LMFDB - Embedded newform 1088.4.a.bg.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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 67x^{5} - 35x^{4} + 893x^{3} + 595x^{2} - 3064x - 2804 $ x^7 - x^6 - 67x^5 - 35x^4 + 893x^3 + 595x^2 - 3064*x - 2804
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.85516$ 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)

$f(q)$	$=$	$q-5.68731 q^{3} +19.9589 q^{5} -26.8171 q^{7} +5.34548 q^{9} +O(q^{10})$	$q-5.68731 q^{3} +19.9589 q^{5} -26.8171 q^{7} +5.34548 q^{9} -29.4527 q^{11} -73.8483 q^{13} -113.513 q^{15} -17.0000 q^{17} -28.2272 q^{19} +152.517 q^{21} -75.7456 q^{23} +273.359 q^{25} +123.156 q^{27} -178.227 q^{29} -7.68875 q^{31} +167.507 q^{33} -535.241 q^{35} +305.170 q^{37} +419.998 q^{39} +435.284 q^{41} -352.362 q^{43} +106.690 q^{45} +621.555 q^{47} +376.158 q^{49} +96.6842 q^{51} +212.611 q^{53} -587.844 q^{55} +160.537 q^{57} -821.233 q^{59} -330.764 q^{61} -143.350 q^{63} -1473.93 q^{65} +403.786 q^{67} +430.789 q^{69} +492.072 q^{71} -914.486 q^{73} -1554.68 q^{75} +789.837 q^{77} +237.292 q^{79} -844.754 q^{81} -160.728 q^{83} -339.302 q^{85} +1013.63 q^{87} +283.622 q^{89} +1980.40 q^{91} +43.7283 q^{93} -563.384 q^{95} +904.085 q^{97} -157.439 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{7} + 67 q^{9}+O(q^{10}) $ 7 * q - 2 * q^7 + 67 * q^9	$ 7 q - 2 q^{7} + 67 q^{9} - 108 q^{11} + 34 q^{13} + 128 q^{15} - 119 q^{17} - 124 q^{19} + 296 q^{21} - 6 q^{23} + 197 q^{25} - 248 q^{29} + 50 q^{31} + 512 q^{33} - 640 q^{35} + 484 q^{37} + 1504 q^{39} - 366 q^{41} - 1412 q^{43} + 80 q^{45} + 1012 q^{47} + 1115 q^{49} + 146 q^{53} + 1024 q^{55} + 48 q^{57} - 2332 q^{59} + 548 q^{61} + 2838 q^{63} - 208 q^{65} - 924 q^{67} + 1672 q^{69} + 1286 q^{71} + 870 q^{73} - 3136 q^{75} + 1344 q^{77} + 1818 q^{79} + 3039 q^{81} - 1772 q^{83} + 384 q^{87} + 1706 q^{89} - 588 q^{91} + 5576 q^{93} + 2048 q^{95} + 1802 q^{97} - 5148 q^{99}+O(q^{100}) $ 7 * q - 2 * q^7 + 67 * q^9 - 108 * q^11 + 34 * q^13 + 128 * q^15 - 119 * q^17 - 124 * q^19 + 296 * q^21 - 6 * q^23 + 197 * q^25 - 248 * q^29 + 50 * q^31 + 512 * q^33 - 640 * q^35 + 484 * q^37 + 1504 * q^39 - 366 * q^41 - 1412 * q^43 + 80 * q^45 + 1012 * q^47 + 1115 * q^49 + 146 * q^53 + 1024 * q^55 + 48 * q^57 - 2332 * q^59 + 548 * q^61 + 2838 * q^63 - 208 * q^65 - 924 * q^67 + 1672 * q^69 + 1286 * q^71 + 870 * q^73 - 3136 * q^75 + 1344 * q^77 + 1818 * q^79 + 3039 * q^81 - 1772 * q^83 + 384 * q^87 + 1706 * q^89 - 588 * q^91 + 5576 * q^93 + 2048 * q^95 + 1802 * q^97 - 5148 * 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
$3$	−5.68731	−1.09452	−0.547262	−	0.836962i	$-0.684330\pi$
$3$	−5.68731	−1.09452	−0.547262	+	0.836962i	$0.684330\pi$
$4$	0	0
$5$	19.9589	1.78518	0.892590	−	0.450868i	$-0.148886\pi$
$5$	19.9589	1.78518	0.892590	+	0.450868i	$0.148886\pi$
$6$	0	0
$7$	−26.8171	−1.44799	−0.723994	−	0.689806i	$-0.757696\pi$
$7$	−26.8171	−1.44799	−0.723994	+	0.689806i	$0.757696\pi$
$8$	0	0
$9$	5.34548	0.197981
$10$	0	0
$11$	−29.4527	−0.807302	−0.403651	−	0.914913i	$-0.632259\pi$
$11$	−29.4527	−0.807302	−0.403651	+	0.914913i	$0.632259\pi$
$12$	0	0
$13$	−73.8483	−1.57553	−0.787763	−	0.615978i	$-0.788761\pi$
$13$	−73.8483	−1.57553	−0.787763	+	0.615978i	$0.788761\pi$
$14$	0	0
$15$	−113.513	−1.95392
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−28.2272	−0.340829	−0.170415	−	0.985372i	$-0.554511\pi$
$19$	−28.2272	−0.340829	−0.170415	+	0.985372i	$0.554511\pi$
$20$	0	0
$21$	152.517	1.58486
$22$	0	0
$23$	−75.7456	−0.686698	−0.343349	−	0.939208i	$-0.611561\pi$
$23$	−75.7456	−0.686698	−0.343349	+	0.939208i	$0.611561\pi$
$24$	0	0
$25$	273.359	2.18687
$26$	0	0
$27$	123.156	0.877829
$28$	0	0
$29$	−178.227	−1.14124	−0.570619	−	0.821215i	$-0.693297\pi$
$29$	−178.227	−1.14124	−0.570619	+	0.821215i	$0.693297\pi$
$30$	0	0
$31$	−7.68875	−0.0445464	−0.0222732	−	0.999752i	$-0.507090\pi$
$31$	−7.68875	−0.0445464	−0.0222732	+	0.999752i	$0.507090\pi$
$32$	0	0
$33$	167.507	0.883611
$34$	0	0
$35$	−535.241	−2.58492
$36$	0	0
$37$	305.170	1.35594	0.677969	−	0.735090i	$-0.262860\pi$
$37$	305.170	1.35594	0.677969	+	0.735090i	$0.262860\pi$
$38$	0	0
$39$	419.998	1.72445
$40$	0	0
$41$	435.284	1.65805	0.829024	−	0.559214i	$-0.188897\pi$
$41$	435.284	1.65805	0.829024	+	0.559214i	$0.188897\pi$
$42$	0	0
$43$	−352.362	−1.24964	−0.624822	−	0.780767i	$-0.714828\pi$
$43$	−352.362	−1.24964	−0.624822	+	0.780767i	$0.714828\pi$
$44$	0	0
$45$	106.690	0.353431
$46$	0	0
$47$	621.555	1.92900	0.964502	−	0.264076i	$-0.0850669\pi$
$47$	621.555	1.92900	0.964502	+	0.264076i	$0.0850669\pi$
$48$	0	0
$49$	376.158	1.09667
$50$	0	0
$51$	96.6842	0.265461
$52$	0	0
$53$	212.611	0.551027	0.275513	−	0.961297i	$-0.411152\pi$
$53$	212.611	0.551027	0.275513	+	0.961297i	$0.411152\pi$
