LMFDB - Embedded newform 1575.4.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.78066700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 18x^{3} + 7x^{2} + 30x - 10 $ x^5 - x^4 - 18x^3 + 7x^2 + 30*x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 5 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.71490$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.18660 q^{2} +18.9008 q^{4} +7.00000 q^{7} -56.5383 q^{8} +O(q^{10})$	$q-5.18660 q^{2} +18.9008 q^{4} +7.00000 q^{7} -56.5383 q^{8} +35.9555 q^{11} +45.2622 q^{13} -36.3062 q^{14} +142.035 q^{16} -113.154 q^{17} +61.5906 q^{19} -186.487 q^{22} +30.6108 q^{23} -234.757 q^{26} +132.306 q^{28} -214.989 q^{29} +164.206 q^{31} -284.372 q^{32} +586.882 q^{34} -410.533 q^{37} -319.446 q^{38} +309.310 q^{41} -29.9519 q^{43} +679.589 q^{44} -158.766 q^{46} -483.790 q^{47} +49.0000 q^{49} +855.493 q^{52} +295.582 q^{53} -395.768 q^{56} +1115.06 q^{58} -416.191 q^{59} -151.196 q^{61} -851.670 q^{62} +338.644 q^{64} +89.5253 q^{67} -2138.70 q^{68} -714.265 q^{71} -1135.58 q^{73} +2129.27 q^{74} +1164.11 q^{76} +251.688 q^{77} -323.347 q^{79} -1604.27 q^{82} -297.898 q^{83} +155.348 q^{86} -2032.86 q^{88} +90.2097 q^{89} +316.835 q^{91} +578.570 q^{92} +2509.23 q^{94} -492.101 q^{97} -254.143 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8}+O(q^{10}) $ 5 * q - q^2 + 27 * q^4 + 35 * q^7 - 33 * q^8	$ 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8} - 66 q^{11} - 2 q^{13} - 7 q^{14} + 155 q^{16} - 108 q^{17} + 174 q^{19} - 506 q^{22} + 116 q^{23} - 446 q^{26} + 189 q^{28} - 370 q^{29} + 342 q^{31} + 55 q^{32} + 112 q^{34} - 408 q^{37} + 34 q^{38} - 802 q^{41} - 584 q^{43} - 290 q^{44} + 640 q^{46} - 716 q^{47} + 245 q^{49} + 338 q^{52} - 98 q^{53} - 231 q^{56} + 482 q^{58} - 704 q^{59} + 650 q^{61} - 2070 q^{62} + 75 q^{64} + 180 q^{67} - 4520 q^{68} - 1470 q^{71} - 534 q^{73} + 1312 q^{74} + 4370 q^{76} - 462 q^{77} - 820 q^{79} - 1338 q^{82} - 1520 q^{83} - 832 q^{86} - 3258 q^{88} - 286 q^{89} - 14 q^{91} - 1288 q^{92} + 2540 q^{94} + 278 q^{97} - 49 q^{98}+O(q^{100}) $ 5 * q - q^2 + 27 * q^4 + 35 * q^7 - 33 * q^8 - 66 * q^11 - 2 * q^13 - 7 * q^14 + 155 * q^16 - 108 * q^17 + 174 * q^19 - 506 * q^22 + 116 * q^23 - 446 * q^26 + 189 * q^28 - 370 * q^29 + 342 * q^31 + 55 * q^32 + 112 * q^34 - 408 * q^37 + 34 * q^38 - 802 * q^41 - 584 * q^43 - 290 * q^44 + 640 * q^46 - 716 * q^47 + 245 * q^49 + 338 * q^52 - 98 * q^53 - 231 * q^56 + 482 * q^58 - 704 * q^59 + 650 * q^61 - 2070 * q^62 + 75 * q^64 + 180 * q^67 - 4520 * q^68 - 1470 * q^71 - 534 * q^73 + 1312 * q^74 + 4370 * q^76 - 462 * q^77 - 820 * q^79 - 1338 * q^82 - 1520 * q^83 - 832 * q^86 - 3258 * q^88 - 286 * q^89 - 14 * q^91 - 1288 * q^92 + 2540 * q^94 + 278 * q^97 - 49 * q^98

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 $

$2$	−5.18660	−1.83374	−0.916870	−	0.399185i	$-0.869293\pi$
$2$	−5.18660	−1.83374	−0.916870	+	0.399185i	$0.869293\pi$
$3$	0	0
$3$	0	0
$4$	18.9008	2.36260
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−56.5383	−2.49866
$9$	0	0
$10$	0	0
$11$	35.9555	0.985545	0.492772	−	0.870158i	$-0.335984\pi$
$11$	35.9555	0.985545	0.492772	+	0.870158i	$0.335984\pi$
$12$	0	0
$13$	45.2622	0.965652	0.482826	−	0.875716i	$-0.339610\pi$
$13$	45.2622	0.965652	0.482826	+	0.875716i	$0.339610\pi$
$14$	−36.3062	−0.693089
$15$	0	0
$16$	142.035	2.21929
$17$	−113.154	−1.61434	−0.807170	−	0.590319i	$-0.799002\pi$
$17$	−113.154	−1.61434	−0.807170	+	0.590319i	$0.799002\pi$
$18$	0	0
$19$	61.5906	0.743677	0.371838	−	0.928297i	$-0.378728\pi$
$19$	61.5906	0.743677	0.371838	+	0.928297i	$0.378728\pi$
$20$	0	0
$21$	0	0
$22$	−186.487	−1.80723
$23$	30.6108	0.277513	0.138756	−	0.990327i	$-0.455690\pi$
$23$	30.6108	0.277513	0.138756	+	0.990327i	$0.455690\pi$
$24$	0	0
$25$	0	0
$26$	−234.757	−1.77075
$27$	0	0
$28$	132.306	0.892980
$29$	−214.989	−1.37663	−0.688317	−	0.725410i	$-0.741650\pi$
$29$	−214.989	−1.37663	−0.688317	+	0.725410i	$0.741650\pi$
$30$	0	0
$31$	164.206	0.951362	0.475681	−	0.879618i	$-0.342202\pi$
$31$	164.206	0.951362	0.475681	+	0.879618i	$0.342202\pi$
$32$	−284.372	−1.57095
$33$	0	0
$34$	586.882	2.96028
$35$	0	0
$36$	0	0
$37$	−410.533	−1.82409	−0.912043	−	0.410095i	$-0.865496\pi$
$37$	−410.533	−1.82409	−0.912043	+	0.410095i	$0.865496\pi$
$38$	−319.446	−1.36371
$39$	0	0
$40$	0	0
$41$	309.310	1.17820	0.589100	−	0.808060i	$-0.299483\pi$
$41$	309.310	1.17820	0.589100	+	0.808060i	$0.299483\pi$
$42$	0	0
$43$	−29.9519	−0.106224	−0.0531118	−	0.998589i	$-0.516914\pi$
$43$	−29.9519	−0.106224	−0.0531118	+	0.998589i	$0.516914\pi$
$44$	679.589	2.32845
$45$	0	0
$46$	−158.766	−0.508886
$47$	−483.790	−1.50145	−0.750724	−	0.660616i	$-0.770295\pi$
$47$	−483.790	−1.50145	−0.750724	+	0.660616i	$0.770295\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	855.493	2.28145
$53$	295.582	0.766063	0.383031	−	0.923735i	$-0.374880\pi$
