LMFDB - Embedded newform 1088.4.a.bh.1.4

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			1.4
Root			$2.67896$ 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)

$f(q)$	$=$	$q-1.23530 q^{3} -8.71094 q^{5} +33.7897 q^{7} -25.4740 q^{9} +O(q^{10})$	$q-1.23530 q^{3} -8.71094 q^{5} +33.7897 q^{7} -25.4740 q^{9} +57.0863 q^{11} +84.2673 q^{13} +10.7606 q^{15} -17.0000 q^{17} +159.552 q^{19} -41.7403 q^{21} -57.9475 q^{23} -49.1195 q^{25} +64.8210 q^{27} -124.693 q^{29} -98.3364 q^{31} -70.5185 q^{33} -294.340 q^{35} -355.131 q^{37} -104.095 q^{39} -67.6160 q^{41} +218.558 q^{43} +221.903 q^{45} +312.426 q^{47} +798.744 q^{49} +21.0000 q^{51} +148.689 q^{53} -497.276 q^{55} -197.094 q^{57} +7.67524 q^{59} +456.058 q^{61} -860.760 q^{63} -734.048 q^{65} -47.9472 q^{67} +71.5823 q^{69} -786.703 q^{71} +670.825 q^{73} +60.6771 q^{75} +1928.93 q^{77} +34.1833 q^{79} +607.726 q^{81} -322.765 q^{83} +148.086 q^{85} +154.033 q^{87} +417.620 q^{89} +2847.37 q^{91} +121.475 q^{93} -1389.85 q^{95} +996.046 q^{97} -1454.22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.23530	−0.237733	−0.118866	−	0.992910i	$-0.537926\pi$
$3$	−1.23530	−0.237733	−0.118866	+	0.992910i	$0.537926\pi$
$4$	0	0
$5$	−8.71094	−0.779130	−0.389565	−	0.920999i	$-0.627375\pi$
$5$	−8.71094	−0.779130	−0.389565	+	0.920999i	$0.627375\pi$
$6$	0	0
$7$	33.7897	1.82447	0.912236	−	0.409665i	$-0.134354\pi$
$7$	33.7897	1.82447	0.912236	+	0.409665i	$0.134354\pi$
$8$	0	0
$9$	−25.4740	−0.943483
$10$	0	0
$11$	57.0863	1.56474	0.782372	−	0.622812i	$-0.214010\pi$
$11$	57.0863	1.56474	0.782372	+	0.622812i	$0.214010\pi$
$12$	0	0
$13$	84.2673	1.79781	0.898906	−	0.438142i	$-0.144363\pi$
$13$	84.2673	1.79781	0.898906	+	0.438142i	$0.144363\pi$
$14$	0	0
$15$	10.7606	0.185225
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	159.552	1.92652	0.963258	−	0.268576i	$-0.0865532\pi$
$19$	159.552	1.92652	0.963258	+	0.268576i	$0.0865532\pi$
$20$	0	0
$21$	−41.7403	−0.433737
$22$	0	0
$23$	−57.9475	−0.525343	−0.262671	−	0.964885i	$-0.584604\pi$
$23$	−57.9475	−0.525343	−0.262671	+	0.964885i	$0.584604\pi$
$24$	0	0
$25$	−49.1195	−0.392956
$26$	0	0
$27$	64.8210	0.462030
$28$	0	0
$29$	−124.693	−0.798445	−0.399223	−	0.916854i	$-0.630720\pi$
$29$	−124.693	−0.798445	−0.399223	+	0.916854i	$0.630720\pi$
$30$	0	0
$31$	−98.3364	−0.569734	−0.284867	−	0.958567i	$-0.591949\pi$
$31$	−98.3364	−0.569734	−0.284867	+	0.958567i	$0.591949\pi$
$32$	0	0
$33$	−70.5185	−0.371991
$34$	0	0
$35$	−294.340	−1.42150
$36$	0	0
$37$	−355.131	−1.57793	−0.788963	−	0.614441i	$-0.789382\pi$
$37$	−355.131	−1.57793	−0.788963	+	0.614441i	$0.789382\pi$
$38$	0	0
$39$	−104.095	−0.427399
$40$	0	0
$41$	−67.6160	−0.257557	−0.128779	−	0.991673i	$-0.541106\pi$
$41$	−67.6160	−0.257557	−0.128779	+	0.991673i	$0.541106\pi$
$42$	0	0
$43$	218.558	0.775111	0.387556	−	0.921846i	$-0.373319\pi$
$43$	218.558	0.775111	0.387556	+	0.921846i	$0.373319\pi$
$44$	0	0
$45$	221.903	0.735096
$46$	0	0
$47$	312.426	0.969618	0.484809	−	0.874620i	$-0.338889\pi$
$47$	312.426	0.969618	0.484809	+	0.874620i	$0.338889\pi$
$48$	0	0
$49$	798.744	2.32870
$50$	0	0
$51$	21.0000	0.0576587
$52$	0	0
$53$	148.689	0.385358	0.192679	−	0.981262i	$-0.438282\pi$
$53$	148.689	0.385358	0.192679	+	0.981262i	$0.438282\pi$
$54$	0	0
$55$	−497.276	−1.21914
$56$	0	0
$57$	−197.094	−0.457996
$58$	0	0
$59$	7.67524	0.0169361	0.00846806	−	0.999964i	$-0.497304\pi$
$59$	7.67524	0.0169361	0.00846806	+	0.999964i	$0.497304\pi$
$60$	0	0
$61$	456.058	0.957251	0.478625	−	0.878019i	$-0.341135\pi$
$61$	456.058	0.957251	0.478625	+	0.878019i	$0.341135\pi$
$62$	0	0
$63$	−860.760	−1.72136
$64$	0	0
$65$	−734.048	−1.40073
$66$	0	0
$67$	−47.9472	−0.0874281	−0.0437141	−	0.999044i	$-0.513919\pi$
$67$	−47.9472	−0.0874281	−0.0437141	+	0.999044i	$0.513919\pi$
$68$	0	0
$69$	71.5823	0.124891
$70$	0	0
$71$	−786.703	−1.31499	−0.657497	−	0.753457i	$-0.728385\pi$
$71$	−786.703	−1.31499	−0.657497	+	0.753457i	$0.728385\pi$
$72$	0	0
$73$	670.825	1.07554	0.537768	−	0.843093i	$-0.319268\pi$
$73$	670.825	1.07554	0.537768	+	0.843093i	$0.319268\pi$
$74$	0	0
$75$	60.6771	0.0934185
$76$	0	0
$77$	1928.93	2.85483
$78$	0	0
$79$	34.1833	0.0486825	0.0243413	−	0.999704i	$-0.492251\pi$
$79$	34.1833	0.0486825	0.0243413	+	0.999704i	$0.492251\pi$
$80$	0	0
$81$	607.726	0.833644
$82$	0	0
$83$	−322.765	−0.426844	−0.213422	−	0.976960i	$-0.568461\pi$
$83$	−322.765	−0.426844	−0.213422	+	0.976960i	$0.568461\pi$
$84$	0	0