$54$	0	0
$55$	−587.844	−1.44118
$56$	0	0
$57$	160.537	0.373045
$58$	0	0
$59$	−821.233	−1.81213	−0.906063	−	0.423142i	$-0.860927\pi$
$59$	−821.233	−1.81213	−0.906063	+	0.423142i	$0.860927\pi$
$60$	0	0
$61$	−330.764	−0.694263	−0.347131	−	0.937816i	$-0.612844\pi$
$61$	−330.764	−0.694263	−0.347131	+	0.937816i	$0.612844\pi$
$62$	0	0
$63$	−143.350	−0.286674
$64$	0	0
$65$	−1473.93	−2.81260
$66$	0	0
$67$	403.786	0.736273	0.368137	−	0.929772i	$-0.379996\pi$
$67$	403.786	0.736273	0.368137	+	0.929772i	$0.379996\pi$
$68$	0	0
$69$	430.789	0.751607
$70$	0	0
$71$	492.072	0.822509	0.411255	−	0.911521i	$-0.365091\pi$
$71$	492.072	0.822509	0.411255	+	0.911521i	$0.365091\pi$
$72$	0	0
$73$	−914.486	−1.46620	−0.733099	−	0.680122i	$-0.761927\pi$
$73$	−914.486	−1.46620	−0.733099	+	0.680122i	$0.761927\pi$
$74$	0	0
$75$	−1554.68	−2.39358
$76$	0	0
$77$	789.837	1.16896
$78$	0	0
$79$	237.292	0.337943	0.168971	−	0.985621i	$-0.445955\pi$
$79$	237.292	0.337943	0.168971	+	0.985621i	$0.445955\pi$
$80$	0	0
$81$	−844.754	−1.15878
$82$	0	0
$83$	−160.728	−0.212556	−0.106278	−	0.994336i	$-0.533893\pi$
$83$	−160.728	−0.212556	−0.106278	+	0.994336i	$0.533893\pi$
$84$	0	0
$85$	−339.302	−0.432970
$86$	0	0
$87$	1013.63	1.24911
$88$	0	0
$89$	283.622	0.337796	0.168898	−	0.985634i	$-0.445979\pi$
$89$	283.622	0.337796	0.168898	+	0.985634i	$0.445979\pi$
$90$	0	0
$91$	1980.40	2.28134
$92$	0	0
$93$	43.7283	0.0487571
$94$	0	0
$95$	−563.384	−0.608442
$96$	0	0
$97$	904.085	0.946350	0.473175	−	0.880969i	$-0.343108\pi$
$97$	904.085	0.946350	0.473175	+	0.880969i	$0.343108\pi$
$98$	0	0
$99$	−157.439	−0.159830
$100$	0	0
$101$	757.075	0.745859	0.372930	−	0.927860i	$-0.378353\pi$
$101$	757.075	0.745859	0.372930	+	0.927860i	$0.378353\pi$
$102$	0	0
$103$	769.172	0.735813	0.367907	−	0.929863i	$-0.380075\pi$
$103$	769.172	0.735813	0.367907	+	0.929863i	$0.380075\pi$
$104$	0	0
$105$	3044.08	2.82925
$106$	0	0
$107$	748.302	0.676085	0.338042	−	0.941131i	$-0.390235\pi$
$107$	748.302	0.676085	0.338042	+	0.941131i	$0.390235\pi$
$108$	0	0
$109$	−1484.70	−1.30466	−0.652331	−	0.757934i	$-0.726209\pi$
$109$	−1484.70	−1.30466	−0.652331	+	0.757934i	$0.726209\pi$
$110$	0	0
$111$	−1735.60	−1.48411
$112$	0	0
$113$	−310.502	−0.258492	−0.129246	−	0.991613i	$-0.541256\pi$
$113$	−310.502	−0.258492	−0.129246	+	0.991613i	$0.541256\pi$
$114$	0	0
$115$	−1511.80	−1.22588
$116$	0	0
$117$	−394.755	−0.311924
$118$	0	0
$119$	455.891	0.351189
$120$	0	0
$121$	−463.538	−0.348263
$122$	0	0
$123$	−2475.59	−1.81477
$124$	0	0
$125$	2961.08	2.11878
$126$	0	0
$127$	−929.939	−0.649754	−0.324877	−	0.945756i	$-0.605323\pi$
$127$	−929.939	−0.649754	−0.324877	+	0.945756i	$0.605323\pi$
$128$	0	0
$129$	2003.99	1.36776
$130$	0	0
$131$	1909.68	1.27366	0.636829	−	0.771005i	$-0.280246\pi$
$131$	1909.68	1.27366	0.636829	+	0.771005i	$0.280246\pi$
$132$	0	0
$133$	756.971	0.493517
$134$	0	0
$135$	2458.06	1.56708
$136$	0	0
$137$	416.886	0.259978	0.129989	−	0.991515i	$-0.458506\pi$
$137$	416.886	0.259978	0.129989	+	0.991515i	$0.458506\pi$
$138$	0	0
$139$	463.131	0.282606	0.141303	−	0.989966i	$-0.454871\pi$
$139$	463.131	0.282606	0.141303	+	0.989966i	$0.454871\pi$
$140$	0	0
$141$	−3534.98	−2.11134
$142$	0	0
$143$	2175.03	1.27193
$144$	0	0
$145$	−3557.22	−2.03732
$146$	0	0
$147$	−2139.32	−1.20033
$148$	0	0
$149$	2051.91	1.12818	0.564090	−	0.825713i	$-0.309227\pi$
$149$	2051.91	1.12818	0.564090	+	0.825713i	$0.309227\pi$
$150$	0	0
$151$	821.177	0.442559	0.221280	−	0.975210i	$-0.428977\pi$
$151$	821.177	0.442559	0.221280	+	0.975210i	$0.428977\pi$
$152$	0	0
$153$	−90.8731	−0.0480174
$154$	0	0
$155$	−153.459	−0.0795234
$156$	0	0
$157$	−70.9859	−0.0360847	−0.0180423	−	0.999837i	$-0.505743\pi$
$157$	−70.9859	−0.0360847	−0.0180423	+	0.999837i	$0.505743\pi$
$158$	0	0
$159$	−1209.19	−0.603111
$160$	0	0
$161$	2031.28	0.994330
$162$	0	0
$163$	1320.46	0.634517	0.317258	−	0.948339i	$-0.397238\pi$
$163$	1320.46	0.634517	0.317258	+	0.948339i	$0.397238\pi$
$164$	0	0
$165$	3343.25	1.57741
$166$	0	0
$167$	3953.18	1.83177	0.915887	−	0.401436i	$-0.131489\pi$
$167$	3953.18	1.83177	0.915887	+	0.401436i	$0.131489\pi$
$168$	0	0
$169$	3256.58	1.48228
$170$	0	0
$171$	−150.888	−0.0674776
$172$	0	0
$173$	3097.73	1.36136	0.680682	−	0.732579i	$-0.261684\pi$
$173$	3097.73	1.36136	0.680682	+	0.732579i	$0.261684\pi$
$174$	0	0
$175$	−7330.69	−3.16656
$176$	0	0
$177$	4670.61	1.98341
$178$	0	0
$179$	−3145.31	−1.31336	−0.656679	−	0.754170i	$-0.728040\pi$
$179$	−3145.31	−1.31336	−0.656679	+	0.754170i	$0.728040\pi$
$180$	0	0
$181$	−1875.56	−0.770217	−0.385108	−	0.922871i	$-0.625836\pi$
$181$	−1875.56	−0.770217	−0.385108	+	0.922871i	$0.625836\pi$
$182$	0	0
$183$	1881.16	0.759887
$184$	0	0
$185$	6090.88	2.42060
$186$	0	0
$187$	500.696	0.195800
$188$	0	0
$189$	−3302.69	−1.27109
$190$	0	0