$53$	295.582	0.766063	0.383031	+	0.923735i	$0.374880\pi$
$54$	0	0
$55$	0	0
$56$	−395.768	−0.944405
$57$	0	0
$58$	1115.06	2.52439
$59$	−416.191	−0.918364	−0.459182	−	0.888342i	$-0.651857\pi$
$59$	−416.191	−0.918364	−0.459182	+	0.888342i	$0.651857\pi$
$60$	0	0
$61$	−151.196	−0.317355	−0.158677	−	0.987330i	$-0.550723\pi$
$61$	−151.196	−0.317355	−0.158677	+	0.987330i	$0.550723\pi$
$62$	−851.670	−1.74455
$63$	0	0
$64$	338.644	0.661415
$65$	0	0
$66$	0	0
$67$	89.5253	0.163243	0.0816213	−	0.996663i	$-0.473990\pi$
$67$	89.5253	0.163243	0.0816213	+	0.996663i	$0.473990\pi$
$68$	−2138.70	−3.81404
$69$	0	0
$70$	0	0
$71$	−714.265	−1.19391	−0.596955	−	0.802274i	$-0.703623\pi$
$71$	−714.265	−1.19391	−0.596955	+	0.802274i	$0.703623\pi$
$72$	0	0
$73$	−1135.58	−1.82068	−0.910340	−	0.413861i	$-0.864180\pi$
$73$	−1135.58	−1.82068	−0.910340	+	0.413861i	$0.864180\pi$
$74$	2129.27	3.34490
$75$	0	0
$76$	1164.11	1.75701
$77$	251.688	0.372501
$78$	0	0
$79$	−323.347	−0.460498	−0.230249	−	0.973132i	$-0.573954\pi$
$79$	−323.347	−0.460498	−0.230249	+	0.973132i	$0.573954\pi$
$80$	0	0
$81$	0	0
$82$	−1604.27	−2.16051
$83$	−297.898	−0.393959	−0.196979	−	0.980408i	$-0.563113\pi$
$83$	−297.898	−0.393959	−0.196979	+	0.980408i	$0.563113\pi$
$84$	0	0
$85$	0	0
$86$	155.348	0.194787
$87$	0	0
$88$	−2032.86	−2.46254
$89$	90.2097	0.107441	0.0537203	−	0.998556i	$-0.482892\pi$
$89$	90.2097	0.107441	0.0537203	+	0.998556i	$0.482892\pi$
$90$	0	0
$91$	316.835	0.364982
$92$	578.570	0.655653
$93$	0	0
$94$	2509.23	2.75326
$95$	0	0
$96$	0	0
$97$	−492.101	−0.515106	−0.257553	−	0.966264i	$-0.582916\pi$
$97$	−492.101	−0.515106	−0.257553	+	0.966264i	$0.582916\pi$
$98$	−254.143	−0.261963
$99$	0	0
$100$	0	0
$101$	272.636	0.268597	0.134299	−	0.990941i	$-0.457122\pi$
$101$	272.636	0.268597	0.134299	+	0.990941i	$0.457122\pi$
$102$	0	0
$103$	−628.179	−0.600936	−0.300468	−	0.953792i	$-0.597143\pi$
$103$	−628.179	−0.600936	−0.300468	+	0.953792i	$0.597143\pi$
$104$	−2559.05	−2.41284
$105$	0	0
$106$	−1533.07	−1.40476
$107$	−908.469	−0.820794	−0.410397	−	0.911907i	$-0.634610\pi$
$107$	−908.469	−0.820794	−0.410397	+	0.911907i	$0.634610\pi$
$108$	0	0
$109$	984.306	0.864948	0.432474	−	0.901646i	$-0.357641\pi$
$109$	984.306	0.864948	0.432474	+	0.901646i	$0.357641\pi$
$110$	0	0
$111$	0	0
$112$	994.244	0.838814
$113$	2228.79	1.85546	0.927730	−	0.373251i	$-0.121757\pi$
$113$	2228.79	1.85546	0.927730	+	0.373251i	$0.121757\pi$
$114$	0	0
$115$	0	0
$116$	−4063.47	−3.25244
$117$	0	0
$118$	2158.62	1.68404
$119$	−792.075	−0.610163
$120$	0	0
$121$	−38.2023	−0.0287019
$122$	784.193	0.581946
$123$	0	0
$124$	3103.63	2.24769
$125$	0	0
$126$	0	0
$127$	−2860.78	−1.99885	−0.999423	−	0.0339770i	$-0.989183\pi$
$127$	−2860.78	−1.99885	−0.999423	+	0.0339770i	$0.989183\pi$
$128$	518.561	0.358084
$129$	0	0
$130$	0	0
$131$	524.971	0.350130	0.175065	−	0.984557i	$-0.443987\pi$
$131$	524.971	0.350130	0.175065	+	0.984557i	$0.443987\pi$
$132$	0	0
$133$	431.134	0.281083
$134$	−464.332	−0.299344
$135$	0	0
$136$	6397.51	4.03369
$137$	2149.03	1.34017	0.670087	−	0.742282i	$-0.266257\pi$
$137$	2149.03	1.34017	0.670087	+	0.742282i	$0.266257\pi$
$138$	0	0
$139$	−1507.78	−0.920060	−0.460030	−	0.887903i	$-0.652161\pi$
$139$	−1507.78	−0.920060	−0.460030	+	0.887903i	$0.652161\pi$
$140$	0	0
$141$	0	0
$142$	3704.61	2.18932
$143$	1627.42	0.951693
$144$	0	0
$145$	0	0
$146$	5889.80	3.33865
$147$	0	0
$148$	−7759.41	−4.30959
$149$	1013.61	0.557304	0.278652	−	0.960392i	$-0.410112\pi$
$149$	1013.61	0.557304	0.278652	+	0.960392i	$0.410112\pi$
$150$	0	0
$151$	1663.70	0.896623	0.448312	−	0.893877i	$-0.352025\pi$
$151$	1663.70	0.896623	0.448312	+	0.893877i	$0.352025\pi$
$152$	−3482.23	−1.85820
$153$	0	0
$154$	−1305.41	−0.683070
$155$	0	0
$156$	0	0
$157$	2399.48	1.21974	0.609872	−	0.792500i	$-0.291221\pi$
$157$	2399.48	1.21974	0.609872	+	0.792500i	$0.291221\pi$
$158$	1677.07	0.844434
$159$	0	0
$160$	0	0
$161$	214.276	0.104890
$162$	0	0
$163$	1415.81	0.680337	0.340168	−	0.940365i	$-0.389516\pi$
$163$	1415.81	0.680337	0.340168	+	0.940365i	$0.389516\pi$
$164$	5846.22	2.78362
$165$	0	0
$166$	1545.08	0.722418
$167$	3543.86	1.64211	0.821055	−	0.570850i	$-0.193386\pi$
$167$	3543.86	1.64211	0.821055	+	0.570850i	$0.193386\pi$
$168$	0	0
$169$	−148.335	−0.0675170
$170$	0	0
$171$	0	0
$172$	−566.115	−0.250964
$173$	95.6649	0.0420420	0.0210210	−	0.999779i	$-0.493308\pi$
$173$	95.6649	0.0420420	0.0210210	+	0.999779i	$0.493308\pi$
$174$	0	0
$175$	0	0
$176$	5106.93	2.18721
$177$	0	0
$178$	−467.882	−0.197018
$179$	−2166.13	−0.904491	−0.452245	−	0.891894i	$-0.649377\pi$
$179$	−2166.13	−0.904491	−0.452245	+	0.891894i	$0.649377\pi$
$180$	0	0
$181$	−1859.24	−0.763514	−0.381757	−	0.924263i	$-0.624681\pi$
$181$	−1859.24	−0.763514	−0.381757	+	0.924263i	$0.624681\pi$
$182$	−1643.30	−0.669282
$183$	0	0
$184$	−1730.68	−0.693411
$185$	0	0
$186$	0	0
$187$	−4068.49	−1.59100
$188$	−9144.03	−3.54733
$189$	0	0
$190$	0	0