$85$	148.086	0.188967
$86$	0	0
$87$	154.033	0.189817
$88$	0	0
$89$	417.620	0.497389	0.248695	−	0.968582i	$-0.419998\pi$
$89$	417.620	0.497389	0.248695	+	0.968582i	$0.419998\pi$
$90$	0	0
$91$	2847.37	3.28006
$92$	0	0
$93$	121.475	0.135444
$94$	0	0
$95$	−1389.85	−1.50101
$96$	0	0
$97$	996.046	1.04261	0.521305	−	0.853370i	$-0.325445\pi$
$97$	996.046	1.04261	0.521305	+	0.853370i	$0.325445\pi$
$98$	0	0
$99$	−1454.22	−1.47631
$100$	0	0
$101$	−1039.92	−1.02452	−0.512259	−	0.858831i	$-0.671191\pi$
$101$	−1039.92	−1.02452	−0.512259	+	0.858831i	$0.671191\pi$
$102$	0	0
$103$	197.777	0.189199	0.0945997	−	0.995515i	$-0.469843\pi$
$103$	197.777	0.189199	0.0945997	+	0.995515i	$0.469843\pi$
$104$	0	0
$105$	363.597	0.337938
$106$	0	0
$107$	1656.13	1.49630	0.748149	−	0.663531i	$-0.230943\pi$
$107$	1656.13	1.49630	0.748149	+	0.663531i	$0.230943\pi$
$108$	0	0
$109$	−1740.18	−1.52916	−0.764582	−	0.644527i	$-0.777054\pi$
$109$	−1740.18	−1.52916	−0.764582	+	0.644527i	$0.777054\pi$
$110$	0	0
$111$	438.692	0.375125
$112$	0	0
$113$	170.469	0.141915	0.0709575	−	0.997479i	$-0.477395\pi$
$113$	170.469	0.141915	0.0709575	+	0.997479i	$0.477395\pi$
$114$	0	0
$115$	504.777	0.409311
$116$	0	0
$117$	−2146.63	−1.69620
$118$	0	0
$119$	−574.425	−0.442500
$120$	0	0
$121$	1927.85	1.44842
$122$	0	0
$123$	83.5257	0.0612297
$124$	0	0
$125$	1516.74	1.08529
$126$	0	0
$127$	−101.593	−0.0709835	−0.0354917	−	0.999370i	$-0.511300\pi$
$127$	−101.593	−0.0709835	−0.0354917	+	0.999370i	$0.511300\pi$
$128$	0	0
$129$	−269.984	−0.184269
$130$	0	0
$131$	1132.47	0.755298	0.377649	−	0.925949i	$-0.376733\pi$
$131$	1132.47	0.755298	0.377649	+	0.925949i	$0.376733\pi$
$132$	0	0
$133$	5391.23	3.51488
$134$	0	0
$135$	−564.652	−0.359981
$136$	0	0
$137$	−2830.26	−1.76500	−0.882502	−	0.470309i	$-0.844142\pi$
$137$	−2830.26	−1.76500	−0.882502	+	0.470309i	$0.844142\pi$
$138$	0	0
$139$	−637.849	−0.389221	−0.194610	−	0.980881i	$-0.562344\pi$
$139$	−637.849	−0.389221	−0.194610	+	0.980881i	$0.562344\pi$
$140$	0	0
$141$	−385.939	−0.230510
$142$	0	0
$143$	4810.51	2.81311
$144$	0	0
$145$	1086.19	0.622093
$146$	0	0
$147$	−986.685	−0.553608
$148$	0	0
$149$	1132.35	0.622589	0.311295	−	0.950313i	$-0.399237\pi$
$149$	1132.35	0.622589	0.311295	+	0.950313i	$0.399237\pi$
$150$	0	0
$151$	627.687	0.338281	0.169141	−	0.985592i	$-0.445901\pi$
$151$	627.687	0.338281	0.169141	+	0.985592i	$0.445901\pi$
$152$	0	0
$153$	433.059	0.228828
$154$	0	0
$155$	856.603	0.443897
$156$	0	0
$157$	1667.49	0.847646	0.423823	−	0.905745i	$-0.360688\pi$
$157$	1667.49	0.847646	0.423823	+	0.905745i	$0.360688\pi$
$158$	0	0
$159$	−183.675	−0.0916123
$160$	0	0
$161$	−1958.03	−0.958474
$162$	0	0
$163$	−362.114	−0.174006	−0.0870030	−	0.996208i	$-0.527729\pi$
$163$	−362.114	−0.174006	−0.0870030	+	0.996208i	$0.527729\pi$
$164$	0	0
$165$	614.283	0.289829
$166$	0	0
$167$	−1235.08	−0.572298	−0.286149	−	0.958185i	$-0.592375\pi$
$167$	−1235.08	−0.572298	−0.286149	+	0.958185i	$0.592375\pi$
$168$	0	0
$169$	4903.98	2.23213
$170$	0	0
$171$	−4064.44	−1.81764
$172$	0	0
$173$	1435.78	0.630984	0.315492	−	0.948928i	$-0.397830\pi$
$173$	1435.78	0.630984	0.315492	+	0.948928i	$0.397830\pi$
$174$	0	0
$175$	−1659.73	−0.716937
$176$	0	0
$177$	−9.48120	−0.00402627
$178$	0	0
$179$	1534.14	0.640598	0.320299	−	0.947316i	$-0.396217\pi$
$179$	1534.14	0.640598	0.320299	+	0.947316i	$0.396217\pi$
$180$	0	0
$181$	1055.30	0.433369	0.216684	−	0.976242i	$-0.430476\pi$
$181$	1055.30	0.433369	0.216684	+	0.976242i	$0.430476\pi$
$182$	0	0
$183$	−563.367	−0.227570
$184$	0	0
$185$	3093.53	1.22941
$186$	0	0
$187$	−970.468	−0.379506
$188$	0	0
$189$	2190.28	0.842960
$190$	0	0
$191$	1030.40	0.390351	0.195175	−	0.980768i	$-0.437472\pi$
$191$	1030.40	0.390351	0.195175	+	0.980768i	$0.437472\pi$
$192$	0	0
$193$	−3212.17	−1.19802	−0.599009	−	0.800743i	$-0.704439\pi$
$193$	−3212.17	−1.19802	−0.599009	+	0.800743i	$0.704439\pi$
$194$	0	0
$195$	906.766	0.332999
$196$	0	0
$197$	−4060.33	−1.46846	−0.734229	−	0.678902i	$-0.762456\pi$
$197$	−4060.33	−1.46846	−0.734229	+	0.678902i	$0.762456\pi$
$198$	0	0
$199$	3447.91	1.22822	0.614110	−	0.789220i	$-0.289515\pi$
$199$	3447.91	1.22822	0.614110	+	0.789220i	$0.289515\pi$
$200$	0	0
$201$	59.2290	0.0207845
$202$	0	0
$203$	−4213.34	−1.45674
$204$	0	0
$205$	588.999	0.200670
$206$	0	0
$207$	1476.16	0.495652
$208$	0	0
$209$	9108.26	3.01451
$210$	0	0
$211$	3884.01	1.26723	0.633617	−	0.773647i	$-0.281569\pi$
$211$	3884.01	1.26723	0.633617	+	0.773647i	$0.281569\pi$