$191$	4565.11	1.72942	0.864712	−	0.502267i	$-0.167501\pi$
$191$	4565.11	1.72942	0.864712	+	0.502267i	$0.167501\pi$
$192$	0	0
$193$	−2054.36	−0.766198	−0.383099	−	0.923707i	$-0.625143\pi$
$193$	−2054.36	−0.766198	−0.383099	+	0.923707i	$0.625143\pi$
$194$	0	0
$195$	8382.72	3.07845
$196$	0	0
$197$	1611.93	0.582972	0.291486	−	0.956575i	$-0.405850\pi$
$197$	1611.93	0.582972	0.291486	+	0.956575i	$0.405850\pi$
$198$	0	0
$199$	−1192.37	−0.424748	−0.212374	−	0.977188i	$-0.568120\pi$
$199$	−1192.37	−0.424748	−0.212374	+	0.977188i	$0.568120\pi$
$200$	0	0
$201$	−2296.46	−0.805868
$202$	0	0
$203$	4779.53	1.65250
$204$	0	0
$205$	8687.80	2.95991
$206$	0	0
$207$	−404.896	−0.135953
$208$	0	0
$209$	831.366	0.275152
$210$	0	0
$211$	−2259.42	−0.737181	−0.368591	−	0.929592i	$-0.620160\pi$
$211$	−2259.42	−0.737181	−0.368591	+	0.929592i	$0.620160\pi$
$212$	0	0
$213$	−2798.56	−0.900255
$214$	0	0
$215$	−7032.77	−2.23084
$216$	0	0
$217$	206.190	0.0645027
$218$	0	0
$219$	5200.96	1.60479
$220$	0	0
$221$	1255.42	0.382121
$222$	0	0
$223$	−403.726	−0.121235	−0.0606176	−	0.998161i	$-0.519307\pi$
$223$	−403.726	−0.121235	−0.0606176	+	0.998161i	$0.519307\pi$
$224$	0	0
$225$	1461.23	0.432958
$226$	0	0
$227$	−1722.15	−0.503539	−0.251770	−	0.967787i	$-0.581013\pi$
$227$	−1722.15	−0.503539	−0.251770	+	0.967787i	$0.581013\pi$
$228$	0	0
$229$	−465.061	−0.134201	−0.0671006	−	0.997746i	$-0.521375\pi$
$229$	−465.061	−0.134201	−0.0671006	+	0.997746i	$0.521375\pi$
$230$	0	0
$231$	−4492.04	−1.27946
$232$	0	0
$233$	3237.48	0.910277	0.455139	−	0.890421i	$-0.349590\pi$
$233$	3237.48	0.910277	0.455139	+	0.890421i	$0.349590\pi$
$234$	0	0
$235$	12405.6	3.44362
$236$	0	0
$237$	−1349.56	−0.369886
$238$	0	0
$239$	1606.44	0.434777	0.217389	−	0.976085i	$-0.430246\pi$
$239$	1606.44	0.434777	0.217389	+	0.976085i	$0.430246\pi$
$240$	0	0
$241$	−6480.40	−1.73211	−0.866056	−	0.499947i	$-0.833353\pi$
$241$	−6480.40	−1.73211	−0.866056	+	0.499947i	$0.833353\pi$
$242$	0	0
$243$	1479.16	0.390487
$244$	0	0
$245$	7507.70	1.95775
$246$	0	0
$247$	2084.53	0.536985
$248$	0	0
$249$	914.109	0.232648
$250$	0	0
$251$	2459.22	0.618424	0.309212	−	0.950993i	$-0.399935\pi$
$251$	2459.22	0.618424	0.309212	+	0.950993i	$0.399935\pi$
$252$	0	0
$253$	2230.91	0.554373
$254$	0	0
$255$	1929.71	0.473896
$256$	0	0
$257$	6122.09	1.48594	0.742968	−	0.669327i	$-0.233417\pi$
$257$	6122.09	1.48594	0.742968	+	0.669327i	$0.233417\pi$
$258$	0	0
$259$	−8183.79	−1.96338
$260$	0	0
$261$	−952.708	−0.225943
$262$	0	0
$263$	−1544.64	−0.362155	−0.181078	−	0.983469i	$-0.557959\pi$
$263$	−1544.64	−0.362155	−0.181078	+	0.983469i	$0.557959\pi$
$264$	0	0
$265$	4243.49	0.983682
$266$	0	0
$267$	−1613.04	−0.369725
$268$	0	0
$269$	−4363.68	−0.989065	−0.494532	−	0.869159i	$-0.664661\pi$
$269$	−4363.68	−0.989065	−0.494532	+	0.869159i	$0.664661\pi$
$270$	0	0
$271$	1327.89	0.297650	0.148825	−	0.988864i	$-0.452451\pi$
$271$	1327.89	0.297650	0.148825	+	0.988864i	$0.452451\pi$
$272$	0	0
$273$	−11263.1	−2.49698
$274$	0	0
$275$	−8051.16	−1.76547
$276$	0	0
$277$	1927.14	0.418017	0.209009	−	0.977914i	$-0.432976\pi$
$277$	1927.14	0.418017	0.209009	+	0.977914i	$0.432976\pi$
$278$	0	0
$279$	−41.1000	−0.00881933
$280$	0	0
$281$	−2613.73	−0.554883	−0.277442	−	0.960742i	$-0.589487\pi$
$281$	−2613.73	−0.554883	−0.277442	+	0.960742i	$0.589487\pi$
$282$	0	0
$283$	664.081	0.139489	0.0697447	−	0.997565i	$-0.477782\pi$
$283$	664.081	0.139489	0.0697447	+	0.997565i	$0.477782\pi$
$284$	0	0
$285$	3204.14	0.665953
$286$	0	0
$287$	−11673.1	−2.40083
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−5141.81	−1.03580
$292$	0	0
$293$	−6918.83	−1.37953	−0.689765	−	0.724033i	$-0.742286\pi$
$293$	−6918.83	−1.37953	−0.689765	+	0.724033i	$0.742286\pi$
$294$	0	0
$295$	−16390.9	−3.23497
$296$	0	0
$297$	−3627.28	−0.708673
$298$	0	0
$299$	5593.69	1.08191
$300$	0	0
$301$	9449.33	1.80947
$302$	0	0
$303$	−4305.72	−0.816360
$304$	0	0
$305$	−6601.70	−1.23938
$306$	0	0
$307$	862.174	0.160283	0.0801415	−	0.996783i	$-0.474463\pi$
$307$	862.174	0.160283	0.0801415	+	0.996783i	$0.474463\pi$
$308$	0	0
$309$	−4374.52	−0.805365
$310$	0	0
$311$	3987.95	0.727125	0.363562	−	0.931570i	$-0.381560\pi$
$311$	3987.95	0.727125	0.363562	+	0.931570i	$0.381560\pi$
$312$	0	0
$313$	2013.88	0.363678	0.181839	−	0.983328i	$-0.441795\pi$
$313$	2013.88	0.363678	0.181839	+	0.983328i	$0.441795\pi$
$314$	0	0
$315$	−2861.12	−0.511764
$316$	0	0
$317$	2185.01	0.387137	0.193569	−	0.981087i	$-0.437994\pi$
$317$	2185.01	0.387137	0.193569	+	0.981087i	$0.437994\pi$
$318$	0	0
$319$	5249.26	0.921324
$320$	0	0
$321$	−4255.82	−0.739991
$322$	0	0
$323$	479.862	0.0826632
$324$	0	0
$325$	−20187.1	−3.44547
$326$	0	0
$327$	8443.92	1.42798
$328$	0	0
$329$	−16668.3	−2.79317
$330$	0	0
$331$	−9447.42	−1.56881	−0.784407	−	0.620246i	$-0.787033\pi$
