0	−0.614353	−0.307177	−	0.951652i	$-0.599384\pi$
$739$	−12342.0	−0.614353	−0.307177	+	0.951652i	$0.599384\pi$
$740$	0	0
$741$	−10301.4	−0.510702
$742$	0	0
$743$	18490.4	0.912985	0.456492	−	0.889727i	$-0.349106\pi$
$743$	18490.4	0.912985	0.456492	+	0.889727i	$0.349106\pi$
$744$	0	0
$745$	−49503.0	−2.43443
$746$	0	0
$747$	−7861.70	−0.385067
$748$	0	0
$749$	35357.8	1.72490
$750$	0	0
$751$	9321.89	0.452944	0.226472	−	0.974018i	$-0.427281\pi$
$751$	9321.89	0.452944	0.226472	+	0.974018i	$0.427281\pi$
$752$	0	0
$753$	−11075.4	−0.536003
$754$	0	0
$755$	−60752.6	−2.92849
$756$	0	0
$757$	−28417.6	−1.36441	−0.682203	−	0.731163i	$-0.738978\pi$
$757$	−28417.6	−1.36441	−0.682203	+	0.731163i	$0.738978\pi$
$758$	0	0
$759$	16792.4	0.803064
$760$	0	0
$761$	19383.8	0.923341	0.461670	−	0.887052i	$-0.347250\pi$
$761$	19383.8	0.923341	0.461670	+	0.887052i	$0.347250\pi$
$762$	0	0
$763$	−3043.24	−0.144394
$764$	0	0
$765$	−6599.32	−0.311894
$766$	0	0
$767$	5757.98	0.271067
$768$	0	0
$769$	−29856.5	−1.40007	−0.700035	−	0.714108i	$-0.746832\pi$
$769$	−29856.5	−1.40007	−0.700035	+	0.714108i	$0.746832\pi$
$770$	0	0
$771$	16016.7	0.748157
$772$	0	0
$773$	−20965.5	−0.975518	−0.487759	−	0.872978i	$-0.662186\pi$
$773$	−20965.5	−0.975518	−0.487759	+	0.872978i	$0.662186\pi$
$774$	0	0
$775$	−75384.5	−3.49405
$776$	0	0
$777$	24375.6	1.12544
$778$	0	0
$779$	−3186.05	−0.146537
$780$	0	0
$781$	−21589.6	−0.989161
$782$	0	0
$783$	3339.46	0.152417
$784$	0	0
$785$	−27986.6	−1.27246
$786$	0	0
$787$	13031.4	0.590240	0.295120	−	0.955460i	$-0.404640\pi$
$787$	13031.4	0.590240	0.295120	+	0.955460i	$0.404640\pi$
$788$	0	0
$789$	−21595.8	−0.974439
$790$	0	0
$791$	−33508.3	−1.50622
$792$	0	0
$793$	559.505	0.0250550
$794$	0	0
$795$	−62636.2	−2.79431
$796$	0	0
$797$	4156.24	0.184719	0.0923597	−	0.995726i	$-0.470559\pi$
$797$	4156.24	0.184719	0.0923597	+	0.995726i	$0.470559\pi$
$798$	0	0
$799$	−3245.93	−0.143720
$800$	0	0
$801$	33636.1	1.48373
$802$	0	0
$803$	402.959	0.0177087
$804$	0	0
$805$	45085.3	1.97397
$806$	0	0
$807$	33628.4	1.46688
$808$	0	0
$809$	38507.6	1.67349	0.836745	−	0.547592i	$-0.184456\pi$
$809$	38507.6	1.67349	0.836745	+	0.547592i	$0.184456\pi$
$810$	0	0
$811$	19824.6	0.858367	0.429183	−	0.903217i	$-0.358801\pi$
$811$	19824.6	0.858367	0.429183	+	0.903217i	$0.358801\pi$
$812$	0	0
$813$	18894.9	0.815096
$814$	0	0
$815$	29427.3	1.26478
$816$	0	0
$817$	−28986.7	−1.24127
$818$	0	0
$819$	−11369.9	−0.485099
$820$	0	0
$821$	−4055.06	−0.172378	−0.0861892	−	0.996279i	$-0.527469\pi$
$821$	−4055.06	−0.172378	−0.0861892	+	0.996279i	$0.527469\pi$
$822$	0	0
$823$	14743.6	0.624460	0.312230	−	0.950007i	$-0.398924\pi$
$823$	14743.6	0.624460	0.312230	+	0.950007i	$0.398924\pi$
$824$	0	0
$825$	58176.4	2.45508
$826$	0	0
$827$	−24850.8	−1.04492	−0.522459	−	0.852664i	$-0.674985\pi$
$827$	−24850.8	−1.04492	−0.522459	+	0.852664i	$0.674985\pi$