$212$	0	0
$213$	971.811	0.312617
$214$	0	0
$215$	−1903.85	−0.603913
$216$	0	0
$217$	−3322.76	−1.03946
$218$	0	0
$219$	−828.667	−0.255690
$220$	0	0
$221$	−1432.54	−0.436033
$222$	0	0
$223$	324.141	0.0973367	0.0486683	−	0.998815i	$-0.484502\pi$
$223$	324.141	0.0973367	0.0486683	+	0.998815i	$0.484502\pi$
$224$	0	0
$225$	1251.27	0.370747
$226$	0	0
$227$	3481.99	1.01810	0.509048	−	0.860738i	$-0.329997\pi$
$227$	3481.99	1.01810	0.509048	+	0.860738i	$0.329997\pi$
$228$	0	0
$229$	282.700	0.0815780	0.0407890	−	0.999168i	$-0.487013\pi$
$229$	282.700	0.0815780	0.0407890	+	0.999168i	$0.487013\pi$
$230$	0	0
$231$	−2382.80	−0.678687
$232$	0	0
$233$	−6404.44	−1.80072	−0.900362	−	0.435141i	$-0.856698\pi$
$233$	−6404.44	−1.80072	−0.900362	+	0.435141i	$0.856698\pi$
$234$	0	0
$235$	−2721.53	−0.755459
$236$	0	0
$237$	−42.2265	−0.0115734
$238$	0	0
$239$	696.413	0.188482	0.0942410	−	0.995549i	$-0.469958\pi$
$239$	696.413	0.188482	0.0942410	+	0.995549i	$0.469958\pi$
$240$	0	0
$241$	−1225.35	−0.327518	−0.163759	−	0.986500i	$-0.552362\pi$
$241$	−1225.35	−0.327518	−0.163759	+	0.986500i	$0.552362\pi$
$242$	0	0
$243$	−2500.89	−0.660214
$244$	0	0
$245$	−6957.81	−1.81436
$246$	0	0
$247$	13445.0	3.46351
$248$	0	0
$249$	398.710	0.101475
$250$	0	0
$251$	−3888.55	−0.977862	−0.488931	−	0.872323i	$-0.662613\pi$
$251$	−3888.55	−0.977862	−0.488931	+	0.872323i	$0.662613\pi$
$252$	0	0
$253$	−3308.01	−0.822027
$254$	0	0
$255$	−182.930	−0.0449236
$256$	0	0
$257$	3627.25	0.880395	0.440197	−	0.897901i	$-0.354908\pi$
$257$	3627.25	0.880395	0.440197	+	0.897901i	$0.354908\pi$
$258$	0	0
$259$	−11999.8	−2.87888
$260$	0	0
$261$	3176.44	0.753320
$262$	0	0
$263$	−5870.64	−1.37642	−0.688212	−	0.725510i	$-0.741604\pi$
$263$	−5870.64	−1.37642	−0.688212	+	0.725510i	$0.741604\pi$
$264$	0	0
$265$	−1295.22	−0.300244
$266$	0	0
$267$	−515.884	−0.118246
$268$	0	0
$269$	1928.91	0.437203	0.218601	−	0.975814i	$-0.429851\pi$
$269$	1928.91	0.437203	0.218601	+	0.975814i	$0.429851\pi$
$270$	0	0
$271$	−5869.49	−1.31567	−0.657835	−	0.753162i	$-0.728527\pi$
$271$	−5869.49	−1.31567	−0.657835	+	0.753162i	$0.728527\pi$
$272$	0	0
$273$	−3517.34	−0.779777
$274$	0	0
$275$	−2804.05	−0.614875
$276$	0	0
$277$	1148.70	0.249165	0.124583	−	0.992209i	$-0.460241\pi$
$277$	1148.70	0.249165	0.124583	+	0.992209i	$0.460241\pi$
$278$	0	0
$279$	2505.03	0.537534
$280$	0	0
$281$	6175.93	1.31112	0.655561	−	0.755143i	$-0.272432\pi$
$281$	6175.93	1.31112	0.655561	+	0.755143i	$0.272432\pi$
$282$	0	0
$283$	5667.54	1.19046	0.595230	−	0.803555i	$-0.297061\pi$
$283$	5667.54	1.19046	0.595230	+	0.803555i	$0.297061\pi$
$284$	0	0
$285$	1716.88	0.356839
$286$	0	0
$287$	−2284.72	−0.469906
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−1230.41	−0.247863
$292$	0	0
$293$	5550.94	1.10679	0.553395	−	0.832919i	$-0.313332\pi$
$293$	5550.94	1.10679	0.553395	+	0.832919i	$0.313332\pi$
$294$	0	0
$295$	−66.8586	−0.0131955
$296$	0	0
$297$	3700.39	0.722958
$298$	0	0
$299$	−4883.08	−0.944468
$300$	0	0
$301$	7385.01	1.41417
$302$	0	0
$303$	1284.61	0.243562
$304$	0	0
$305$	−3972.70	−0.745823
$306$	0	0
$307$	6386.14	1.18722	0.593610	−	0.804753i	$-0.297702\pi$
$307$	6386.14	1.18722	0.593610	+	0.804753i	$0.297702\pi$
$308$	0	0
$309$	−244.313	−0.0449789
$310$	0	0
$311$	6579.21	1.19959	0.599795	−	0.800153i	$-0.295249\pi$
$311$	6579.21	1.19959	0.599795	+	0.800153i	$0.295249\pi$
$312$	0	0
$313$	−4374.24	−0.789925	−0.394963	−	0.918697i	$-0.629242\pi$
$313$	−4374.24	−0.789925	−0.394963	+	0.918697i	$0.629242\pi$
$314$	0	0
$315$	7498.03	1.34116
$316$	0	0
$317$	−3199.03	−0.566800	−0.283400	−	0.959002i	$-0.591462\pi$
$317$	−3199.03	−0.566800	−0.283400	+	0.959002i	$0.591462\pi$
$318$	0	0
$319$	−7118.27	−1.24936
$320$	0	0
$321$	−2045.81	−0.355719
$322$	0	0
$323$	−2712.39	−0.467249
$324$	0	0
$325$	−4139.17	−0.706461
$326$	0	0
$327$	2149.63	0.363532
$328$	0	0
$329$	10556.8	1.76904
$330$	0	0
$331$	−6156.86	−1.02239	−0.511195	−	0.859464i	$-0.670797\pi$
$331$	−6156.86	−1.02239	−0.511195	+	0.859464i	$0.670797\pi$
$332$	0	0
$333$	9046.63	1.48875
$334$	0	0
$335$	417.665	0.0681179
$336$	0	0
$337$	928.428	0.150073	0.0750367	−	0.997181i	$-0.476093\pi$
$337$	928.428	0.150073	0.0750367	+	0.997181i	$0.476093\pi$
$338$	0	0
$339$	−210.580	−0.0337379
$340$	0	0
$341$	−5613.67	−0.891487
$342$	0	0
$343$	15399.4	2.42417
$344$	0	0
$345$	−623.549	−0.0973066
$346$	0	0
$347$	6838.98	1.05803	0.529014	−	0.848613i	$-0.322562\pi$