$331$	−9447.42	−1.56881	−0.784407	+	0.620246i	$0.787033\pi$
$332$	0	0
$333$	1631.28	0.268450
$334$	0	0
$335$	8059.13	1.31438
$336$	0	0
$337$	−124.735	−0.0201625	−0.0100813	−	0.999949i	$-0.503209\pi$
$337$	−124.735	−0.0201625	−0.0100813	+	0.999949i	$0.503209\pi$
$338$	0	0
$339$	1765.92	0.282926
$340$	0	0
$341$	226.454	0.0359624
$342$	0	0
$343$	−889.189	−0.139976
$344$	0	0
$345$	8598.08	1.34175
$346$	0	0
$347$	−12368.5	−1.91348	−0.956741	−	0.290941i	$-0.906032\pi$
$347$	−12368.5	−1.91348	−0.956741	+	0.290941i	$0.906032\pi$
$348$	0	0
$349$	−5867.39	−0.899926	−0.449963	−	0.893047i	$-0.648563\pi$
$349$	−5867.39	−0.899926	−0.449963	+	0.893047i	$0.648563\pi$
$350$	0	0
$351$	−9094.86	−1.38304
$352$	0	0
$353$	3863.87	0.582587	0.291293	−	0.956634i	$-0.405914\pi$
$353$	3863.87	0.582587	0.291293	+	0.956634i	$0.405914\pi$
$354$	0	0
$355$	9821.22	1.46833
$356$	0	0
$357$	−2592.79	−0.384384
$358$	0	0
$359$	11554.0	1.69860	0.849302	−	0.527907i	$-0.177023\pi$
$359$	11554.0	1.69860	0.849302	+	0.527907i	$0.177023\pi$
$360$	0	0
$361$	−6062.23	−0.883835
$362$	0	0
$363$	2636.28	0.381182
$364$	0	0
$365$	−18252.2	−2.61743
$366$	0	0
$367$	5419.29	0.770803	0.385402	−	0.922749i	$-0.374063\pi$
$367$	5419.29	0.770803	0.385402	+	0.922749i	$0.374063\pi$
$368$	0	0
$369$	2326.80	0.328261
$370$	0	0
$371$	−5701.62	−0.797880
$372$	0	0
$373$	11720.3	1.62695	0.813477	−	0.581597i	$-0.197572\pi$
$373$	11720.3	1.62695	0.813477	+	0.581597i	$0.197572\pi$
$374$	0	0
$375$	−16840.6	−2.31905
$376$	0	0
$377$	13161.8	1.79805
$378$	0	0
$379$	−11268.9	−1.52729	−0.763645	−	0.645636i	$-0.776592\pi$
$379$	−11268.9	−1.52729	−0.763645	+	0.645636i	$0.776592\pi$
$380$	0	0
$381$	5288.85	0.711170
$382$	0	0
$383$	−99.7673	−0.0133104	−0.00665518	−	0.999978i	$-0.502118\pi$
$383$	−99.7673	−0.0133104	−0.00665518	+	0.999978i	$0.502118\pi$
$384$	0	0
$385$	15764.3	2.08681
$386$	0	0
$387$	−1883.54	−0.247405
$388$	0	0
$389$	−14982.7	−1.95283	−0.976416	−	0.215900i	$-0.930732\pi$
$389$	−14982.7	−1.95283	−0.976416	+	0.215900i	$0.930732\pi$
$390$	0	0
$391$	1287.68	0.166549
$392$	0	0
$393$	−10860.9	−1.39405
$394$	0	0
$395$	4736.10	0.603289
$396$	0	0
$397$	1478.34	0.186891	0.0934453	−	0.995624i	$-0.470212\pi$
$397$	1478.34	0.186891	0.0934453	+	0.995624i	$0.470212\pi$
$398$	0	0
$399$	−4305.13	−0.540165
$400$	0	0
$401$	6853.74	0.853515	0.426758	−	0.904366i	$-0.359656\pi$
$401$	6853.74	0.853515	0.426758	+	0.904366i	$0.359656\pi$
$402$	0	0
$403$	567.801	0.0701841
$404$	0	0
$405$	−16860.4	−2.06864
$406$	0	0
$407$	−8988.10	−1.09465
$408$	0	0
$409$	4149.68	0.501684	0.250842	−	0.968028i	$-0.419293\pi$
$409$	4149.68	0.501684	0.250842	+	0.968028i	$0.419293\pi$
$410$	0	0
$411$	−2370.96	−0.284552
$412$	0	0
$413$	22023.1	2.62394
$414$	0	0
$415$	−3207.96	−0.379451
$416$	0	0
$417$	−2633.97	−0.309319
$418$	0	0
$419$	3960.01	0.461716	0.230858	−	0.972987i	$-0.425847\pi$
$419$	3960.01	0.461716	0.230858	+	0.972987i	$0.425847\pi$
$420$	0	0
$421$	12113.0	1.40226	0.701131	−	0.713032i	$-0.252679\pi$
$421$	12113.0	1.40226	0.701131	+	0.713032i	$0.252679\pi$
$422$	0	0
$423$	3322.51	0.381905
$424$	0	0
$425$	−4647.10	−0.530394
$426$	0	0
$427$	8870.15	1.00528
$428$	0	0
$429$	−12370.1	−1.39215
$430$	0	0
$431$	12015.9	1.34289	0.671443	−	0.741056i	$-0.265675\pi$
$431$	12015.9	1.34289	0.671443	+	0.741056i	$0.265675\pi$
$432$	0	0
$433$	−4238.61	−0.470427	−0.235213	−	0.971944i	$-0.575579\pi$
$433$	−4238.61	−0.470427	−0.235213	+	0.971944i	$0.575579\pi$
$434$	0	0
$435$	20231.0	2.22989
$436$	0	0
$437$	2138.08	0.234047
$438$	0	0
$439$	−2634.79	−0.286450	−0.143225	−	0.989690i	$-0.545747\pi$
$439$	−2634.79	−0.286450	−0.143225	+	0.989690i	$0.545747\pi$
$440$	0	0
$441$	2010.74	0.217119
$442$	0	0
$443$	841.234	0.0902217	0.0451109	−	0.998982i	$-0.485636\pi$
$443$	841.234	0.0902217	0.0451109	+	0.998982i	$0.485636\pi$
$444$	0	0
$445$	5660.78	0.603027
$446$	0	0
$447$	−11669.8	−1.23482
$448$	0	0
$449$	4377.66	0.460122	0.230061	−	0.973176i	$-0.426107\pi$
$449$	4377.66	0.460122	0.230061	+	0.973176i	$0.426107\pi$
$450$	0	0
$451$	−12820.3	−1.33855
$452$	0	0
$453$	−4670.29	−0.484391
$454$	0	0
$455$	39526.6	4.07261
$456$	0	0
$457$	−13420.3	−1.37369	−0.686846	−	0.726803i	$-0.741005\pi$
$457$	−13420.3	−1.37369	−0.686846	+	0.726803i	$0.741005\pi$
$458$	0	0
$459$	−2093.65	−0.212905
$460$	0	0
$461$	10591.1	1.07001	0.535007	−	0.844848i	$-0.320309\pi$
$461$	10591.1	1.07001	0.535007	+	0.844848i	$0.320309\pi$
$462$	0	0
$463$	17491.2	1.75569	0.877846	−	0.478943i	$-0.158980\pi$
$463$	17491.2	1.75569	0.877846	+	0.478943i	$0.158980\pi$
$464$	0	0
$465$	872.769	0.0870402
$466$	0	0
$467$	7697.27	0.762714	0.381357	−	0.924428i	$-0.375457\pi$
$467$	7697.27	0.762714	0.381357	+	0.924428i	$0.375457\pi$
$468$	0	0
$469$	−10828.4	−1.06611
$470$	0	0
$471$	403.719	0.0394955
$472$	0	0