$828$	0	0
$829$	12731.1	0.533377	0.266688	−	0.963783i	$-0.414071\pi$
$829$	12731.1	0.533377	0.266688	+	0.963783i	$0.414071\pi$
$830$	0	0
$831$	60153.8	2.51109
$832$	0	0
$833$	14002.6	0.582427
$834$	0	0
$835$	−54813.5	−2.27173
$836$	0	0
$837$	−14437.3	−0.596210
$838$	0	0
$839$	−11989.1	−0.493337	−0.246669	−	0.969100i	$-0.579336\pi$
$839$	−11989.1	−0.493337	−0.246669	+	0.969100i	$0.579336\pi$
$840$	0	0
$841$	−19089.0	−0.782690
$842$	0	0
$843$	42752.6	1.74671
$844$	0	0
$845$	36828.5	1.49934
$846$	0	0
$847$	2888.87	0.117193
$848$	0	0
$849$	−42233.0	−1.70722
$850$	0	0
$851$	7171.31	0.288871
$852$	0	0
$853$	−24744.0	−0.993224	−0.496612	−	0.867973i	$-0.665423\pi$
$853$	−24744.0	−0.993224	−0.496612	+	0.867973i	$0.665423\pi$
$854$	0	0
$855$	−35504.0	−1.42013
$856$	0	0
$857$	−6396.71	−0.254968	−0.127484	−	0.991841i	$-0.540690\pi$
$857$	−6396.71	−0.254968	−0.127484	+	0.991841i	$0.540690\pi$
$858$	0	0
$859$	−5535.27	−0.219862	−0.109931	−	0.993939i	$-0.535063\pi$
$859$	−5535.27	−0.219862	−0.109931	+	0.993939i	$0.535063\pi$
$860$	0	0
$861$	−8186.19	−0.324024
$862$	0	0
$863$	−2478.07	−0.0977458	−0.0488729	−	0.998805i	$-0.515563\pi$
$863$	−2478.07	−0.0977458	−0.0488729	+	0.998805i	$0.515563\pi$
$864$	0	0
$865$	14867.5	0.584404
$866$	0	0
$867$	−1988.28	−0.0778842
$868$	0	0
$869$	9395.88	0.366782
$870$	0	0
$871$	1872.27	0.0728351
$872$	0	0
$873$	28597.7	1.10869
$874$	0	0
$875$	74679.1	2.88527
$876$	0	0
$877$	12732.5	0.490246	0.245123	−	0.969492i	$-0.421172\pi$
$877$	12732.5	0.490246	0.245123	+	0.969492i	$0.421172\pi$
$878$	0	0
$879$	−13987.3	−0.536723
$880$	0	0
$881$	−4514.92	−0.172658	−0.0863289	−	0.996267i	$-0.527514\pi$
$881$	−4514.92	−0.172658	−0.0863289	+	0.996267i	$0.527514\pi$
$882$	0	0
$883$	−29843.9	−1.13740	−0.568702	−	0.822544i	$-0.692554\pi$
$883$	−29843.9	−1.13740	−0.568702	+	0.822544i	$0.692554\pi$
$884$	0	0
$885$	46197.6	1.75471
$886$	0	0
$887$	9244.69	0.349951	0.174975	−	0.984573i	$-0.444015\pi$
$887$	9244.69	0.349951	0.174975	+	0.984573i	$0.444015\pi$
$888$	0	0
$889$	−3511.41	−0.132473
$890$	0	0
$891$	30523.3	1.14766
$892$	0	0
$893$	−17462.9	−0.654395
$894$	0	0
$895$	60707.4	2.26729
$896$	0	0
$897$	−7786.97	−0.289854
$898$	0	0
$899$	−22913.2	−0.850053
$900$	0	0
$901$	−8106.55	−0.299743
$902$	0	0
$903$	−74477.9	−2.74471
$904$	0	0
$905$	−2819.39	−0.103558
$906$	0	0
$907$	−17024.8	−0.623263	−0.311632	−	0.950203i	$-0.600875\pi$
$907$	−17024.8	−0.623263	−0.311632	+	0.950203i	$0.600875\pi$
$908$	0	0
$909$	7063.68	0.257742
$910$	0	0
$911$	−21104.9	−0.767547	−0.383773	−	0.923427i	$-0.625376\pi$
$911$	−21104.9	−0.767547	−0.383773	+	0.923427i	$0.625376\pi$
$912$	0	0
$913$	13650.8	0.494825
$914$	0	0
$915$	4489.04	0.162189
$916$	0	0
$917$	−9724.07	−0.350182
$918$	0	0
$919$	−16052.0	−0.576176	−0.288088	−	0.957604i	$-0.593020\pi$
$919$	−16052.0	−0.576176	−0.288088	+	0.957604i	$0.593020\pi$