$347$	6838.98	1.05803	0.529014	+	0.848613i	$0.322562\pi$
$348$	0	0
$349$	6613.51	1.01436	0.507182	−	0.861839i	$-0.330687\pi$
$349$	6613.51	1.01436	0.507182	+	0.861839i	$0.330687\pi$
$350$	0	0
$351$	5462.29	0.830642
$352$	0	0
$353$	−6005.98	−0.905569	−0.452785	−	0.891620i	$-0.649569\pi$
$353$	−6005.98	−0.905569	−0.452785	+	0.891620i	$0.649569\pi$
$354$	0	0
$355$	6852.93	1.02455
$356$	0	0
$357$	709.585	0.105197
$358$	0	0
$359$	8803.08	1.29417	0.647087	−	0.762416i	$-0.275987\pi$
$359$	8803.08	1.29417	0.647087	+	0.762416i	$0.275987\pi$
$360$	0	0
$361$	18598.0	2.71147
$362$	0	0
$363$	−2381.47	−0.344338
$364$	0	0
$365$	−5843.52	−0.837982
$366$	0	0
$367$	2185.60	0.310865	0.155432	−	0.987847i	$-0.450323\pi$
$367$	2185.60	0.310865	0.155432	+	0.987847i	$0.450323\pi$
$368$	0	0
$369$	1722.45	0.243001
$370$	0	0
$371$	5024.15	0.703076
$372$	0	0
$373$	−7003.78	−0.972231	−0.486115	−	0.873895i	$-0.661587\pi$
$373$	−7003.78	−0.972231	−0.486115	+	0.873895i	$0.661587\pi$
$374$	0	0
$375$	−1873.63	−0.258010
$376$	0	0
$377$	−10507.5	−1.43545
$378$	0	0
$379$	4187.90	0.567593	0.283797	−	0.958885i	$-0.408406\pi$
$379$	4187.90	0.567593	0.283797	+	0.958885i	$0.408406\pi$
$380$	0	0
$381$	125.497	0.0168751
$382$	0	0
$383$	−9788.94	−1.30598	−0.652992	−	0.757365i	$-0.726486\pi$
$383$	−9788.94	−1.30598	−0.652992	+	0.757365i	$0.726486\pi$
$384$	0	0
$385$	−16802.8	−2.22429
$386$	0	0
$387$	−5567.56	−0.731305
$388$	0	0
$389$	1752.48	0.228417	0.114208	−	0.993457i	$-0.463567\pi$
$389$	1752.48	0.228417	0.114208	+	0.993457i	$0.463567\pi$
$390$	0	0
$391$	985.107	0.127414
$392$	0	0
$393$	−1398.93	−0.179559
$394$	0	0
$395$	−297.769	−0.0379300
$396$	0	0
$397$	−8850.84	−1.11892	−0.559459	−	0.828858i	$-0.688991\pi$
$397$	−8850.84	−1.11892	−0.559459	+	0.828858i	$0.688991\pi$
$398$	0	0
$399$	−6659.76	−0.835601
$400$	0	0
$401$	−12751.9	−1.58802	−0.794012	−	0.607902i	$-0.792011\pi$
$401$	−12751.9	−1.58802	−0.794012	+	0.607902i	$0.792011\pi$
$402$	0	0
$403$	−8286.55	−1.02427
$404$	0	0
$405$	−5293.87	−0.649517
$406$	0	0
$407$	−20273.2	−2.46905
$408$	0	0
$409$	7139.77	0.863175	0.431588	−	0.902071i	$-0.357954\pi$
$409$	7139.77	0.863175	0.431588	+	0.902071i	$0.357954\pi$
$410$	0	0
$411$	3496.21	0.419599
$412$	0	0
$413$	259.344	0.0308995
$414$	0	0
$415$	2811.59	0.332567
$416$	0	0
$417$	787.932	0.0925305
$418$	0	0
$419$	7216.09	0.841358	0.420679	−	0.907210i	$-0.361792\pi$
$419$	7216.09	0.841358	0.420679	+	0.907210i	$0.361792\pi$
$420$	0	0
$421$	−10474.9	−1.21263	−0.606313	−	0.795226i	$-0.707352\pi$
$421$	−10474.9	−1.21263	−0.606313	+	0.795226i	$0.707352\pi$
$422$	0	0
$423$	−7958.76	−0.914818
$424$	0	0
$425$	835.031	0.0953058
$426$	0	0
$427$	15410.1	1.74648
$428$	0	0
$429$	−5942.41	−0.668769
$430$	0	0
$431$	−6971.04	−0.779079	−0.389540	−	0.921010i	$-0.627366\pi$
$431$	−6971.04	−0.779079	−0.389540	+	0.921010i	$0.627366\pi$
$432$	0	0
$433$	9974.94	1.10708	0.553539	−	0.832823i	$-0.313277\pi$
$433$	9974.94	1.10708	0.553539	+	0.832823i	$0.313277\pi$
$434$	0	0
$435$	−1341.77	−0.147892
$436$	0	0
$437$	−9245.66	−1.01208
$438$	0	0
$439$	−7803.12	−0.848343	−0.424171	−	0.905582i	$-0.639435\pi$
$439$	−7803.12	−0.848343	−0.424171	+	0.905582i	$0.639435\pi$
$440$	0	0
$441$	−20347.2	−2.19709
$442$	0	0
$443$	−1867.98	−0.200340	−0.100170	−	0.994970i	$-0.531939\pi$
$443$	−1867.98	−0.200340	−0.100170	+	0.994970i	$0.531939\pi$
$444$	0	0
$445$	−3637.86	−0.387531
$446$	0	0
$447$	−1398.79	−0.148010
$448$	0	0
$449$	−13622.1	−1.43178	−0.715890	−	0.698213i	$-0.753979\pi$
$449$	−13622.1	−1.43178	−0.715890	+	0.698213i	$0.753979\pi$
$450$	0	0
$451$	−3859.95	−0.403011
$452$	0	0
$453$	−775.379	−0.0804205
$454$	0	0
$455$	−24803.3	−2.55559
$456$	0	0
$457$	−1721.45	−0.176205	−0.0881027	−	0.996111i	$-0.528080\pi$
$457$	−1721.45	−0.176205	−0.0881027	+	0.996111i	$0.528080\pi$
$458$	0	0
$459$	−1101.96	−0.112059
$460$	0	0
$461$	6388.22	0.645399	0.322700	−	0.946501i	$-0.395410\pi$
$461$	6388.22	0.645399	0.322700	+	0.946501i	$0.395410\pi$
$462$	0	0
$463$	−13824.4	−1.38763	−0.693815	−	0.720153i	$-0.744072\pi$
$463$	−13824.4	−1.38763	−0.693815	+	0.720153i	$0.744072\pi$
$464$	0	0
$465$	−1058.16	−0.105529
$466$	0	0
$467$	11881.8	1.17735	0.588677	−	0.808369i	$-0.299649\pi$
$467$	11881.8	1.17735	0.588677	+	0.808369i	$0.299649\pi$
$468$	0	0
$469$	−1620.12	−0.159510
$470$	0	0
$471$	−2059.85	−0.201513
$472$	0	0
$473$	12476.7	1.21285
$474$	0	0
$475$	−7837.13	−0.757036
$476$	0	0
$477$	−3787.71	−0.363579
$478$	0	0
$479$	8283.13	0.790116	0.395058	−	0.918656i	$-0.370724\pi$