$473$	10378.0	1.00884
$474$	0	0
$475$	−7716.14	−0.745349
$476$	0	0
$477$	1136.51	0.109093
$478$	0	0
$479$	−9718.85	−0.927068	−0.463534	−	0.886079i	$-0.653419\pi$
$479$	−9718.85	−0.927068	−0.463534	+	0.886079i	$0.653419\pi$
$480$	0	0
$481$	−22536.3	−2.13632
$482$	0	0
$483$	−11552.5	−1.08832
$484$	0	0
$485$	18044.6	1.68941
$486$	0	0
$487$	9626.93	0.895766	0.447883	−	0.894092i	$-0.352178\pi$
$487$	9626.93	0.895766	0.447883	+	0.894092i	$0.352178\pi$
$488$	0	0
$489$	−7509.85	−0.694493
$490$	0	0
$491$	6069.20	0.557839	0.278920	−	0.960314i	$-0.410024\pi$
$491$	6069.20	0.557839	0.278920	+	0.960314i	$0.410024\pi$
$492$	0	0
$493$	3029.86	0.276791
$494$	0	0
$495$	−3142.31	−0.285326
$496$	0	0
$497$	−13195.9	−1.19098
$498$	0	0
$499$	−5046.71	−0.452749	−0.226374	−	0.974040i	$-0.572687\pi$
$499$	−5046.71	−0.452749	−0.226374	+	0.974040i	$0.572687\pi$
$500$	0	0
$501$	−22483.0	−2.00492
$502$	0	0
$503$	20967.0	1.85859	0.929295	−	0.369337i	$-0.120415\pi$
$503$	20967.0	1.85859	0.929295	+	0.369337i	$0.120415\pi$
$504$	0	0
$505$	15110.4	1.33149
$506$	0	0
$507$	−18521.2	−1.62239
$508$	0	0
$509$	16658.3	1.45062	0.725309	−	0.688423i	$-0.241697\pi$
$509$	16658.3	1.45062	0.725309	+	0.688423i	$0.241697\pi$
$510$	0	0
$511$	24523.9	2.12304
$512$	0	0
$513$	−3476.34	−0.299190
$514$	0	0
$515$	15351.8	1.31356
$516$	0	0
$517$	−18306.5	−1.55729
$518$	0	0
$519$	−17617.7	−1.49004
$520$	0	0
$521$	−8897.98	−0.748230	−0.374115	−	0.927382i	$-0.622053\pi$
$521$	−8897.98	−0.748230	−0.374115	+	0.927382i	$0.622053\pi$
$522$	0	0
$523$	9790.56	0.818568	0.409284	−	0.912407i	$-0.365778\pi$
$523$	9790.56	0.818568	0.409284	+	0.912407i	$0.365778\pi$
$524$	0	0
$525$	41691.9	3.46588
$526$	0	0
$527$	130.709	0.0108041
$528$	0	0
$529$	−6429.60	−0.528446
$530$	0	0
$531$	−4389.88	−0.358766
$532$	0	0
$533$	−32145.0	−2.61230
$534$	0	0
$535$	14935.3	1.20693
$536$	0	0
$537$	17888.3	1.43750
$538$	0	0
$539$	−11078.9	−0.885343
$540$	0	0
$541$	−3627.61	−0.288287	−0.144143	−	0.989557i	$-0.546043\pi$
$541$	−3627.61	−0.288287	−0.144143	+	0.989557i	$0.546043\pi$
$542$	0	0
$543$	10666.9	0.843020
$544$	0	0
$545$	−29632.9	−2.32906
$546$	0	0
$547$	8761.49	0.684852	0.342426	−	0.939545i	$-0.388751\pi$
$547$	8761.49	0.684852	0.342426	+	0.939545i	$0.388751\pi$
$548$	0	0
$549$	−1768.09	−0.137451
$550$	0	0
$551$	5030.84	0.388967
$552$	0	0
$553$	−6363.50	−0.489337
$554$	0	0
$555$	−34640.7	−2.64940
$556$	0	0
$557$	−4950.01	−0.376551	−0.188275	−	0.982116i	$-0.560290\pi$
$557$	−4950.01	−0.376551	−0.188275	+	0.982116i	$0.560290\pi$
$558$	0	0
$559$	26021.3	1.96885
$560$	0	0
$561$	−2847.61	−0.214307
$562$	0	0
$563$	−8404.91	−0.629173	−0.314587	−	0.949229i	$-0.601866\pi$
$563$	−8404.91	−0.629173	−0.314587	+	0.949229i	$0.601866\pi$
$564$	0	0
$565$	−6197.30	−0.461455
$566$	0	0
$567$	22653.9	1.67791
$568$	0	0
$569$	1290.85	0.0951061	0.0475530	−	0.998869i	$-0.484858\pi$
$569$	1290.85	0.0951061	0.0475530	+	0.998869i	$0.484858\pi$
$570$	0	0
$571$	17502.9	1.28279	0.641395	−	0.767211i	$-0.278356\pi$
$571$	17502.9	1.28279	0.641395	+	0.767211i	$0.278356\pi$
$572$	0	0
$573$	−25963.2	−1.89290
$574$	0	0
$575$	−20705.7	−1.50172
$576$	0	0
$577$	16298.3	1.17592	0.587962	−	0.808888i	$-0.299930\pi$
$577$	16298.3	1.17592	0.587962	+	0.808888i	$0.299930\pi$
$578$	0	0
$579$	11683.8	0.838621
$580$	0	0
$581$	4310.26	0.307779
$582$	0	0
$583$	−6261.98	−0.444845
$584$	0	0
$585$	−7878.88	−0.556840
$586$	0	0
$587$	15111.5	1.06255	0.531277	−	0.847198i	$-0.321712\pi$
$587$	15111.5	1.06255	0.531277	+	0.847198i	$0.321712\pi$
$588$	0	0
$589$	217.031	0.0151827
$590$	0	0
$591$	−9167.56	−0.638076
$592$	0	0
$593$	−573.060	−0.0396842	−0.0198421	−	0.999803i	$-0.506316\pi$
$593$	−573.060	−0.0396842	−0.0198421	+	0.999803i	$0.506316\pi$
$594$	0	0
$595$	9099.09	0.626935
$596$	0	0
$597$	6781.38	0.464897
$598$	0	0
$599$	−24314.5	−1.65853	−0.829267	−	0.558853i	$-0.811242\pi$
$599$	−24314.5	−1.65853	−0.829267	+	0.558853i	$0.811242\pi$
$600$	0	0
$601$	−24660.7	−1.67376	−0.836881	−	0.547385i	$-0.815623\pi$
$601$	−24660.7	−1.67376	−0.836881	+	0.547385i	$0.815623\pi$
$602$	0	0
$603$	2158.43	0.145768
$604$	0	0
$605$	−9251.73	−0.621713
$606$	0	0
$607$	−22539.3	−1.50715	−0.753576	−	0.657361i	$-0.771673\pi$
$607$	−22539.3	−1.50715	−0.753576	+	0.657361i	$0.771673\pi$
$608$	0	0
$609$	−27182.7	−1.80870
$610$	0	0
$611$	−45900.8	−3.03920
$612$	0	0
$613$	−15592.5	−1.02737	−0.513683	−	0.857980i	$-0.671719\pi$
$613$	−15592.5	−1.02737	−0.513683	+	0.857980i	$0.671719\pi$
$614$	0	0
$615$	−49410.2	−3.23969
$616$	0	0
$617$	−11433.8	−0.746039	−0.373019	−	0.927824i	$-0.621677\pi$
$617$	−11433.8	−0.746039	−0.373019	+	0.927824i	$0.621677\pi$
$618$	0	0
$619$	−7651.66	−0.496844	−0.248422	−	0.968652i	$-0.579912\pi$
$619$	−7651.66	−0.496844	−0.248422	+	0.968652i	$0.579912\pi$
$620$	0	0
$621$	−9328.52	−0.602803