$191$	2661.60	1.00831	0.504154	−	0.863614i	$-0.331805\pi$
$191$	2661.60	1.00831	0.504154	+	0.863614i	$0.331805\pi$
$192$	0	0
$193$	1952.03	0.728033	0.364016	−	0.931393i	$-0.381405\pi$
$193$	1952.03	0.728033	0.364016	+	0.931393i	$0.381405\pi$
$194$	2552.33	0.944570
$195$	0	0
$196$	926.141	0.337515
$197$	3587.80	1.29757	0.648783	−	0.760974i	$-0.275278\pi$
$197$	3587.80	1.29757	0.648783	+	0.760974i	$0.275278\pi$
$198$	0	0
$199$	−767.691	−0.273468	−0.136734	−	0.990608i	$-0.543661\pi$
$199$	−767.691	−0.273468	−0.136734	+	0.990608i	$0.543661\pi$
$200$	0	0
$201$	0	0
$202$	−1414.06	−0.492538
$203$	−1504.92	−0.520319
$204$	0	0
$205$	0	0
$206$	3258.12	1.10196
$207$	0	0
$208$	6428.81	2.14306
$209$	2214.52	0.732927
$210$	0	0
$211$	−1199.84	−0.391472	−0.195736	−	0.980657i	$-0.562710\pi$
$211$	−1199.84	−0.391472	−0.195736	+	0.980657i	$0.562710\pi$
$212$	5586.75	1.80990
$213$	0	0
$214$	4711.87	1.50512
$215$	0	0
$216$	0	0
$217$	1149.44	0.359581
$218$	−5105.20	−1.58609
$219$	0	0
$220$	0	0
$221$	−5121.58	−1.55889
$222$	0	0
$223$	−2917.84	−0.876203	−0.438101	−	0.898926i	$-0.644349\pi$
$223$	−2917.84	−0.876203	−0.438101	+	0.898926i	$0.644349\pi$
$224$	−1990.60	−0.593762
$225$	0	0
$226$	−11559.9	−3.40243
$227$	612.679	0.179141	0.0895703	−	0.995981i	$-0.471451\pi$
$227$	612.679	0.179141	0.0895703	+	0.995981i	$0.471451\pi$
$228$	0	0
$229$	2641.51	0.762253	0.381126	−	0.924523i	$-0.375536\pi$
$229$	2641.51	0.762253	0.381126	+	0.924523i	$0.375536\pi$
$230$	0	0
$231$	0	0
$232$	12155.1	3.43974
$233$	−2322.87	−0.653117	−0.326558	−	0.945177i	$-0.605889\pi$
$233$	−2322.87	−0.653117	−0.326558	+	0.945177i	$0.605889\pi$
$234$	0	0
$235$	0	0
$236$	−7866.36	−2.16973
$237$	0	0
$238$	4108.18	1.11888
$239$	−5372.54	−1.45406	−0.727030	−	0.686605i	$-0.759100\pi$
$239$	−5372.54	−1.45406	−0.727030	+	0.686605i	$0.759100\pi$
$240$	0	0
$241$	−1412.33	−0.377494	−0.188747	−	0.982026i	$-0.560443\pi$
$241$	−1412.33	−0.377494	−0.188747	+	0.982026i	$0.560443\pi$
$242$	198.140	0.0526319
$243$	0	0
$244$	−2857.73	−0.749784
$245$	0	0
$246$	0	0
$247$	2787.73	0.718133
$248$	−9283.91	−2.37713
$249$	0	0
$250$	0	0
$251$	−5676.06	−1.42737	−0.713684	−	0.700467i	$-0.752975\pi$
$251$	−5676.06	−1.42737	−0.713684	+	0.700467i	$0.752975\pi$
$252$	0	0
$253$	1100.63	0.273501
$254$	14837.7	3.66536
$255$	0	0
$256$	−5398.72	−1.31805
$257$	−3939.60	−0.956207	−0.478104	−	0.878303i	$-0.658676\pi$
$257$	−3939.60	−0.956207	−0.478104	+	0.878303i	$0.658676\pi$
$258$	0	0
$259$	−2873.73	−0.689440
$260$	0	0
$261$	0	0
$262$	−2722.82	−0.642047
$263$	−5397.68	−1.26553	−0.632766	−	0.774343i	$-0.718081\pi$
$263$	−5397.68	−1.26553	−0.632766	+	0.774343i	$0.718081\pi$
$264$	0	0
$265$	0	0
$266$	−2236.12	−0.515434
$267$	0	0
$268$	1692.10	0.385678
$269$	−4973.60	−1.12731	−0.563654	−	0.826011i	$-0.690605\pi$
$269$	−4973.60	−1.12731	−0.563654	+	0.826011i	$0.690605\pi$
$270$	0	0
$271$	4147.48	0.929672	0.464836	−	0.885397i	$-0.346113\pi$
$271$	4147.48	0.929672	0.464836	+	0.885397i	$0.346113\pi$
$272$	−16071.7	−3.58269
$273$	0	0
$274$	−11146.2	−2.45753
$275$	0	0
$276$	0	0
$277$	2616.79	0.567609	0.283805	−	0.958882i	$-0.408403\pi$
$277$	2616.79	0.567609	0.283805	+	0.958882i	$0.408403\pi$
$278$	7820.26	1.68715
$279$	0	0
$280$	0	0
$281$	−2866.89	−0.608628	−0.304314	−	0.952572i	$-0.598427\pi$
$281$	−2866.89	−0.608628	−0.304314	+	0.952572i	$0.598427\pi$
$282$	0	0
$283$	−3015.40	−0.633381	−0.316691	−	0.948529i	$-0.602572\pi$
$283$	−3015.40	−0.633381	−0.316691	+	0.948529i	$0.602572\pi$
$284$	−13500.2	−2.82074
$285$	0	0
$286$	−8440.80	−1.74516
$287$	2165.17	0.445318
$288$	0	0
$289$	7890.73	1.60609
$290$	0	0
$291$	0	0
$292$	−21463.4	−4.30155
$293$	3580.41	0.713890	0.356945	−	0.934125i	$-0.383818\pi$
$293$	3580.41	0.713890	0.356945	+	0.934125i	$0.383818\pi$
$294$	0	0
$295$	0	0
$296$	23210.8	4.55777
$297$	0	0
$298$	−5257.20	−1.02195
$299$	1385.51	0.267981
$300$	0	0
$301$	−209.663	−0.0401488
$302$	−8628.96	−1.64417
$303$	0	0
$304$	8748.01	1.65044
$305$	0	0
$306$	0	0
$307$	1432.92	0.266389	0.133194	−	0.991090i	$-0.457477\pi$
$307$	1432.92	0.266389	0.133194	+	0.991090i	$0.457477\pi$
$308$	4757.12	0.880072
$309$	0	0
$310$	0	0
$311$	−5488.33	−1.00069	−0.500345	−	0.865826i	$-0.666794\pi$
$311$	−5488.33	−1.00069	−0.500345	+	0.865826i	$0.666794\pi$
$312$	0	0
$313$	−2461.68	−0.444545	−0.222272	−	0.974985i	$-0.571347\pi$
$313$	−2461.68	−0.444545	−0.222272	+	0.974985i	$0.571347\pi$
$314$	−12445.2	−2.23669
$315$	0	0
$316$	−6111.53	−1.08798
$317$	−3113.12	−0.551579	−0.275789	−	0.961218i	$-0.588939\pi$
$317$	−3113.12	−0.551579	−0.275789	+	0.961218i	$0.588939\pi$
$318$	0	0
$319$	−7730.03	−1.35673
$320$	0	0
$321$	0	0
$322$	−1111.36	−0.192341
$323$	−6969.20	−1.20055
$324$	0	0
$325$	0	0
$326$	−7343.25	−1.24756
$327$	0	0
$328$	−17487.9	−2.94392
$329$	−3386.53	−0.567494
$330$	0	0
$331$	−6364.21	−1.05682	−0.528411	−	0.848988i	$-0.677212\pi$
$331$	−6364.21	−1.05682	−0.528411	+	0.848988i	$0.677212\pi$