$479$	8283.13	0.790116	0.395058	+	0.918656i	$0.370724\pi$
$480$	0	0
$481$	−29926.0	−2.83681
$482$	0	0
$483$	2418.74	0.227861
$484$	0	0
$485$	−8676.50	−0.812329
$486$	0	0
$487$	15007.2	1.39639	0.698195	−	0.715907i	$-0.253987\pi$
$487$	15007.2	1.39639	0.698195	+	0.715907i	$0.253987\pi$
$488$	0	0
$489$	447.318	0.0413669
$490$	0	0
$491$	−5549.90	−0.510109	−0.255054	−	0.966927i	$-0.582093\pi$
$491$	−5549.90	−0.510109	−0.255054	+	0.966927i	$0.582093\pi$
$492$	0	0
$493$	2119.78	0.193651
$494$	0	0
$495$	12667.6	1.15024
$496$	0	0
$497$	−26582.5	−2.39917
$498$	0	0
$499$	6964.17	0.624768	0.312384	−	0.949956i	$-0.398872\pi$
$499$	6964.17	0.624768	0.312384	+	0.949956i	$0.398872\pi$
$500$	0	0
$501$	1525.69	0.136054
$502$	0	0
$503$	−13271.1	−1.17640	−0.588202	−	0.808714i	$-0.700164\pi$
$503$	−13271.1	−1.17640	−0.588202	+	0.808714i	$0.700164\pi$
$504$	0	0
$505$	9058.72	0.798233
$506$	0	0
$507$	−6057.87	−0.530650
$508$	0	0
$509$	9357.36	0.814848	0.407424	−	0.913239i	$-0.366427\pi$
$509$	9357.36	0.814848	0.407424	+	0.913239i	$0.366427\pi$
$510$	0	0
$511$	22667.0	1.96228
$512$	0	0
$513$	10342.3	0.890108
$514$	0	0
$515$	−1722.82	−0.147411
$516$	0	0
$517$	17835.3	1.51720
$518$	0	0
$519$	−1773.61	−0.150006
$520$	0	0
$521$	12174.5	1.02375	0.511876	−	0.859060i	$-0.328951\pi$
$521$	12174.5	1.02375	0.511876	+	0.859060i	$0.328951\pi$
$522$	0	0
$523$	−19595.2	−1.63831	−0.819157	−	0.573569i	$-0.805558\pi$
$523$	−19595.2	−1.63831	−0.819157	+	0.573569i	$0.805558\pi$
$524$	0	0
$525$	2050.26	0.170439
$526$	0	0
$527$	1671.72	0.138181
$528$	0	0
$529$	−8809.09	−0.724015
$530$	0	0
$531$	−195.520	−0.0159790
$532$	0	0
$533$	−5697.82	−0.463039
$534$	0	0
$535$	−14426.4	−1.16581
$536$	0	0
$537$	−1895.12	−0.152291
$538$	0	0
$539$	45597.4	3.64382
$540$	0	0
$541$	−6705.41	−0.532880	−0.266440	−	0.963851i	$-0.585847\pi$
$541$	−6705.41	−0.532880	−0.266440	+	0.963851i	$0.585847\pi$
$542$	0	0
$543$	−1303.61	−0.103026
$544$	0	0
$545$	15158.6	1.19142
$546$	0	0
$547$	−3323.39	−0.259777	−0.129888	−	0.991529i	$-0.541462\pi$
$547$	−3323.39	−0.259777	−0.129888	+	0.991529i	$0.541462\pi$
$548$	0	0
$549$	−11617.7	−0.903150
$550$	0	0
$551$	−19895.1	−1.53822
$552$	0	0
$553$	1155.04	0.0888199
$554$	0	0
$555$	−3821.42	−0.292271
$556$	0	0
$557$	−10443.0	−0.794407	−0.397204	−	0.917730i	$-0.630019\pi$
$557$	−10443.0	−0.794407	−0.397204	+	0.917730i	$0.630019\pi$
$558$	0	0
$559$	18417.3	1.39350
$560$	0	0
$561$	1198.81	0.0902210
$562$	0	0
$563$	−5310.18	−0.397509	−0.198754	−	0.980049i	$-0.563690\pi$
$563$	−5310.18	−0.397509	−0.198754	+	0.980049i	$0.563690\pi$
$564$	0	0
$565$	−1484.95	−0.110570
$566$	0	0
$567$	20534.9	1.52096
$568$	0	0
$569$	−1320.87	−0.0973180	−0.0486590	−	0.998815i	$-0.515495\pi$
$569$	−1320.87	−0.0973180	−0.0486590	+	0.998815i	$0.515495\pi$
$570$	0	0
$571$	−1791.22	−0.131279	−0.0656396	−	0.997843i	$-0.520909\pi$
$571$	−1791.22	−0.131279	−0.0656396	+	0.997843i	$0.520909\pi$
$572$	0	0
$573$	−1272.85	−0.0927992
$574$	0	0
$575$	2846.35	0.206437
$576$	0	0
$577$	12691.0	0.915656	0.457828	−	0.889041i	$-0.348628\pi$
$577$	12691.0	0.915656	0.457828	+	0.889041i	$0.348628\pi$
$578$	0	0
$579$	3967.98	0.284808
$580$	0	0
$581$	−10906.1	−0.778765
$582$	0	0
$583$	8488.11	0.602987
$584$	0	0
$585$	18699.2	1.32156
$586$	0	0
$587$	23534.6	1.65482	0.827408	−	0.561601i	$-0.189814\pi$
$587$	23534.6	1.65482	0.827408	+	0.561601i	$0.189814\pi$
$588$	0	0
$589$	−15689.8	−1.09760
$590$	0	0
$591$	5015.70	0.349101
$592$	0	0
$593$	−23599.9	−1.63429	−0.817144	−	0.576434i	$-0.804444\pi$
$593$	−23599.9	−1.63429	−0.817144	+	0.576434i	$0.804444\pi$
$594$	0	0
$595$	5003.78	0.344765
$596$	0	0
$597$	−4259.19	−0.291988
$598$	0	0
$599$	−22197.4	−1.51412	−0.757061	−	0.653344i	$-0.773366\pi$
$599$	−22197.4	−1.51412	−0.757061	+	0.653344i	$0.773366\pi$
$600$	0	0
$601$	8802.87	0.597465	0.298733	−	0.954337i	$-0.403436\pi$
$601$	8802.87	0.597465	0.298733	+	0.954337i	$0.403436\pi$
$602$	0	0
$603$	1221.41	0.0824870
$604$	0	0
$605$	−16793.4	−1.12851
$606$	0	0
$607$	−15846.1	−1.05960	−0.529798	−	0.848124i	$-0.677732\pi$
$607$	−15846.1	−1.05960	−0.529798	+	0.848124i	$0.677732\pi$
$608$	0	0
$609$	5204.72	0.346315
$610$	0	0
$611$	26327.3	1.74319
$612$	0	0
$613$	−22879.9	−1.50752	−0.753761	−	0.657149i	$-0.771762\pi$
$613$	−22879.9	−1.50752	−0.753761	+	0.657149i	$0.771762\pi$
$614$	0	0
$615$	−727.587	−0.0477059
$616$	0	0
$617$	−510.114	−0.0332843	−0.0166422	−	0.999862i	$-0.505298\pi$