$622$	0	0
$623$	−7605.91	−0.489124
$624$	0	0
$625$	24930.2	1.59553
$626$	0	0
$627$	−4728.24	−0.301160
$628$	0	0
$629$	−5187.90	−0.328863
$630$	0	0
$631$	−18034.4	−1.13778	−0.568889	−	0.822414i	$-0.692627\pi$
$631$	−18034.4	−1.13778	−0.568889	+	0.822414i	$0.692627\pi$
$632$	0	0
$633$	12850.0	0.806862
$634$	0	0
$635$	−18560.6	−1.15993
$636$	0	0
$637$	−27778.6	−1.72783
$638$	0	0
$639$	2630.36	0.162841
$640$	0	0
$641$	9882.60	0.608954	0.304477	−	0.952520i	$-0.401518\pi$
$641$	9882.60	0.608954	0.304477	+	0.952520i	$0.401518\pi$
$642$	0	0
$643$	28633.7	1.75615	0.878074	−	0.478524i	$-0.158828\pi$
$643$	28633.7	1.75615	0.878074	+	0.478524i	$0.158828\pi$
$644$	0	0
$645$	39997.5	2.44171
$646$	0	0
$647$	−3735.91	−0.227007	−0.113504	−	0.993538i	$-0.536207\pi$
$647$	−3735.91	−0.227007	−0.113504	+	0.993538i	$0.536207\pi$
$648$	0	0
$649$	24187.5	1.46293
$650$	0	0
$651$	−1172.67	−0.0705997
$652$	0	0
$653$	−19905.9	−1.19292	−0.596462	−	0.802642i	$-0.703427\pi$
$653$	−19905.9	−1.19292	−0.596462	+	0.802642i	$0.703427\pi$
$654$	0	0
$655$	38115.1	2.27371
$656$	0	0
$657$	−4888.36	−0.290279
$658$	0	0
$659$	1179.76	0.0697375	0.0348688	−	0.999392i	$-0.488899\pi$
$659$	1179.76	0.0697375	0.0348688	+	0.999392i	$0.488899\pi$
$660$	0	0
$661$	22433.8	1.32008	0.660042	−	0.751229i	$-0.270539\pi$
$661$	22433.8	1.32008	0.660042	+	0.751229i	$0.270539\pi$
$662$	0	0
$663$	−7139.97	−0.418241
$664$	0	0
$665$	15108.3	0.881016
$666$	0	0
$667$	13499.9	0.783686
$668$	0	0
$669$	2296.11	0.132695
$670$	0	0
$671$	9741.91	0.560480
$672$	0	0
$673$	9015.80	0.516395	0.258197	−	0.966092i	$-0.416872\pi$
$673$	9015.80	0.516395	0.258197	+	0.966092i	$0.416872\pi$
$674$	0	0
$675$	33665.8	1.91970
$676$	0	0
$677$	−25703.2	−1.45916	−0.729582	−	0.683894i	$-0.760285\pi$
$677$	−25703.2	−1.45916	−0.729582	+	0.683894i	$0.760285\pi$
$678$	0	0
$679$	−24244.9	−1.37030
$680$	0	0
$681$	9794.43	0.551135
$682$	0	0
$683$	−7053.48	−0.395160	−0.197580	−	0.980287i	$-0.563308\pi$
$683$	−7053.48	−0.395160	−0.197580	+	0.980287i	$0.563308\pi$
$684$	0	0
$685$	8320.61	0.464108
$686$	0	0
$687$	2644.94	0.146886
$688$	0	0
$689$	−15701.0	−0.868157
$690$	0	0
$691$	25767.0	1.41856	0.709279	−	0.704928i	$-0.249021\pi$
$691$	25767.0	1.41856	0.709279	+	0.704928i	$0.249021\pi$
$692$	0	0
$693$	4222.05	0.231432
$694$	0	0
$695$	9243.61	0.504503
$696$	0	0
$697$	−7399.83	−0.402136
$698$	0	0
$699$	−18412.6	−0.996319
$700$	0	0
$701$	31678.1	1.70680	0.853400	−	0.521257i	$-0.174537\pi$
$701$	31678.1	1.70680	0.853400	+	0.521257i	$0.174537\pi$
$702$	0	0
$703$	−8614.10	−0.462143
$704$	0	0
$705$	−70554.4	−3.76912
$706$	0	0
$707$	−20302.6	−1.07999
$708$	0	0
$709$	20171.9	1.06851	0.534253	−	0.845325i	$-0.320593\pi$
$709$	20171.9	1.06851	0.534253	+	0.845325i	$0.320593\pi$
$710$	0	0
$711$	1268.44	0.0669061
$712$	0	0
$713$	582.389	0.0305899
$714$	0	0
$715$	43411.3	2.27062
$716$	0	0
$717$	−9136.31	−0.475874
$718$	0	0
$719$	−2026.27	−0.105100	−0.0525502	−	0.998618i	$-0.516735\pi$
$719$	−2026.27	−0.105100	−0.0525502	+	0.998618i	$0.516735\pi$
$720$	0	0
$721$	−20627.0	−1.06545
$722$	0	0
$723$	36856.0	1.89584
$724$	0	0
$725$	−48719.9	−2.49574
$726$	0	0
$727$	−7126.24	−0.363545	−0.181773	−	0.983341i	$-0.558184\pi$
$727$	−7126.24	−0.363545	−0.181773	+	0.983341i	$0.558184\pi$
$728$	0	0
$729$	14395.9	0.731387
$730$	0	0
$731$	5990.15	0.303083
$732$	0	0
$733$	−2660.98	−0.134087	−0.0670433	−	0.997750i	$-0.521357\pi$
$733$	−2660.98	−0.134087	−0.0670433	+	0.997750i	$0.521357\pi$
$734$	0	0
$735$	−42698.6	−2.14281
$736$	0	0
$737$	−11892.6	−0.594395
$738$	0	0
$739$	−36239.0	−1.80389	−0.901943	−	0.431855i	$-0.857859\pi$
$739$	−36239.0	−1.80389	−0.901943	+	0.431855i	$0.857859\pi$
$740$	0	0
$741$	−11855.4	−0.587743
$742$	0	0
$743$	23036.0	1.13743	0.568713	−	0.822536i	$-0.307441\pi$
$743$	23036.0	1.13743	0.568713	+	0.822536i	$0.307441\pi$
$744$	0	0
$745$	40953.9	2.01401
$746$	0	0
$747$	−859.167	−0.0420820
$748$	0	0
$749$	−20067.3	−0.978963
$750$	0	0
$751$	−34563.6	−1.67942	−0.839708	−	0.543038i	$-0.817274\pi$
$751$	−34563.6	−1.67942	−0.839708	+	0.543038i	$0.817274\pi$
$752$	0	0
$753$	−13986.3	−0.676879
$754$	0	0
$755$	16389.8	0.790048
$756$	0	0
$757$	5567.28	0.267300	0.133650	−	0.991029i	$-0.457330\pi$
$757$	5567.28	0.267300	0.133650	+	0.991029i	$0.457330\pi$
$758$	0	0
$759$	−12687.9	−0.606774
$760$	0	0
$761$	23973.3	1.14196	0.570980	−	0.820964i	$-0.306563\pi$
$761$	23973.3	1.14196	0.570980	+	0.820964i	$0.306563\pi$
$762$	0	0
$763$	39815.3	1.88913
$764$	0	0
$765$	−1813.73	−0.0857197
$766$	0	0
$767$	60646.7	2.85505
$768$	0	0
$769$	12135.2	0.569058	0.284529	−	0.958667i	$-0.408163\pi$
$769$	12135.2	0.569058	0.284529	+	0.958667i	$0.408163\pi$
$770$	0	0
$771$	−34818.2	−1.62639
$772$	0	0
$773$	20259.5	0.942668	0.471334	−	0.881955i	$-0.343773\pi$