$332$	−5630.52	−0.930768
$333$	0	0
$334$	−18380.6	−3.01120
$335$	0	0
$336$	0	0
$337$	−5163.25	−0.834600	−0.417300	−	0.908769i	$-0.637024\pi$
$337$	−5163.25	−0.834600	−0.417300	+	0.908769i	$0.637024\pi$
$338$	769.353	0.123809
$339$	0	0
$340$	0	0
$341$	5904.10	0.937610
$342$	0	0
$343$	343.000	0.0539949
$344$	1693.43	0.265417
$345$	0	0
$346$	−496.176	−0.0770941
$347$	−305.000	−0.0471851	−0.0235926	−	0.999722i	$-0.507510\pi$
$347$	−305.000	−0.0471851	−0.0235926	+	0.999722i	$0.507510\pi$
$348$	0	0
$349$	−8233.50	−1.26283	−0.631417	−	0.775443i	$-0.717526\pi$
$349$	−8233.50	−1.26283	−0.631417	+	0.775443i	$0.717526\pi$
$350$	0	0
$351$	0	0
$352$	−10224.7	−1.54824
$353$	1064.37	0.160483	0.0802415	−	0.996775i	$-0.474431\pi$
$353$	1064.37	0.160483	0.0802415	+	0.996775i	$0.474431\pi$
$354$	0	0
$355$	0	0
$356$	1705.04	0.253839
$357$	0	0
$358$	11234.8	1.65860
$359$	3915.59	0.575646	0.287823	−	0.957684i	$-0.407069\pi$
$359$	3915.59	0.575646	0.287823	+	0.957684i	$0.407069\pi$
$360$	0	0
$361$	−3065.59	−0.446945
$362$	9643.13	1.40009
$363$	0	0
$364$	5988.45	0.862308
$365$	0	0
$366$	0	0
$367$	7430.70	1.05689	0.528446	−	0.848967i	$-0.322775\pi$
$367$	7430.70	1.05689	0.528446	+	0.848967i	$0.322775\pi$
$368$	4347.80	0.615882
$369$	0	0
$370$	0	0
$371$	2069.07	0.289544
$372$	0	0
$373$	−3275.71	−0.454718	−0.227359	−	0.973811i	$-0.573009\pi$
$373$	−3275.71	−0.454718	−0.227359	+	0.973811i	$0.573009\pi$
$374$	21101.6	2.91749
$375$	0	0
$376$	27352.7	3.75161
$377$	−9730.86	−1.32935
$378$	0	0
$379$	5745.18	0.778655	0.389327	−	0.921099i	$-0.372707\pi$
$379$	5745.18	0.778655	0.389327	+	0.921099i	$0.372707\pi$
$380$	0	0
$381$	0	0
$382$	−13804.7	−1.84897
$383$	−101.844	−0.0135874	−0.00679372	−	0.999977i	$-0.502163\pi$
$383$	−101.844	−0.0135874	−0.00679372	+	0.999977i	$0.502163\pi$
$384$	0	0
$385$	0	0
$386$	−10124.4	−1.33502
$387$	0	0
$388$	−9301.11	−1.21699
$389$	−3047.72	−0.397237	−0.198619	−	0.980077i	$-0.563646\pi$
$389$	−3047.72	−0.397237	−0.198619	+	0.980077i	$0.563646\pi$
$390$	0	0
$391$	−3463.72	−0.448000
$392$	−2770.38	−0.356952
$393$	0	0
$394$	−18608.5	−2.37940
$395$	0	0
$396$	0	0
$397$	9830.23	1.24273	0.621367	−	0.783520i	$-0.286578\pi$
$397$	9830.23	1.24273	0.621367	+	0.783520i	$0.286578\pi$
$398$	3981.71	0.501470
$399$	0	0
$400$	0	0
$401$	−3693.01	−0.459900	−0.229950	−	0.973202i	$-0.573856\pi$
$401$	−3693.01	−0.459900	−0.229950	+	0.973202i	$0.573856\pi$
$402$	0	0
$403$	7432.31	0.918684
$404$	5153.06	0.634589
$405$	0	0
$406$	7805.43	0.954130
$407$	−14760.9	−1.79772
$408$	0	0
$409$	8815.07	1.06571	0.532857	−	0.846205i	$-0.321118\pi$
$409$	8815.07	1.06571	0.532857	+	0.846205i	$0.321118\pi$
$410$	0	0
$411$	0	0
$412$	−11873.1	−1.41977
$413$	−2913.34	−0.347109
$414$	0	0
$415$	0	0
$416$	−12871.3	−1.51699
$417$	0	0
$418$	−11485.8	−1.34400
$419$	−3746.84	−0.436862	−0.218431	−	0.975852i	$-0.570094\pi$
$419$	−3746.84	−0.436862	−0.218431	+	0.975852i	$0.570094\pi$
$420$	0	0
$421$	−10554.5	−1.22184	−0.610922	−	0.791690i	$-0.709201\pi$
$421$	−10554.5	−1.22184	−0.610922	+	0.791690i	$0.709201\pi$
$422$	6223.10	0.717858
$423$	0	0
$424$	−16711.7	−1.91413
$425$	0	0
$426$	0	0
$427$	−1058.37	−0.119949
$428$	−17170.8	−1.93921
$429$	0	0
$430$	0	0
$431$	14029.9	1.56797	0.783986	−	0.620778i	$-0.213183\pi$
$431$	14029.9	1.56797	0.783986	+	0.620778i	$0.213183\pi$
$432$	0	0
$433$	−9639.08	−1.06980	−0.534901	−	0.844915i	$-0.679651\pi$
$433$	−9639.08	−1.06980	−0.534901	+	0.844915i	$0.679651\pi$
$434$	−5961.69	−0.659378
$435$	0	0
$436$	18604.2	2.04353
$437$	1885.34	0.206380
$438$	0	0
$439$	3936.53	0.427974	0.213987	−	0.976837i	$-0.431355\pi$
$439$	3936.53	0.427974	0.213987	+	0.976837i	$0.431355\pi$
$440$	0	0
$441$	0	0
$442$	26563.6	2.85860
$443$	10812.5	1.15963	0.579817	−	0.814746i	$-0.303124\pi$
$443$	10812.5	1.15963	0.579817	+	0.814746i	$0.303124\pi$
$444$	0	0
$445$	0	0
$446$	15133.7	1.60673
$447$	0	0
$448$	2370.51	0.249991
$449$	−8154.86	−0.857131	−0.428565	−	0.903511i	$-0.640981\pi$
$449$	−8154.86	−0.857131	−0.428565	+	0.903511i	$0.640981\pi$
$450$	0	0
$451$	11121.4	1.16117
$452$	42126.0	4.38372
$453$	0	0
$454$	−3177.72	−0.328497
$455$	0	0
$456$	0	0
$457$	−17787.5	−1.82071	−0.910353	−	0.413833i	$-0.864190\pi$
$457$	−17787.5	−1.82071	−0.910353	+	0.413833i	$0.864190\pi$
$458$	−13700.5	−1.39777
$459$	0	0
$460$	0	0
$461$	−13192.1	−1.33280	−0.666398	−	0.745596i	$-0.732165\pi$
$461$	−13192.1	−1.33280	−0.666398	+	0.745596i	$0.732165\pi$
$462$	0	0
$463$	−542.568	−0.0544607	−0.0272303	−	0.999629i	$-0.508669\pi$
$463$	−542.568	−0.0544607	−0.0272303	+	0.999629i	$0.508669\pi$
$464$	−30535.9	−3.05516
$465$	0	0
$466$	12047.8	1.19765
$467$	5962.51	0.590818	0.295409	−	0.955371i	$-0.404544\pi$
$467$	5962.51	0.590818	0.295409	+	0.955371i	$0.404544\pi$
$468$	0	0
$469$	626.677	0.0616999
$470$	0	0
$471$	0	0
$472$	23530.7	2.29468
$473$	−1076.93	−0.104688
$474$	0	0
$475$	0	0
$476$	−14970.9	−1.44157
$477$	0	0
$478$	27865.2	2.66637