$617$	−510.114	−0.0332843	−0.0166422	+	0.999862i	$0.505298\pi$
$618$	0	0
$619$	313.054	0.0203274	0.0101637	−	0.999948i	$-0.496765\pi$
$619$	313.054	0.0203274	0.0101637	+	0.999948i	$0.496765\pi$
$620$	0	0
$621$	−3756.21	−0.242724
$622$	0	0
$623$	14111.3	0.907473
$624$	0	0
$625$	−7072.34	−0.452630
$626$	0	0
$627$	−11251.4	−0.716647
$628$	0	0
$629$	6037.23	0.382703
$630$	0	0
$631$	−11605.2	−0.732165	−0.366082	−	0.930582i	$-0.619301\pi$
$631$	−11605.2	−0.732165	−0.366082	+	0.930582i	$0.619301\pi$
$632$	0	0
$633$	−4797.90	−0.301263
$634$	0	0
$635$	884.969	0.0553054
$636$	0	0
$637$	67308.0	4.18656
$638$	0	0
$639$	20040.5	1.24067
$640$	0	0
$641$	−4030.93	−0.248381	−0.124191	−	0.992258i	$-0.539633\pi$
$641$	−4030.93	−0.248381	−0.124191	+	0.992258i	$0.539633\pi$
$642$	0	0
$643$	−27197.0	−1.66803	−0.834017	−	0.551739i	$-0.813964\pi$
$643$	−27197.0	−1.66803	−0.834017	+	0.551739i	$0.813964\pi$
$644$	0	0
$645$	2351.81	0.143570
$646$	0	0
$647$	20712.9	1.25859	0.629295	−	0.777167i	$-0.283344\pi$
$647$	20712.9	1.25859	0.629295	+	0.777167i	$0.283344\pi$
$648$	0	0
$649$	438.152	0.0265007
$650$	0	0
$651$	4104.59	0.247114
$652$	0	0
$653$	7413.63	0.444285	0.222142	−	0.975014i	$-0.428695\pi$
$653$	7413.63	0.444285	0.222142	+	0.975014i	$0.428695\pi$
$654$	0	0
$655$	−9864.85	−0.588476
$656$	0	0
$657$	−17088.6	−1.01475
$658$	0	0
$659$	27427.6	1.62128	0.810642	−	0.585542i	$-0.199118\pi$
$659$	27427.6	1.62128	0.810642	+	0.585542i	$0.199118\pi$
$660$	0	0
$661$	28996.0	1.70623	0.853113	−	0.521727i	$-0.174712\pi$
$661$	28996.0	1.70623	0.853113	+	0.521727i	$0.174712\pi$
$662$	0	0
$663$	1769.62	0.103659
$664$	0	0
$665$	−46962.7	−2.73855
$666$	0	0
$667$	7225.65	0.419458
$668$	0	0
$669$	−400.410	−0.0231401
$670$	0	0
$671$	26034.7	1.49785
$672$	0	0
$673$	25272.4	1.44752	0.723759	−	0.690053i	$-0.242413\pi$
$673$	25272.4	1.44752	0.723759	+	0.690053i	$0.242413\pi$
$674$	0	0
$675$	−3183.97	−0.181557
$676$	0	0
$677$	−1656.49	−0.0940383	−0.0470192	−	0.998894i	$-0.514972\pi$
$677$	−1656.49	−0.0940383	−0.0470192	+	0.998894i	$0.514972\pi$
$678$	0	0
$679$	33656.1	1.90221
$680$	0	0
$681$	−4301.29	−0.242035
$682$	0	0
$683$	−19763.8	−1.10724	−0.553618	−	0.832771i	$-0.686753\pi$
$683$	−19763.8	−1.10724	−0.553618	+	0.832771i	$0.686753\pi$
$684$	0	0
$685$	24654.2	1.37517
$686$	0	0
$687$	−349.218	−0.0193938
$688$	0	0
$689$	12529.6	0.692802
$690$	0	0
$691$	−4478.98	−0.246582	−0.123291	−	0.992371i	$-0.539345\pi$
$691$	−4478.98	−0.246582	−0.123291	+	0.992371i	$0.539345\pi$
$692$	0	0
$693$	−49137.7	−2.69349
$694$	0	0
$695$	5556.27	0.303254
$696$	0	0
$697$	1149.47	0.0624668
$698$	0	0
$699$	7911.38	0.428091
$700$	0	0
$701$	29171.4	1.57174	0.785870	−	0.618391i	$-0.212215\pi$
$701$	29171.4	1.57174	0.785870	+	0.618391i	$0.212215\pi$
$702$	0	0
$703$	−56662.0	−3.03990
$704$	0	0
$705$	3361.89	0.179597
$706$	0	0
$707$	−35138.7	−1.86921
$708$	0	0
$709$	−12877.4	−0.682115	−0.341058	−	0.940042i	$-0.610785\pi$
$709$	−12877.4	−0.682115	−0.341058	+	0.940042i	$0.610785\pi$
$710$	0	0
$711$	−870.787	−0.0459311
$712$	0	0
$713$	5698.35	0.299306
$714$	0	0
$715$	−41904.1	−2.19178
$716$	0	0
$717$	−860.276	−0.0448084
$718$	0	0
$719$	−22034.4	−1.14290	−0.571451	−	0.820636i	$-0.693619\pi$
$719$	−22034.4	−1.14290	−0.571451	+	0.820636i	$0.693619\pi$
$720$	0	0
$721$	6682.82	0.345189
$722$	0	0
$723$	1513.67	0.0778618
$724$	0	0
$725$	6124.86	0.313754
$726$	0	0
$727$	−32522.1	−1.65912	−0.829558	−	0.558421i	$-0.811407\pi$
$727$	−32522.1	−1.65912	−0.829558	+	0.558421i	$0.811407\pi$
$728$	0	0
$729$	−13319.3	−0.676689
$730$	0	0
$731$	−3715.49	−0.187992
$732$	0	0
$733$	−11134.4	−0.561060	−0.280530	−	0.959845i	$-0.590510\pi$
$733$	−11134.4	−0.561060	−0.280530	+	0.959845i	$0.590510\pi$
$734$	0	0
$735$	8594.95	0.431333
$736$	0	0
$737$	−2737.13	−0.136803
$738$	0	0
$739$	−19567.1	−0.974004	−0.487002	−	0.873401i	$-0.661909\pi$
$739$	−19567.1	−0.974004	−0.487002	+	0.873401i	$0.661909\pi$
$740$	0	0
$741$	−16608.6	−0.823391
$742$	0	0
$743$	37194.1	1.83650	0.918249	−	0.396003i	$-0.129603\pi$
$743$	37194.1	1.83650	0.918249	+	0.396003i	$0.129603\pi$
$744$	0	0
$745$	−9863.85	−0.485078
$746$	0	0
$747$	8222.13	0.402720
$748$	0	0
$749$	55960.1	2.72995
$750$	0	0
$751$	−10072.2	−0.489402	−0.244701	−	0.969599i	$-0.578690\pi$
$751$	−10072.2	−0.489402	−0.244701	+	0.969599i	$0.578690\pi$
$752$	0	0
$753$	4803.51	0.232470
$754$	0	0
$755$	−5467.74	−0.263565
$756$	0	0
$757$	−1189.27	−0.0570999	−0.0285499	−	0.999592i	$-0.509089\pi$