$773$	20259.5	0.942668	0.471334	+	0.881955i	$0.343773\pi$
$774$	0	0
$775$	−2101.79	−0.0974173
$776$	0	0
$777$	46543.7	2.14897
$778$	0	0
$779$	−12286.8	−0.565111
$780$	0	0
$781$	−14492.8	−0.664013
$782$	0	0
$783$	−21949.7	−1.00181
$784$	0	0
$785$	−1416.80	−0.0644176
$786$	0	0
$787$	−12068.5	−0.546629	−0.273314	−	0.961925i	$-0.588120\pi$
$787$	−12068.5	−0.546629	−0.273314	+	0.961925i	$0.588120\pi$
$788$	0	0
$789$	8784.87	0.396387
$790$	0	0
$791$	8326.78	0.374294
$792$	0	0
$793$	24426.4	1.09383
$794$	0	0
$795$	−24134.1	−1.07666
$796$	0	0
$797$	3518.65	0.156382	0.0781912	−	0.996938i	$-0.475086\pi$
$797$	3518.65	0.156382	0.0781912	+	0.996938i	$0.475086\pi$
$798$	0	0
$799$	−10566.4	−0.467852
$800$	0	0
$801$	1516.09	0.0668770
$802$	0	0
$803$	26934.1	1.18367
$804$	0	0
$805$	40542.1	1.77506
$806$	0	0
$807$	24817.6	1.08255
$808$	0	0
$809$	13999.2	0.608387	0.304193	−	0.952610i	$-0.401613\pi$
$809$	13999.2	0.608387	0.304193	+	0.952610i	$0.401613\pi$
$810$	0	0
$811$	−22166.1	−0.959749	−0.479874	−	0.877337i	$-0.659318\pi$
$811$	−22166.1	−0.959749	−0.479874	+	0.877337i	$0.659318\pi$
$812$	0	0
$813$	−7552.09	−0.325785
$814$	0	0
$815$	26354.9	1.13273
$816$	0	0
$817$	9946.18	0.425915
$818$	0	0
$819$	10586.2	0.451662
$820$	0	0
$821$	−2829.06	−0.120262	−0.0601310	−	0.998190i	$-0.519152\pi$
$821$	−2829.06	−0.120262	−0.0601310	+	0.998190i	$0.519152\pi$
$822$	0	0
$823$	−21071.4	−0.892469	−0.446235	−	0.894916i	$-0.647235\pi$
$823$	−21071.4	−0.892469	−0.446235	+	0.894916i	$0.647235\pi$
$824$	0	0
$825$	45789.4	1.93234
$826$	0	0
$827$	−26448.6	−1.11210	−0.556051	−	0.831148i	$-0.687684\pi$
$827$	−26448.6	−1.11210	−0.556051	+	0.831148i	$0.687684\pi$
$828$	0	0
$829$	20166.7	0.844894	0.422447	−	0.906388i	$-0.361171\pi$
$829$	20166.7	0.844894	0.422447	+	0.906388i	$0.361171\pi$
$830$	0	0
$831$	−10960.3	−0.457530
$832$	0	0
$833$	−6394.68	−0.265981
$834$	0	0
$835$	78901.2	3.27005
$836$	0	0
$837$	−946.915	−0.0391041
$838$	0	0
$839$	2005.72	0.0825330	0.0412665	−	0.999148i	$-0.486861\pi$
$839$	2005.72	0.0825330	0.0412665	+	0.999148i	$0.486861\pi$
$840$	0	0
$841$	7375.83	0.302425
$842$	0	0
$843$	14865.1	0.607333
$844$	0	0
$845$	64997.8	2.64614
$846$	0	0
$847$	12430.8	0.504281
$848$	0	0
$849$	−3776.83	−0.152674
$850$	0	0
$851$	−23115.3	−0.931120
$852$	0	0
$853$	−1319.83	−0.0529778	−0.0264889	−	0.999649i	$-0.508433\pi$
$853$	−1319.83	−0.0529778	−0.0264889	+	0.999649i	$0.508433\pi$
$854$	0	0
$855$	−3011.56	−0.120460
$856$	0	0
$857$	−24666.9	−0.983202	−0.491601	−	0.870820i	$-0.663588\pi$
$857$	−24666.9	−0.983202	−0.491601	+	0.870820i	$0.663588\pi$
$858$	0	0
$859$	10670.9	0.423848	0.211924	−	0.977286i	$-0.432027\pi$
$859$	10670.9	0.423848	0.211924	+	0.977286i	$0.432027\pi$
$860$	0	0
$861$	66388.3	2.62777
$862$	0	0
$863$	4808.70	0.189676	0.0948378	−	0.995493i	$-0.469767\pi$
$863$	4808.70	0.189676	0.0948378	+	0.995493i	$0.469767\pi$
$864$	0	0
$865$	61827.3	2.43028
$866$	0	0
$867$	−1643.63	−0.0643837
$868$	0	0
$869$	−6988.90	−0.272822
$870$	0	0
$871$	−29818.9	−1.16002
$872$	0	0
$873$	4832.77	0.187359
$874$	0	0
$875$	−79407.7	−3.06797
$876$	0	0
$877$	26224.7	1.00974	0.504871	−	0.863195i	$-0.331540\pi$
$877$	26224.7	1.00974	0.504871	+	0.863195i	$0.331540\pi$
$878$	0	0
$879$	39349.5	1.50993
$880$	0	0
$881$	−15165.9	−0.579968	−0.289984	−	0.957031i	$-0.593650\pi$
$881$	−15165.9	−0.579968	−0.289984	+	0.957031i	$0.593650\pi$
$882$	0	0
$883$	−50530.4	−1.92580	−0.962901	−	0.269855i	$-0.913024\pi$
$883$	−50530.4	−1.92580	−0.962901	+	0.269855i	$0.913024\pi$
$884$	0	0
$885$	93220.3	3.54075
$886$	0	0
$887$	−36785.1	−1.39247	−0.696235	−	0.717814i	$-0.745143\pi$
$887$	−36785.1	−1.39247	−0.696235	+	0.717814i	$0.745143\pi$
$888$	0	0
$889$	24938.3	0.940835
$890$	0	0
$891$	24880.3	0.935489
$892$	0	0
$893$	−17544.7	−0.657461
$894$	0	0
$895$	−62776.9	−2.34458
$896$	0	0
$897$	−31813.0	−1.18418
$898$	0	0
$899$	1370.34	0.0508381
$900$	0	0
$901$	−3614.39	−0.133644
$902$	0	0
$903$	−53741.3	−1.98051
$904$	0	0
$905$	−37434.2	−1.37498
$906$	0	0
$907$	−9326.27	−0.341426	−0.170713	−	0.985321i	$-0.554607\pi$
$907$	−9326.27	−0.341426	−0.170713	+	0.985321i	$0.554607\pi$
$908$	0	0
$909$	4046.93	0.147666
$910$	0	0
$911$	−19471.5	−0.708146	−0.354073	−	0.935218i	$-0.615204\pi$
$911$	−19471.5	−0.708146	−0.354073	+	0.935218i	$0.615204\pi$
$912$	0	0
$913$	4733.87	0.171597
$914$	0	0
$915$	37545.9	1.35654
$916$	0	0
$917$	−51212.0	−1.84424
$918$	0	0
$919$	−37800.6	−1.35683	−0.678415	−	0.734679i	$-0.737333\pi$
$919$	−37800.6	−1.35683	−0.678415	+	0.734679i	$0.737333\pi$
$920$	0	0
$921$	−4903.45	−0.175433
$922$	0	0
$923$	−36338.7	−1.29588
$924$	0	0
$925$	83421.0	2.96526
$926$	0	0
$927$	4111.59	0.145677
$928$	0	0
$929$	5788.41	0.204426	0.102213	−	0.994763i	$-0.467408\pi$
$929$	5788.41	0.204426	0.102213	+	0.994763i	$0.467408\pi$
$930$	0	0