$920$	0	0
$921$	−21602.1	−0.772870
$922$	0	0
$923$	10011.5	0.357023
$924$	0	0
$925$	24844.6	0.883120
$926$	0	0
$927$	−19466.8	−0.689723
$928$	0	0
$929$	−8279.43	−0.292400	−0.146200	−	0.989255i	$-0.546704\pi$
$929$	−8279.43	−0.292400	−0.146200	+	0.989255i	$0.546704\pi$
$930$	0	0
$931$	75333.3	2.65193
$932$	0	0
$933$	−68164.6	−2.39186
$934$	0	0
$935$	11458.8	0.400795
$936$	0	0
$937$	−47460.0	−1.65470	−0.827348	−	0.561689i	$-0.810152\pi$
$937$	−47460.0	−1.65470	−0.827348	+	0.561689i	$0.810152\pi$
$938$	0	0
$939$	−9942.73	−0.345547
$940$	0	0
$941$	−47914.5	−1.65990	−0.829950	−	0.557837i	$-0.811631\pi$
$941$	−47914.5	−1.65990	−0.829950	+	0.557837i	$0.811631\pi$
$942$	0	0
$943$	−2408.38	−0.0831683
$944$	0	0
$945$	29913.9	1.02974
$946$	0	0
$947$	1854.15	0.0636238	0.0318119	−	0.999494i	$-0.489872\pi$
$947$	1854.15	0.0636238	0.0318119	+	0.999494i	$0.489872\pi$
$948$	0	0
$949$	−186.860	−0.00639170
$950$	0	0
$951$	9049.21	0.308560
$952$	0	0
$953$	18073.1	0.614319	0.307159	−	0.951658i	$-0.400622\pi$
$953$	18073.1	0.614319	0.307159	+	0.951658i	$0.400622\pi$
$954$	0	0
$955$	446.148	0.0151173
$956$	0	0
$957$	17682.8	0.597286
$958$	0	0
$959$	−22082.1	−0.743555
$960$	0	0
$961$	69268.6	2.32515
$962$	0	0
$963$	−21047.5	−0.704307
$964$	0	0
$965$	−7534.37	−0.251337
$966$	0	0
$967$	−38564.7	−1.28248	−0.641239	−	0.767341i	$-0.721579\pi$
$967$	−38564.7	−1.28248	−0.641239	+	0.767341i	$0.721579\pi$
$968$	0	0
$969$	−10696.9	−0.354626
$970$	0	0
$971$	30157.3	0.996698	0.498349	−	0.866976i	$-0.333940\pi$
$971$	30157.3	0.996698	0.498349	+	0.866976i	$0.333940\pi$
$972$	0	0
$973$	30819.2	1.01544
$974$	0	0
$975$	−26977.5	−0.886125
$976$	0	0
$977$	4765.98	0.156067	0.0780334	−	0.996951i	$-0.475136\pi$
$977$	4765.98	0.156067	0.0780334	+	0.996951i	$0.475136\pi$
$978$	0	0
$979$	−58404.4	−1.90665
$980$	0	0
$981$	1811.56	0.0589588
$982$	0	0
$983$	8485.97	0.275341	0.137671	−	0.990478i	$-0.456038\pi$
$983$	8485.97	0.275341	0.137671	+	0.990478i	$0.456038\pi$
$984$	0	0
$985$	−69819.2	−2.25850
$986$	0	0
$987$	−44869.0	−1.44701
$988$	0	0
$989$	−21911.5	−0.704493
$990$	0	0
$991$	−24385.9	−0.781680	−0.390840	−	0.920459i	$-0.627815\pi$
$991$	−24385.9	−0.781680	−0.390840	+	0.920459i	$0.627815\pi$
$992$	0	0
$993$	11554.5	0.369257
$994$	0	0
$995$	49874.1	1.58906
$996$	0	0
$997$	−33548.8	−1.06570	−0.532848	−	0.846211i	$-0.678878\pi$
$997$	−33548.8	−1.06570	−0.532848	+	0.846211i	$0.678878\pi$
$998$	0	0
$999$	4758.14	0.150692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.t.1.1		3
4.3	odd	2		1088.4.a.w.1.3		3
8.3	odd	2		272.4.a.i.1.1		3
8.5	even	2		68.4.a.b.1.3	✓	3
24.5	odd	2		612.4.a.g.1.1		3
24.11	even	2		2448.4.a.ba.1.1		3
40.13	odd	4		1700.4.e.d.749.5		6
40.29	even	2		1700.4.a.d.1.1		3
40.37	odd	4		1700.4.e.d.749.2		6
136.13	even	4		1156.4.b.e.577.5		6
136.21	even	4		1156.4.b.e.577.2		6