$757$	−1189.27	−0.0570999	−0.0285499	+	0.999592i	$0.509089\pi$
$758$	0	0
$759$	4086.37	0.195423
$760$	0	0
$761$	16283.3	0.775648	0.387824	−	0.921734i	$-0.373227\pi$
$761$	16283.3	0.775648	0.387824	+	0.921734i	$0.373227\pi$
$762$	0	0
$763$	−58800.1	−2.78992
$764$	0	0
$765$	−3772.35	−0.178287
$766$	0	0
$767$	646.772	0.0304480
$768$	0	0
$769$	14990.1	0.702933	0.351467	−	0.936200i	$-0.385683\pi$
$769$	14990.1	0.702933	0.351467	+	0.936200i	$0.385683\pi$
$770$	0	0
$771$	−4480.72	−0.209299
$772$	0	0
$773$	−5522.07	−0.256940	−0.128470	−	0.991713i	$-0.541007\pi$
$773$	−5522.07	−0.256940	−0.128470	+	0.991713i	$0.541007\pi$
$774$	0	0
$775$	4830.23	0.223880
$776$	0	0
$777$	14823.3	0.684404
$778$	0	0
$779$	−10788.3	−0.496188
$780$	0	0
$781$	−44910.0	−2.05763
$782$	0	0
$783$	−8082.72	−0.368905
$784$	0	0
$785$	−14525.4	−0.660427
$786$	0	0
$787$	−19764.4	−0.895202	−0.447601	−	0.894233i	$-0.647722\pi$
$787$	−19764.4	−0.895202	−0.447601	+	0.894233i	$0.647722\pi$
$788$	0	0
$789$	7251.98	0.327221
$790$	0	0
$791$	5760.11	0.258920
$792$	0	0
$793$	38430.8	1.72096
$794$	0	0
$795$	1599.98	0.0713779
$796$	0	0
$797$	−10824.8	−0.481099	−0.240549	−	0.970637i	$-0.577328\pi$
$797$	−10824.8	−0.481099	−0.240549	+	0.970637i	$0.577328\pi$
$798$	0	0
$799$	−5311.24	−0.235167
$800$	0	0
$801$	−10638.5	−0.469278
$802$	0	0
$803$	38294.9	1.68294
$804$	0	0
$805$	17056.3	0.746776
$806$	0	0
$807$	−2382.77	−0.103937
$808$	0	0
$809$	2525.12	0.109739	0.0548693	−	0.998494i	$-0.482526\pi$
$809$	2525.12	0.109739	0.0548693	+	0.998494i	$0.482526\pi$
$810$	0	0
$811$	7987.78	0.345856	0.172928	−	0.984935i	$-0.444677\pi$
$811$	7987.78	0.345856	0.172928	+	0.984935i	$0.444677\pi$
$812$	0	0
$813$	7250.56	0.312778
$814$	0	0
$815$	3154.36	0.135573
$816$	0	0
$817$	34871.4	1.49327
$818$	0	0
$819$	−72534.0	−3.09468
$820$	0	0
$821$	−44558.7	−1.89417	−0.947083	−	0.320990i	$-0.895984\pi$
$821$	−44558.7	−1.89417	−0.947083	+	0.320990i	$0.895984\pi$
$822$	0	0
$823$	−10000.4	−0.423565	−0.211782	−	0.977317i	$-0.567927\pi$
$823$	−10000.4	−0.423565	−0.211782	+	0.977317i	$0.567927\pi$
$824$	0	0
$825$	3463.83	0.146176
$826$	0	0
$827$	1853.28	0.0779261	0.0389631	−	0.999241i	$-0.487595\pi$
$827$	1853.28	0.0779261	0.0389631	+	0.999241i	$0.487595\pi$
$828$	0	0
$829$	31827.2	1.33342	0.666710	−	0.745318i	$-0.267702\pi$
$829$	31827.2	1.33342	0.666710	+	0.745318i	$0.267702\pi$
$830$	0	0
$831$	−1418.99	−0.0592348
$832$	0	0
$833$	−13578.6	−0.564792
$834$	0	0
$835$	10758.7	0.445894
$836$	0	0
$837$	−6374.26	−0.263234
$838$	0	0
$839$	18401.2	0.757189	0.378594	−	0.925563i	$-0.376408\pi$
$839$	18401.2	0.757189	0.378594	+	0.925563i	$0.376408\pi$
$840$	0	0
$841$	−8840.65	−0.362485
$842$	0	0
$843$	−7629.10	−0.311696
$844$	0	0
$845$	−42718.3	−1.73912
$846$	0	0
$847$	65141.5	2.64261
$848$	0	0
$849$	−7001.09	−0.283012
$850$	0	0
$851$	20579.0	0.828952
$852$	0	0
$853$	−5199.25	−0.208698	−0.104349	−	0.994541i	$-0.533276\pi$
$853$	−5199.25	−0.208698	−0.104349	+	0.994541i	$0.533276\pi$
$854$	0	0
$855$	35405.1	1.41618
$856$	0	0
$857$	26839.1	1.06978	0.534892	−	0.844920i	$-0.320352\pi$
$857$	26839.1	1.06978	0.534892	+	0.844920i	$0.320352\pi$
$858$	0	0
$859$	−37240.0	−1.47917	−0.739587	−	0.673060i	$-0.764979\pi$
$859$	−37240.0	−1.47917	−0.739587	+	0.673060i	$0.764979\pi$
$860$	0	0
$861$	2822.31	0.111712
$862$	0	0
$863$	24363.1	0.960984	0.480492	−	0.876999i	$-0.340458\pi$
$863$	24363.1	0.960984	0.480492	+	0.876999i	$0.340458\pi$
$864$	0	0
$865$	−12507.0	−0.491619
$866$	0	0
$867$	−357.000	−0.0139843
$868$	0	0
$869$	1951.40	0.0761757
$870$	0	0
$871$	−4040.38	−0.157179
$872$	0	0
$873$	−25373.3	−0.983685
$874$	0	0
$875$	51250.4	1.98009
$876$	0	0
$877$	36019.2	1.38687	0.693433	−	0.720521i	$-0.256097\pi$
$877$	36019.2	1.38687	0.693433	+	0.720521i	$0.256097\pi$
$878$	0	0
$879$	−6857.05	−0.263120
$880$	0	0
$881$	26378.0	1.00874	0.504369	−	0.863488i	$-0.331725\pi$
$881$	26378.0	1.00874	0.504369	+	0.863488i	$0.331725\pi$
$882$	0	0
$883$	32109.0	1.22373	0.611866	−	0.790961i	$-0.290419\pi$
$883$	32109.0	1.22373	0.611866	+	0.790961i	$0.290419\pi$
$884$	0	0
$885$	82.5901	0.00313699
$886$	0	0
$887$	−9039.00	−0.342165	−0.171082	−	0.985257i	$-0.554726\pi$
$887$	−9039.00	−0.342165	−0.171082	+	0.985257i	$0.554726\pi$
$888$	0	0
$889$	−3432.79	−0.129507
$890$	0	0
$891$	34692.9	1.30444
$892$	0	0
$893$	49848.3	1.86798
$894$	0	0
$895$	−13363.8	−0.499110
$896$	0	0
$897$	6032.05	0.224531
$898$	0	0
$899$	12261.9	0.454901
$900$	0	0