$479$	1317.66	0.125690	0.0628451	−	0.998023i	$-0.479983\pi$
$479$	1317.66	0.125690	0.0628451	+	0.998023i	$0.479983\pi$
$480$	0	0
$481$	−18581.6	−1.76143
$482$	7325.19	0.692227
$483$	0	0
$484$	−722.054	−0.0678113
$485$	0	0
$486$	0	0
$487$	5976.21	0.556074	0.278037	−	0.960570i	$-0.410316\pi$
$487$	5976.21	0.556074	0.278037	+	0.960570i	$0.410316\pi$
$488$	8548.35	0.792962
$489$	0	0
$490$	0	0
$491$	−10338.9	−0.950285	−0.475143	−	0.879909i	$-0.657604\pi$
$491$	−10338.9	−0.950285	−0.475143	+	0.879909i	$0.657604\pi$
$492$	0	0
$493$	24326.7	2.22236
$494$	−14458.8	−1.31687
$495$	0	0
$496$	23322.9	2.11135
$497$	−4999.85	−0.451256
$498$	0	0
$499$	−4161.74	−0.373357	−0.186678	−	0.982421i	$-0.559772\pi$
$499$	−4161.74	−0.373357	−0.186678	+	0.982421i	$0.559772\pi$
$500$	0	0
$501$	0	0
$502$	29439.4	2.61742
$503$	−7556.24	−0.669813	−0.334907	−	0.942251i	$-0.608705\pi$
$503$	−7556.24	−0.669813	−0.334907	+	0.942251i	$0.608705\pi$
$504$	0	0
$505$	0	0
$506$	−5708.51	−0.501530
$507$	0	0
$508$	−54071.2	−4.72248
$509$	−7231.12	−0.629693	−0.314847	−	0.949143i	$-0.601953\pi$
$509$	−7231.12	−0.629693	−0.314847	+	0.949143i	$0.601953\pi$
$510$	0	0
$511$	−7949.06	−0.688152
$512$	23852.5	2.05887
$513$	0	0
$514$	20433.1	1.75344
$515$	0	0
$516$	0	0
$517$	−17394.9	−1.47974
$518$	14904.9	1.26425
$519$	0	0
$520$	0	0
$521$	7378.54	0.620460	0.310230	−	0.950661i	$-0.399594\pi$
$521$	7378.54	0.620460	0.310230	+	0.950661i	$0.399594\pi$
$522$	0	0
$523$	2200.53	0.183982	0.0919909	−	0.995760i	$-0.470677\pi$
$523$	2200.53	0.183982	0.0919909	+	0.995760i	$0.470677\pi$
$524$	9922.40	0.827217
$525$	0	0
$526$	27995.6	2.32066
$527$	−18580.5	−1.53582
$528$	0	0
$529$	−11230.0	−0.922987
$530$	0	0
$531$	0	0
$532$	8148.80	0.664089
$533$	14000.1	1.13773
$534$	0	0
$535$	0	0
$536$	−5061.60	−0.407888
$537$	0	0
$538$	25796.1	2.06719
$539$	1761.82	0.140792
$540$	0	0
$541$	−23797.3	−1.89118	−0.945588	−	0.325365i	$-0.894513\pi$
$541$	−23797.3	−1.89118	−0.945588	+	0.325365i	$0.894513\pi$
$542$	−21511.3	−1.70478
$543$	0	0
$544$	32177.7	2.53604
$545$	0	0
$546$	0	0
$547$	6586.46	0.514839	0.257419	−	0.966300i	$-0.417128\pi$
$547$	6586.46	0.514839	0.257419	+	0.966300i	$0.417128\pi$
$548$	40618.4	3.16630
$549$	0	0
$550$	0	0
$551$	−13241.3	−1.02377
$552$	0	0
$553$	−2263.43	−0.174052
$554$	−13572.3	−1.04085
$555$	0	0
$556$	−28498.3	−2.17374
$557$	−23522.6	−1.78938	−0.894691	−	0.446686i	$-0.852604\pi$
$557$	−23522.6	−1.78938	−0.894691	+	0.446686i	$0.852604\pi$
$558$	0	0
$559$	−1355.69	−0.102575
$560$	0	0
$561$	0	0
$562$	14869.4	1.11607
$563$	−17654.3	−1.32156	−0.660782	−	0.750578i	$-0.729775\pi$
$563$	−17654.3	−1.32156	−0.660782	+	0.750578i	$0.729775\pi$
$564$	0	0
$565$	0	0
$566$	15639.7	1.16146
$567$	0	0
$568$	40383.3	2.98318
$569$	13453.6	0.991223	0.495611	−	0.868544i	$-0.334944\pi$
$569$	13453.6	0.991223	0.495611	+	0.868544i	$0.334944\pi$
$570$	0	0
$571$	9019.88	0.661069	0.330534	−	0.943794i	$-0.392771\pi$
$571$	9019.88	0.661069	0.330534	+	0.943794i	$0.392771\pi$
$572$	30759.7	2.24847
$573$	0	0
$574$	−11229.9	−0.816597
$575$	0	0
$576$	0	0
$577$	1328.73	0.0958680	0.0479340	−	0.998851i	$-0.484736\pi$
$577$	1328.73	0.0958680	0.0479340	+	0.998851i	$0.484736\pi$
$578$	−40926.1	−2.94516
$579$	0	0
$580$	0	0
$581$	−2085.29	−0.148902
$582$	0	0
$583$	10627.8	0.754989
$584$	64203.8	4.54926
$585$	0	0
$586$	−18570.2	−1.30909
$587$	−13927.3	−0.979283	−0.489642	−	0.871924i	$-0.662872\pi$
$587$	−13927.3	−0.979283	−0.489642	+	0.871924i	$0.662872\pi$
$588$	0	0
$589$	10113.5	0.707506
$590$	0	0
$591$	0	0
$592$	−58309.9	−4.04818
$593$	5684.69	0.393663	0.196831	−	0.980437i	$-0.436935\pi$
$593$	5684.69	0.393663	0.196831	+	0.980437i	$0.436935\pi$
$594$	0	0
$595$	0	0
$596$	19158.1	1.31669
$597$	0	0
$598$	−7186.10	−0.491407
$599$	−27961.5	−1.90730	−0.953651	−	0.300914i	$-0.902708\pi$
$599$	−27961.5	−1.90730	−0.953651	+	0.300914i	$0.902708\pi$
$600$	0	0
$601$	−12396.7	−0.841386	−0.420693	−	0.907203i	$-0.638213\pi$
$601$	−12396.7	−0.841386	−0.420693	+	0.907203i	$0.638213\pi$
$602$	1087.44	0.0736224
$603$	0	0
$604$	31445.3	2.11837
$605$	0	0
$606$	0	0
$607$	3124.52	0.208930	0.104465	−	0.994529i	$-0.466687\pi$
$607$	3124.52	0.208930	0.104465	+	0.994529i	$0.466687\pi$
$608$	−17514.6	−1.16828
$609$	0	0
$610$	0	0
$611$	−21897.4	−1.44988
$612$	0	0
$613$	−9531.29	−0.628002	−0.314001	−	0.949423i	$-0.601670\pi$
$613$	−9531.29	−0.628002	−0.314001	+	0.949423i	$0.601670\pi$
$614$	−7432.01	−0.488488
$615$	0	0
$616$	−14230.0	−0.930754
$617$	5410.37	0.353020	0.176510	−	0.984299i	$-0.443519\pi$
$617$	5410.37	0.353020	0.176510	+	0.984299i	$0.443519\pi$
$618$	0	0
$619$	−10244.9	−0.665228	−0.332614	−	0.943063i	$-0.607931\pi$
$619$	−10244.9	−0.665228	−0.332614	+	0.943063i	$0.607931\pi$
$620$	0	0
$621$	0	0
$622$	28465.8	1.83501
$623$	631.468	0.0406087
$624$	0	0
$625$	0	0
$626$	12767.8	0.815180
$627$	0	0
$628$	45352.3	2.88177
$629$	46453.2	2.94469
$630$	0	0
$631$	−10342.8	−0.652522	−0.326261	−	0.945280i	$-0.605789\pi$