136.101	even	2		1156.4.a.g.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.4.a.b.1.3	✓	3	8.5	even	2
272.4.a.i.1.1		3	8.3	odd	2
612.4.a.g.1.1		3	24.5	odd	2
1088.4.a.t.1.1		3	1.1	even	1	trivial
1088.4.a.w.1.3		3	4.3	odd	2
1156.4.a.g.1.1		3	136.101	even	2
1156.4.b.e.577.2		6	136.21	even	4
1156.4.b.e.577.5		6	136.13	even	4
1700.4.a.d.1.1		3	40.29	even	2
1700.4.e.d.749.2		6	40.37	odd	4
1700.4.e.d.749.5		6	40.13	odd	4
2448.4.a.ba.1.1		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)

\(f(q)\)	\(=\)	\(q-6.87987 q^{3} -19.0923 q^{5} -34.1567 q^{7} +20.3326 q^{9} +O(q^{10})\)	\(q-6.87987 q^{3} -19.0923 q^{5} -34.1567 q^{7} +20.3326 q^{9} -35.3047 q^{11} +16.3715 q^{13} +131.352 q^{15} +17.0000 q^{17} +91.4591 q^{19} +234.994 q^{21} +69.1353 q^{23} +239.516 q^{25} +45.8711 q^{27} +72.8010 q^{29} -314.737 q^{31} +242.892 q^{33} +652.130 q^{35} +103.729 q^{37} -112.634 q^{39} -34.8358 q^{41} -316.936 q^{43} -388.195 q^{45} -190.937 q^{47} +823.683 q^{49} -116.958 q^{51} -476.856 q^{53} +674.048 q^{55} -629.227 q^{57} +351.708 q^{59} +34.1756 q^{61} -694.494 q^{63} -312.569 q^{65} +114.361 q^{67} -475.642 q^{69} +611.520 q^{71} -11.4137 q^{73} -1647.84 q^{75} +1205.89 q^{77} -266.137 q^{79} -864.566 q^{81} -386.656 q^{83} -324.569 q^{85} -500.861 q^{87} +1654.30 q^{89} -559.197 q^{91} +2165.35 q^{93} -1746.16 q^{95} +1406.50 q^{97} -717.835 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} - 26 q^{5} + 8 q^{7} + 63 q^{9}+O(q^{10}) \) 3 * q - 4 * q^3 - 26 * q^5 + 8 * q^7 + 63 * q^9	\( 3 q - 4 q^{3} - 26 q^{5} + 8 q^{7} + 63 q^{9} - 60 q^{11} - 82 q^{13} + 40 q^{15} + 51 q^{17} + 124 q^{19} + 144 q^{21} - 120 q^{23} + 85 q^{25} + 296 q^{27} + 46 q^{29} - 272 q^{31} - 48 q^{33} + 640 q^{35} - 186 q^{37} - 728 q^{39} - 82 q^{41} - 300 q^{43} - 770 q^{45} - 224 q^{47} + 1275 q^{49} - 68 q^{51} - 202 q^{53} + 984 q^{55} + 624 q^{57} + 612 q^{59} - 786 q^{61} - 232 q^{63} + 444 q^{65} - 916 q^{67} + 240 q^{69} - 1272 q^{71} - 306 q^{73} - 1308 q^{75} + 1104 q^{77} + 1568 q^{79} - 1797 q^{81} - 948 q^{83} - 442 q^{85} - 2264 q^{87} + 478 q^{89} - 1856 q^{91} + 4688 q^{93} - 2920 q^{95} + 1318 q^{97} - 1980 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 - 26 * q^5 + 8 * q^7 + 63 * q^9 - 60 * q^11 - 82 * q^13 + 40 * q^15 + 51 * q^17 + 124 * q^19 + 144 * q^21 - 120 * q^23 + 85 * q^25 + 296 * q^27 + 46 * q^29 - 272 * q^31 - 48 * q^33 + 640 * q^35 - 186 * q^37 - 728 * q^39 - 82 * q^41 - 300 * q^43 - 770 * q^45 - 224 * q^47 + 1275 * q^49 - 68 * q^51 - 202 * q^53 + 984 * q^55 + 624 * q^57 + 612 * q^59 - 786 * q^61 - 232 * q^63 + 444 * q^65 - 916 * q^67 + 240 * q^69 - 1272 * q^71 - 306 * q^73 - 1308 * q^75 + 1104 * q^77 + 1568 * q^79 - 1797 * q^81 - 948 * q^83 - 442 * q^85 - 2264 * q^87 + 478 * q^89 - 1856 * q^91 + 4688 * q^93 - 2920 * q^95 + 1318 * q^97 - 1980 * q^99

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