$931$	−10617.9	−0.373777
$932$	0	0
$933$	−22680.7	−0.795855
$934$	0	0
$935$	9993.36	0.349538
$936$	0	0
$937$	1424.84	0.0496770	0.0248385	−	0.999691i	$-0.492093\pi$
$937$	1424.84	0.0496770	0.0248385	+	0.999691i	$0.492093\pi$
$938$	0	0
$939$	−11453.6	−0.398054
$940$	0	0
$941$	−43148.8	−1.49480	−0.747402	−	0.664372i	$-0.768699\pi$
$941$	−43148.8	−1.49480	−0.747402	+	0.664372i	$0.768699\pi$
$942$	0	0
$943$	−32970.8	−1.13858
$944$	0	0
$945$	−65918.1	−2.26912
$946$	0	0
$947$	−5908.66	−0.202751	−0.101376	−	0.994848i	$-0.532324\pi$
$947$	−5908.66	−0.202751	−0.101376	+	0.994848i	$0.532324\pi$
$948$	0	0
$949$	67533.3	2.31003
$950$	0	0
$951$	−12426.8	−0.423731
$952$	0	0
$953$	−22811.7	−0.775388	−0.387694	−	0.921788i	$-0.626728\pi$
$953$	−22811.7	−0.775388	−0.387694	+	0.921788i	$0.626728\pi$
$954$	0	0
$955$	91114.8	3.08734
$956$	0	0
$957$	−29854.2	−1.00841
$958$	0	0
$959$	−11179.7	−0.376445
$960$	0	0
$961$	−29731.9	−0.998016
$962$	0	0
$963$	4000.03	0.133852
$964$	0	0
$965$	−41002.9	−1.36780
$966$	0	0
$967$	2684.44	0.0892716	0.0446358	−	0.999003i	$-0.485787\pi$
$967$	2684.44	0.0892716	0.0446358	+	0.999003i	$0.485787\pi$
$968$	0	0
$969$	−2729.12	−0.0904768
$970$	0	0
$971$	37995.1	1.25574	0.627868	−	0.778320i	$-0.283928\pi$
$971$	37995.1	1.25574	0.627868	+	0.778320i	$0.283928\pi$
$972$	0	0
$973$	−12419.8	−0.409211
$974$	0	0
$975$	114810.	3.77115
$976$	0	0
$977$	37877.6	1.24034	0.620170	−	0.784468i	$-0.287064\pi$
$977$	37877.6	1.24034	0.620170	+	0.784468i	$0.287064\pi$
$978$	0	0
$979$	−8353.43	−0.272703
$980$	0	0
$981$	−7936.41	−0.258298
$982$	0	0
$983$	−21617.0	−0.701399	−0.350699	−	0.936488i	$-0.614056\pi$
$983$	−21617.0	−0.701399	−0.350699	+	0.936488i	$0.614056\pi$
$984$	0	0
$985$	32172.4	1.04071
$986$	0	0
$987$	94797.9	3.05719
$988$	0	0
$989$	26689.9	0.858128
$990$	0	0
$991$	24005.8	0.769495	0.384748	−	0.923022i	$-0.374288\pi$
$991$	24005.8	0.769495	0.384748	+	0.923022i	$0.374288\pi$
$992$	0	0
$993$	53730.4	1.71710
$994$	0	0
$995$	−23798.4	−0.758252
$996$	0	0
$997$	−23206.7	−0.737175	−0.368587	−	0.929593i	$-0.620158\pi$
$997$	−23206.7	−0.737175	−0.368587	+	0.929593i	$0.620158\pi$
$998$	0	0
$999$	37583.6	1.19028

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.bg.1.2		7
4.3	odd	2		1088.4.a.bh.1.6		7
8.3	odd	2		544.4.a.l.1.2	yes	7
8.5	even	2		544.4.a.k.1.6	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.4.a.k.1.6	✓	7	8.5	even	2
544.4.a.l.1.2	yes	7	8.3	odd	2
1088.4.a.bg.1.2		7	1.1	even	1	trivial
1088.4.a.bh.1.6		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-5.68731 q^{3} +19.9589 q^{5} -26.8171 q^{7} +5.34548 q^{9} +O(q^{10})\)	\(q-5.68731 q^{3} +19.9589 q^{5} -26.8171 q^{7} +5.34548 q^{9} -29.4527 q^{11} -73.8483 q^{13} -113.513 q^{15} -17.0000 q^{17} -28.2272 q^{19} +152.517 q^{21} -75.7456 q^{23} +273.359 q^{25} +123.156 q^{27} -178.227 q^{29} -7.68875 q^{31} +167.507 q^{33} -535.241 q^{35} +305.170 q^{37} +419.998 q^{39} +435.284 q^{41} -352.362 q^{43} +106.690 q^{45} +621.555 q^{47} +376.158 q^{49} +96.6842 q^{51} +212.611 q^{53} -587.844 q^{55} +160.537 q^{57} -821.233 q^{59} -330.764 q^{61} -143.350 q^{63} -1473.93 q^{65} +403.786 q^{67} +430.789 q^{69} +492.072 q^{71} -914.486 q^{73} -1554.68 q^{75} +789.837 q^{77} +237.292 q^{79} -844.754 q^{81} -160.728 q^{83} -339.302 q^{85} +1013.63 q^{87} +283.622 q^{89} +1980.40 q^{91} +43.7283 q^{93} -563.384 q^{95} +904.085 q^{97} -157.439 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{7} + 67 q^{9}+O(q^{10}) \) 7 * q - 2 * q^7 + 67 * q^9	\( 7 q - 2 q^{7} + 67 q^{9} - 108 q^{11} + 34 q^{13} + 128 q^{15} - 119 q^{17} - 124 q^{19} + 296 q^{21} - 6 q^{23} + 197 q^{25} - 248 q^{29} + 50 q^{31} + 512 q^{33} - 640 q^{35} + 484 q^{37} + 1504 q^{39} - 366 q^{41} - 1412 q^{43} + 80 q^{45} + 1012 q^{47} + 1115 q^{49} + 146 q^{53} + 1024 q^{55} + 48 q^{57} - 2332 q^{59} + 548 q^{61} + 2838 q^{63} - 208 q^{65} - 924 q^{67} + 1672 q^{69} + 1286 q^{71} + 870 q^{73} - 3136 q^{75} + 1344 q^{77} + 1818 q^{79} + 3039 q^{81} - 1772 q^{83} + 384 q^{87} + 1706 q^{89} - 588 q^{91} + 5576 q^{93} + 2048 q^{95} + 1802 q^{97} - 5148 q^{99}+O(q^{100}) \) 7 * q - 2 * q^7 + 67 * q^9 - 108 * q^11 + 34 * q^13 + 128 * q^15 - 119 * q^17 - 124 * q^19 + 296 * q^21 - 6 * q^23 + 197 * q^25 - 248 * q^29 + 50 * q^31 + 512 * q^33 - 640 * q^35 + 484 * q^37 + 1504 * q^39 - 366 * q^41 - 1412 * q^43 + 80 * q^45 + 1012 * q^47 + 1115 * q^49 + 146 * q^53 + 1024 * q^55 + 48 * q^57 - 2332 * q^59 + 548 * q^61 + 2838 * q^63 - 208 * q^65 - 924 * q^67 + 1672 * q^69 + 1286 * q^71 + 870 * q^73 - 3136 * q^75 + 1344 * q^77 + 1818 * q^79 + 3039 * q^81 - 1772 * q^83 + 384 * q^87 + 1706 * q^89 - 588 * q^91 + 5576 * q^93 + 2048 * q^95 + 1802 * q^97 - 5148 * q^99

Introduction

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