$631$	−10342.8	−0.652522	−0.326261	+	0.945280i	$0.605789\pi$
$632$	18281.5	1.15063
$633$	0	0
$634$	16146.5	1.01145
$635$	0	0
$636$	0	0
$637$	2217.85	0.137950
$638$	40092.6	2.48790
$639$	0	0
$640$	0	0
$641$	981.309	0.0604671	0.0302335	−	0.999543i	$-0.490375\pi$
$641$	981.309	0.0604671	0.0302335	+	0.999543i	$0.490375\pi$
$642$	0	0
$643$	−16289.5	−0.999062	−0.499531	−	0.866296i	$-0.666494\pi$
$643$	−16289.5	−0.999062	−0.499531	+	0.866296i	$0.666494\pi$
$644$	4049.99	0.247813
$645$	0	0
$646$	36146.5	2.20149
$647$	11349.5	0.689634	0.344817	−	0.938670i	$-0.387941\pi$
$647$	11349.5	0.689634	0.344817	+	0.938670i	$0.387941\pi$
$648$	0	0
$649$	−14964.4	−0.905088
$650$	0	0
$651$	0	0
$652$	26760.0	1.60737
$653$	−20510.7	−1.22917	−0.614583	−	0.788852i	$-0.710676\pi$
$653$	−20510.7	−1.22917	−0.614583	+	0.788852i	$0.710676\pi$
$654$	0	0
$655$	0	0
$656$	43932.8	2.61477
$657$	0	0
$658$	17564.6	1.04064
$659$	−15480.4	−0.915070	−0.457535	−	0.889191i	$-0.651268\pi$
$659$	−15480.4	−0.915070	−0.457535	+	0.889191i	$0.651268\pi$
$660$	0	0
$661$	3720.51	0.218928	0.109464	−	0.993991i	$-0.465087\pi$
$661$	3720.51	0.218928	0.109464	+	0.993991i	$0.465087\pi$
$662$	33008.6	1.93794
$663$	0	0
$664$	16842.6	0.984369
$665$	0	0
$666$	0	0
$667$	−6580.98	−0.382034
$668$	66981.9	3.87965
$669$	0	0
$670$	0	0
$671$	−5436.32	−0.312767
$672$	0	0
$673$	17977.2	1.02967	0.514837	−	0.857288i	$-0.327853\pi$
$673$	17977.2	1.02967	0.514837	+	0.857288i	$0.327853\pi$
$674$	26779.7	1.53044
$675$	0	0
$676$	−2803.65	−0.159516
$677$	13079.2	0.742505	0.371253	−	0.928532i	$-0.378928\pi$
$677$	13079.2	0.742505	0.371253	+	0.928532i	$0.378928\pi$
$678$	0	0
$679$	−3444.70	−0.194692
$680$	0	0
$681$	0	0
$682$	−30622.2	−1.71933
$683$	22608.2	1.26659	0.633293	−	0.773912i	$-0.281703\pi$
$683$	22608.2	1.26659	0.633293	+	0.773912i	$0.281703\pi$
$684$	0	0
$685$	0	0
$686$	−1779.00	−0.0990127
$687$	0	0
$688$	−4254.21	−0.235742
$689$	13378.7	0.739749
$690$	0	0
$691$	26262.2	1.44582	0.722910	−	0.690942i	$-0.242804\pi$
$691$	26262.2	1.44582	0.722910	+	0.690942i	$0.242804\pi$
$692$	1808.15	0.0993286
$693$	0	0
$694$	1581.91	0.0865253
$695$	0	0
$696$	0	0
$697$	−34999.6	−1.90201
$698$	42703.9	2.31571
$699$	0	0
$700$	0	0
$701$	15831.3	0.852982	0.426491	−	0.904492i	$-0.359750\pi$
$701$	15831.3	0.852982	0.426491	+	0.904492i	$0.359750\pi$
$702$	0	0
$703$	−25285.0	−1.35653
$704$	12176.1	0.651853
$705$	0	0
$706$	−5520.44	−0.294284
$707$	1908.46	0.101520
$708$	0	0
$709$	−874.284	−0.0463109	−0.0231554	−	0.999732i	$-0.507371\pi$
$709$	−874.284	−0.0463109	−0.0231554	+	0.999732i	$0.507371\pi$
$710$	0	0
$711$	0	0
$712$	−5100.30	−0.268458
$713$	5026.47	0.264015
$714$	0	0
$715$	0	0
$716$	−40941.6	−2.13695
$717$	0	0
$718$	−20308.6	−1.05559
$719$	−103.934	−0.00539091	−0.00269546	−	0.999996i	$-0.500858\pi$
$719$	−103.934	−0.00539091	−0.00269546	+	0.999996i	$0.500858\pi$
$720$	0	0
$721$	−4397.26	−0.227132
$722$	15900.0	0.819580
$723$	0	0
$724$	−35141.2	−1.80388
$725$	0	0
$726$	0	0
$727$	20972.0	1.06989	0.534945	−	0.844887i	$-0.320332\pi$
$727$	20972.0	1.06989	0.534945	+	0.844887i	$0.320332\pi$
$728$	−17913.3	−0.911967
$729$	0	0
$730$	0	0
$731$	3389.16	0.171481
$732$	0	0
$733$	32298.2	1.62750	0.813751	−	0.581213i	$-0.197422\pi$
$733$	32298.2	1.62750	0.813751	+	0.581213i	$0.197422\pi$
$734$	−38540.1	−1.93807
$735$	0	0
$736$	−8704.85	−0.435958
$737$	3218.93	0.160883
$738$	0	0
$739$	19402.4	0.965804	0.482902	−	0.875674i	$-0.339583\pi$
$739$	19402.4	0.965804	0.482902	+	0.875674i	$0.339583\pi$
$740$	0	0
$741$	0	0
$742$	−10731.5	−0.530949
$743$	28784.0	1.42124	0.710621	−	0.703575i	$-0.248414\pi$
$743$	28784.0	1.42124	0.710621	+	0.703575i	$0.248414\pi$
$744$	0	0
$745$	0	0
$746$	16989.8	0.833834
$747$	0	0
$748$	−76897.9	−3.75891
$749$	−6359.28	−0.310231
$750$	0	0
$751$	4382.70	0.212952	0.106476	−	0.994315i	$-0.466043\pi$
$751$	4382.70	0.212952	0.106476	+	0.994315i	$0.466043\pi$
$752$	−68715.0	−3.33215
$753$	0	0
$754$	50470.1	2.43768
$755$	0	0
$756$	0	0
$757$	13669.8	0.656326	0.328163	−	0.944621i	$-0.393570\pi$
$757$	13669.8	0.656326	0.328163	+	0.944621i	$0.393570\pi$
$758$	−29798.0	−1.42785
$759$	0	0
$760$	0	0
$761$	25216.4	1.20117	0.600586	−	0.799560i	$-0.294934\pi$
$761$	25216.4	1.20117	0.600586	+	0.799560i	$0.294934\pi$
$762$	0	0
$763$	6890.14	0.326920
$764$	50306.5	2.38223
$765$	0	0
$766$	528.225	0.0249158
$767$	−18837.7	−0.886819
$768$	0	0
$769$	15930.6	0.747038	0.373519	−	0.927622i	$-0.378151\pi$
$769$	15930.6	0.747038	0.373519	+	0.927622i	$0.378151\pi$
$770$	0	0
$771$	0	0
$772$	36895.0	1.72005
$773$	−28154.5	−1.31002	−0.655012	−	0.755618i	$-0.727336\pi$
$773$	−28154.5	−1.31002	−0.655012	+	0.755618i	$0.727336\pi$
$774$	0	0
$775$	0	0
$776$	27822.5	1.28708
$777$	0	0
$778$	15807.3	0.728430
$779$	19050.6	0.876200
$780$	0	0
$781$	−25681.8	−1.17665
$782$	17964.9	0.821515
$783$	0	0
$784$	6959.71	0.317042
$785$	0	0
$786$	0	0