$901$	−2527.71	−0.0934631
$902$	0	0
$903$	−9122.67	−0.336194
$904$	0	0
$905$	−9192.64	−0.337651
$906$	0	0
$907$	10461.0	0.382967	0.191484	−	0.981496i	$-0.438670\pi$
$907$	10461.0	0.382967	0.191484	+	0.981496i	$0.438670\pi$
$908$	0	0
$909$	26491.1	0.966616
$910$	0	0
$911$	11801.1	0.429184	0.214592	−	0.976704i	$-0.431158\pi$
$911$	11801.1	0.429184	0.214592	+	0.976704i	$0.431158\pi$
$912$	0	0
$913$	−18425.5	−0.667902
$914$	0	0
$915$	4907.46	0.177307
$916$	0	0
$917$	38265.7	1.37802
$918$	0	0
$919$	41851.9	1.50225	0.751125	−	0.660160i	$-0.229511\pi$
$919$	41851.9	1.50225	0.751125	+	0.660160i	$0.229511\pi$
$920$	0	0
$921$	−7888.78	−0.282241
$922$	0	0
$923$	−66293.4	−2.36411
$924$	0	0
$925$	17443.9	0.620055
$926$	0	0
$927$	−5038.18	−0.178507
$928$	0	0
$929$	−33442.4	−1.18106	−0.590532	−	0.807014i	$-0.701082\pi$
$929$	−33442.4	−1.18106	−0.590532	+	0.807014i	$0.701082\pi$
$930$	0	0
$931$	127441.	4.48628
$932$	0	0
$933$	−8127.27	−0.285182
$934$	0	0
$935$	8453.69	0.295685
$936$	0	0
$937$	55394.1	1.93132	0.965661	−	0.259807i	$-0.0836590\pi$
$937$	55394.1	1.93132	0.965661	+	0.259807i	$0.0836590\pi$
$938$	0	0
$939$	5403.48	0.187791
$940$	0	0
$941$	−41190.8	−1.42697	−0.713487	−	0.700669i	$-0.752885\pi$
$941$	−41190.8	−1.42697	−0.713487	+	0.700669i	$0.752885\pi$
$942$	0	0
$943$	3918.18	0.135306
$944$	0	0
$945$	−19079.4	−0.656776
$946$	0	0
$947$	−30768.7	−1.05580	−0.527902	−	0.849305i	$-0.677021\pi$
$947$	−30768.7	−1.05580	−0.527902	+	0.849305i	$0.677021\pi$
$948$	0	0
$949$	56528.6	1.93361
$950$	0	0
$951$	3951.75	0.134747
$952$	0	0
$953$	19210.8	0.652988	0.326494	−	0.945199i	$-0.394133\pi$
$953$	19210.8	0.652988	0.326494	+	0.945199i	$0.394133\pi$
$954$	0	0
$955$	−8975.74	−0.304134
$956$	0	0
$957$	8793.17	0.297014
$958$	0	0
$959$	−95633.7	−3.22020
$960$	0	0
$961$	−20121.0	−0.675404
$962$	0	0
$963$	−42188.3	−1.41173
$964$	0	0
$965$	27981.1	0.933412
$966$	0	0
$967$	33449.1	1.11236	0.556180	−	0.831062i	$-0.312267\pi$
$967$	33449.1	1.11236	0.556180	+	0.831062i	$0.312267\pi$
$968$	0	0
$969$	3350.60	0.111080
$970$	0	0
$971$	34817.5	1.15072	0.575359	−	0.817901i	$-0.304862\pi$
$971$	34817.5	1.15072	0.575359	+	0.817901i	$0.304862\pi$
$972$	0	0
$973$	−21552.7	−0.710122
$974$	0	0
$975$	5113.10	0.167949
$976$	0	0
$977$	36868.8	1.20731	0.603653	−	0.797247i	$-0.293711\pi$
$977$	36868.8	1.20731	0.603653	+	0.797247i	$0.293711\pi$
$978$	0	0
$979$	23840.4	0.778286
$980$	0	0
$981$	44329.4	1.44274
$982$	0	0
$983$	−29655.9	−0.962234	−0.481117	−	0.876656i	$-0.659769\pi$
$983$	−29655.9	−0.962234	−0.481117	+	0.876656i	$0.659769\pi$
$984$	0	0
$985$	35369.3	1.14412
$986$	0	0
$987$	−13040.7	−0.420559
$988$	0	0
$989$	−12664.9	−0.407199
$990$	0	0
$991$	11170.8	0.358074	0.179037	−	0.983842i	$-0.442702\pi$
$991$	11170.8	0.358074	0.179037	+	0.983842i	$0.442702\pi$
$992$	0	0
$993$	7605.54	0.243056
$994$	0	0
$995$	−30034.5	−0.956944
$996$	0	0
$997$	−2300.45	−0.0730752	−0.0365376	−	0.999332i	$-0.511633\pi$
$997$	−2300.45	−0.0730752	−0.0365376	+	0.999332i	$0.511633\pi$
$998$	0	0
$999$	−23020.0	−0.729048

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.bh.1.4		7
4.3	odd	2		1088.4.a.bg.1.4		7
8.3	odd	2		544.4.a.k.1.4	✓	7
8.5	even	2		544.4.a.l.1.4	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.4.a.k.1.4	✓	7	8.3	odd	2
544.4.a.l.1.4	yes	7	8.5	even	2
1088.4.a.bg.1.4		7	4.3	odd	2
1088.4.a.bh.1.4		7	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.23530 q^{3} -8.71094 q^{5} +33.7897 q^{7} -25.4740 q^{9} +O(q^{10})\)	\(q-1.23530 q^{3} -8.71094 q^{5} +33.7897 q^{7} -25.4740 q^{9} +57.0863 q^{11} +84.2673 q^{13} +10.7606 q^{15} -17.0000 q^{17} +159.552 q^{19} -41.7403 q^{21} -57.9475 q^{23} -49.1195 q^{25} +64.8210 q^{27} -124.693 q^{29} -98.3364 q^{31} -70.5185 q^{33} -294.340 q^{35} -355.131 q^{37} -104.095 q^{39} -67.6160 q^{41} +218.558 q^{43} +221.903 q^{45} +312.426 q^{47} +798.744 q^{49} +21.0000 q^{51} +148.689 q^{53} -497.276 q^{55} -197.094 q^{57} +7.67524 q^{59} +456.058 q^{61} -860.760 q^{63} -734.048 q^{65} -47.9472 q^{67} +71.5823 q^{69} -786.703 q^{71} +670.825 q^{73} +60.6771 q^{75} +1928.93 q^{77} +34.1833 q^{79} +607.726 q^{81} -322.765 q^{83} +148.086 q^{85} +154.033 q^{87} +417.620 q^{89} +2847.37 q^{91} +121.475 q^{93} -1389.85 q^{95} +996.046 q^{97} -1454.22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{7} + 67 q^{9}+O(q^{10}) \) 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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