$787$	−16392.1	−0.742461	−0.371231	−	0.928541i	$-0.621064\pi$
$787$	−16392.1	−0.742461	−0.371231	+	0.928541i	$0.621064\pi$
$788$	67812.5	3.06563
$789$	0	0
$790$	0	0
$791$	15601.5	0.701298
$792$	0	0
$793$	−6843.45	−0.306454
$794$	−50985.5	−2.27885
$795$	0	0
$796$	−14510.0	−0.646097
$797$	−2966.12	−0.131826	−0.0659129	−	0.997825i	$-0.520996\pi$
$797$	−2966.12	−0.131826	−0.0659129	+	0.997825i	$0.520996\pi$
$798$	0	0
$799$	54742.6	2.42385
$800$	0	0
$801$	0	0
$802$	19154.2	0.843338
$803$	−40830.4	−1.79436
$804$	0	0
$805$	0	0
$806$	−38548.4	−1.68463
$807$	0	0
$808$	−15414.4	−0.671134
$809$	14712.3	0.639378	0.319689	−	0.947523i	$-0.396422\pi$
$809$	14712.3	0.639378	0.319689	+	0.947523i	$0.396422\pi$
$810$	0	0
$811$	5794.69	0.250899	0.125450	−	0.992100i	$-0.459963\pi$
$811$	5794.69	0.250899	0.125450	+	0.992100i	$0.459963\pi$
$812$	−28444.3	−1.22931
$813$	0	0
$814$	76559.0	3.29655
$815$	0	0
$816$	0	0
$817$	−1844.75	−0.0789961
$818$	−45720.3	−1.95424
$819$	0	0
$820$	0	0
$821$	−34345.7	−1.46002	−0.730009	−	0.683438i	$-0.760484\pi$
$821$	−34345.7	−1.46002	−0.730009	+	0.683438i	$0.760484\pi$
$822$	0	0
$823$	−17454.4	−0.739274	−0.369637	−	0.929176i	$-0.620518\pi$
$823$	−17454.4	−0.739274	−0.369637	+	0.929176i	$0.620518\pi$
$824$	35516.2	1.50153
$825$	0	0
$826$	15110.3	0.636508
$827$	1378.54	0.0579645	0.0289822	−	0.999580i	$-0.490773\pi$
$827$	1378.54	0.0579645	0.0289822	+	0.999580i	$0.490773\pi$
$828$	0	0
$829$	36714.7	1.53818	0.769092	−	0.639138i	$-0.220709\pi$
$829$	36714.7	1.53818	0.769092	+	0.639138i	$0.220709\pi$
$830$	0	0
$831$	0	0
$832$	15327.8	0.638696
$833$	−5544.52	−0.230620
$834$	0	0
$835$	0	0
$836$	41856.3	1.73162
$837$	0	0
$838$	19433.4	0.801091
$839$	−3191.34	−0.131320	−0.0656598	−	0.997842i	$-0.520915\pi$
$839$	−3191.34	−0.131320	−0.0656598	+	0.997842i	$0.520915\pi$
$840$	0	0
$841$	21831.2	0.895123
$842$	54742.2	2.24055
$843$	0	0
$844$	−22678.0	−0.924893
$845$	0	0
$846$	0	0
$847$	−267.416	−0.0108483
$848$	41982.9	1.70012
$849$	0	0
$850$	0	0
$851$	−12566.7	−0.506207
$852$	0	0
$853$	−42221.5	−1.69477	−0.847384	−	0.530980i	$-0.821824\pi$
$853$	−42221.5	−1.69477	−0.847384	+	0.530980i	$0.821824\pi$
$854$	5489.35	0.219955
$855$	0	0
$856$	51363.3	2.05089
$857$	5849.17	0.233143	0.116572	−	0.993182i	$-0.462810\pi$
$857$	5849.17	0.233143	0.116572	+	0.993182i	$0.462810\pi$
$858$	0	0
$859$	−45756.1	−1.81744	−0.908719	−	0.417409i	$-0.862938\pi$
$859$	−45756.1	−1.81744	−0.908719	+	0.417409i	$0.862938\pi$
$860$	0	0
$861$	0	0
$862$	−72767.5	−2.87526
$863$	33012.0	1.30213	0.651067	−	0.759020i	$-0.274322\pi$
$863$	33012.0	1.30213	0.651067	+	0.759020i	$0.274322\pi$
$864$	0	0
$865$	0	0
$866$	49994.1	1.96174
$867$	0	0
$868$	21725.4	0.849548
$869$	−11626.1	−0.453842
$870$	0	0
$871$	4052.11	0.157635
$872$	−55650.9	−2.16121
$873$	0	0
$874$	−9778.50	−0.378447
$875$	0	0
$876$	0	0
$877$	−20709.9	−0.797404	−0.398702	−	0.917080i	$-0.630539\pi$
$877$	−20709.9	−0.797404	−0.398702	+	0.917080i	$0.630539\pi$
$878$	−20417.2	−0.784793
$879$	0	0
$880$	0	0
$881$	13604.5	0.520258	0.260129	−	0.965574i	$-0.416235\pi$
$881$	13604.5	0.520258	0.260129	+	0.965574i	$0.416235\pi$
$882$	0	0
$883$	−47582.7	−1.81346	−0.906729	−	0.421713i	$-0.861429\pi$
$883$	−47582.7	−1.81346	−0.906729	+	0.421713i	$0.861429\pi$
$884$	−96802.1	−3.68304
$885$	0	0
$886$	−56080.2	−2.12647
$887$	−25132.9	−0.951386	−0.475693	−	0.879611i	$-0.657803\pi$
$887$	−25132.9	−0.951386	−0.475693	+	0.879611i	$0.657803\pi$
$888$	0	0
$889$	−20025.5	−0.755492
$890$	0	0
$891$	0	0
$892$	−55149.7	−2.07012
$893$	−29796.9	−1.11659
$894$	0	0
$895$	0	0
$896$	3629.93	0.135343
$897$	0	0
$898$	42296.0	1.57176
$899$	−35302.4	−1.30968
$900$	0	0
$901$	−33446.2	−1.23668
$902$	−57682.3	−2.12928
$903$	0	0
$904$	−126012.	−4.63617
$905$	0	0
$906$	0	0
$907$	−36233.4	−1.32647	−0.663235	−	0.748411i	$-0.730817\pi$
$907$	−36233.4	−1.32647	−0.663235	+	0.748411i	$0.730817\pi$
$908$	11580.1	0.423238
$909$	0	0
$910$	0	0
$911$	−16912.4	−0.615076	−0.307538	−	0.951536i	$-0.599505\pi$
$911$	−16912.4	−0.615076	−0.307538	+	0.951536i	$0.599505\pi$
$912$	0	0
$913$	−10711.1	−0.388264
$914$	92256.5	3.33870
$915$	0	0
$916$	49926.7	1.80090
$917$	3674.80	0.132337
$918$	0	0
$919$	−49589.9	−1.78000	−0.890001	−	0.455959i	$-0.849296\pi$
$919$	−49589.9	−1.78000	−0.890001	+	0.455959i	$0.849296\pi$
$920$	0	0
$921$	0	0
$922$	68422.4	2.44400
$923$	−32329.2	−1.15290
$924$	0	0
$925$	0	0
$926$	2814.09	0.0998668
$927$	0	0
$928$	61136.7	2.16262
$929$	40649.1	1.43558	0.717789	−	0.696261i	$-0.245154\pi$
$929$	40649.1	1.43558	0.717789	+	0.696261i	$0.245154\pi$
$930$	0	0
$931$	3017.94	0.106240
$932$	−43904.2	−1.54306
$933$	0	0
$934$	−30925.1	−1.08341
$935$	0	0
$936$	0	0
$937$	−3678.18	−0.128240	−0.0641200	−	0.997942i	$-0.520424\pi$
$937$	−3678.18	−0.128240	−0.0641200	+	0.997942i	$0.520424\pi$
$938$	−3250.32	−0.113142
$939$	0	0
$940$	0	0
$941$	4835.12	0.167503	0.0837516	−	0.996487i	$-0.473310\pi$
$941$	4835.12	0.167503	0.0837516	+	0.996487i	$0.473310\pi$
$942$	0	0
$943$	9468.24	0.326965
$944$	−59113.6	−2.03812
$945$	0	0
$946$	5585.63	0.191971
$947$	31497.4	1.08081	0.540405	−	0.841405i	$-0.318271\pi$
$947$	31497.4	1.08081	0.540405	+	0.841405i	$0.318271\pi$
$948$	0	0
$949$	−51398.9	−1.75814
$950$	0	0
$951$	0	0
$952$	44782.5	1.52459
$953$	−26365.5	−0.896183	−0.448092	−	0.893988i	$-0.647896\pi$
$953$	−26365.5	−0.896183	−0.448092	+	0.893988i	$0.647896\pi$
$954$	0	0
$955$	0	0
$956$	−101545.	−3.43537
$957$	0	0
$958$	−6834.20	−0.230483
$959$	15043.2	0.506538
$960$	0	0
$961$	−2827.48	−0.0949104
$962$	96375.4	3.23001
$963$	0	0
$964$	−26694.2	−0.891870
$965$	0	0
$966$	0	0
$967$	−2307.07	−0.0767221	−0.0383611	−	0.999264i	$-0.512214\pi$
$967$	−2307.07	−0.0767221	−0.0383611	+	0.999264i	$0.512214\pi$
$968$	2159.89	0.0717164
$969$	0	0
$970$	0	0
$971$	48345.5	1.59782	0.798909	−	0.601452i	$-0.205411\pi$
$971$	48345.5	1.59782	0.798909	+	0.601452i	$0.205411\pi$
$972$	0	0
$973$	−10554.5	−0.347750
$974$	−30996.2	−1.01969
$975$	0	0
$976$	−21475.1	−0.704304
$977$	−25534.5	−0.836153	−0.418076	−	0.908412i	$-0.637296\pi$
$977$	−25534.5	−0.836153	−0.418076	+	0.908412i	$0.637296\pi$
$978$	0	0
$979$	3243.53	0.105887
$980$	0	0
$981$	0	0
$982$	53624.0	1.74258
$983$	41611.3	1.35015	0.675074	−	0.737750i	$-0.264112\pi$
$983$	41611.3	1.35015	0.675074	+	0.737750i	$0.264112\pi$
$984$	0	0
$985$	0	0
$986$	−126173.	−4.07522
$987$	0	0
$988$	52690.4	1.69666
$989$	−916.851	−0.0294784
$990$	0	0
$991$	9963.65	0.319380	0.159690	−	0.987167i	$-0.448950\pi$
$991$	9963.65	0.319380	0.159690	+	0.987167i	$0.448950\pi$
$992$	−46695.5	−1.49454
$993$	0	0
$994$	25932.3	0.827486
$995$	0	0
$996$	0	0
$997$	15960.8	0.507005	0.253502	−	0.967335i	$-0.418417\pi$
$997$	15960.8	0.507005	0.253502	+	0.967335i	$0.418417\pi$
$998$	21585.3	0.684639
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bo.1.1		5
3.2	odd	2		525.4.a.x.1.5		5
5.2	odd	4		315.4.d.b.64.1		10
5.3	odd	4		315.4.d.b.64.10		10
5.4	even	2		1575.4.a.bp.1.5		5
15.2	even	4		105.4.d.b.64.10	yes	10
15.8	even	4		105.4.d.b.64.1	✓	10
15.14	odd	2		525.4.a.w.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.d.b.64.1	✓	10	15.8	even	4
105.4.d.b.64.10	yes	10	15.2	even	4
315.4.d.b.64.1		10	5.2	odd	4
315.4.d.b.64.10		10	5.3	odd	4
525.4.a.w.1.1		5	15.14	odd	2
525.4.a.x.1.5		5	3.2	odd	2
1575.4.a.bo.1.1		5	1.1	even	1	trivial
1575.4.a.bp.1.5		5	5.4	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-5.18660 q^{2} +18.9008 q^{4} +7.00000 q^{7} -56.5383 q^{8} +O(q^{10})\)	\(q-5.18660 q^{2} +18.9008 q^{4} +7.00000 q^{7} -56.5383 q^{8} +35.9555 q^{11} +45.2622 q^{13} -36.3062 q^{14} +142.035 q^{16} -113.154 q^{17} +61.5906 q^{19} -186.487 q^{22} +30.6108 q^{23} -234.757 q^{26} +132.306 q^{28} -214.989 q^{29} +164.206 q^{31} -284.372 q^{32} +586.882 q^{34} -410.533 q^{37} -319.446 q^{38} +309.310 q^{41} -29.9519 q^{43} +679.589 q^{44} -158.766 q^{46} -483.790 q^{47} +49.0000 q^{49} +855.493 q^{52} +295.582 q^{53} -395.768 q^{56} +1115.06 q^{58} -416.191 q^{59} -151.196 q^{61} -851.670 q^{62} +338.644 q^{64} +89.5253 q^{67} -2138.70 q^{68} -714.265 q^{71} -1135.58 q^{73} +2129.27 q^{74} +1164.11 q^{76} +251.688 q^{77} -323.347 q^{79} -1604.27 q^{82} -297.898 q^{83} +155.348 q^{86} -2032.86 q^{88} +90.2097 q^{89} +316.835 q^{91} +578.570 q^{92} +2509.23 q^{94} -492.101 q^{97} -254.143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8}+O(q^{10}) \) 5 * q - q^2 + 27 * q^4 + 35 * q^7 - 33 * q^8	\( 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8} - 66 q^{11} - 2 q^{13} - 7 q^{14} + 155 q^{16} - 108 q^{17} + 174 q^{19} - 506 q^{22} + 116 q^{23} - 446 q^{26} + 189 q^{28} - 370 q^{29} + 342 q^{31} + 55 q^{32} + 112 q^{34} - 408 q^{37} + 34 q^{38} - 802 q^{41} - 584 q^{43} - 290 q^{44} + 640 q^{46} - 716 q^{47} + 245 q^{49} + 338 q^{52} - 98 q^{53} - 231 q^{56} + 482 q^{58} - 704 q^{59} + 650 q^{61} - 2070 q^{62} + 75 q^{64} + 180 q^{67} - 4520 q^{68} - 1470 q^{71} - 534 q^{73} + 1312 q^{74} + 4370 q^{76} - 462 q^{77} - 820 q^{79} - 1338 q^{82} - 1520 q^{83} - 832 q^{86} - 3258 q^{88} - 286 q^{89} - 14 q^{91} - 1288 q^{92} + 2540 q^{94} + 278 q^{97} - 49 q^{98}+O(q^{100}) \) 5 * q - q^2 + 27 * q^4 + 35 * q^7 - 33 * q^8 - 66 * q^11 - 2 * q^13 - 7 * q^14 + 155 * q^16 - 108 * q^17 + 174 * q^19 - 506 * q^22 + 116 * q^23 - 446 * q^26 + 189 * q^28 - 370 * q^29 + 342 * q^31 + 55 * q^32 + 112 * q^34 - 408 * q^37 + 34 * q^38 - 802 * q^41 - 584 * q^43 - 290 * q^44 + 640 * q^46 - 716 * q^47 + 245 * q^49 + 338 * q^52 - 98 * q^53 - 231 * q^56 + 482 * q^58 - 704 * q^59 + 650 * q^61 - 2070 * q^62 + 75 * q^64 + 180 * q^67 - 4520 * q^68 - 1470 * q^71 - 534 * q^73 + 1312 * q^74 + 4370 * q^76 - 462 * q^77 - 820 * q^79 - 1338 * q^82 - 1520 * q^83 - 832 * q^86 - 3258 * q^88 - 286 * q^89 - 14 * q^91 - 1288 * q^92 + 2540 * q^94 + 278 * q^97 - 49 * q^98

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