LMFDB - Embedded newform 1575.4.a.y.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:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.70156$ 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)

$f(q)$	$=$	$q-1.70156 q^{2} -5.10469 q^{4} -7.00000 q^{7} +22.2984 q^{8} +O(q^{10})$	$q-1.70156 q^{2} -5.10469 q^{4} -7.00000 q^{7} +22.2984 q^{8} -37.4031 q^{11} -29.0156 q^{13} +11.9109 q^{14} +2.89531 q^{16} +58.4187 q^{17} -54.5969 q^{19} +63.6437 q^{22} +161.675 q^{23} +49.3719 q^{26} +35.7328 q^{28} -137.581 q^{29} +154.659 q^{31} -183.314 q^{32} -99.4031 q^{34} +350.125 q^{37} +92.9000 q^{38} -353.769 q^{41} +518.156 q^{43} +190.931 q^{44} -275.100 q^{46} -542.219 q^{47} +49.0000 q^{49} +148.116 q^{52} +305.309 q^{53} -156.089 q^{56} +234.103 q^{58} -14.6813 q^{59} -171.069 q^{61} -263.163 q^{62} +288.758 q^{64} -551.956 q^{67} -298.209 q^{68} +120.334 q^{71} -284.659 q^{73} -595.759 q^{74} +278.700 q^{76} +261.822 q^{77} +941.612 q^{79} +601.959 q^{82} +377.150 q^{83} -881.675 q^{86} -834.031 q^{88} +677.725 q^{89} +203.109 q^{91} -825.300 q^{92} +922.619 q^{94} +1225.03 q^{97} -83.3765 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 9 q^{4} - 14 q^{7} + 51 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 + 9 * q^4 - 14 * q^7 + 51 * q^8	$ 2 q + 3 q^{2} + 9 q^{4} - 14 q^{7} + 51 q^{8} - 62 q^{11} + 6 q^{13} - 21 q^{14} + 25 q^{16} + 40 q^{17} - 122 q^{19} - 52 q^{22} + 16 q^{23} + 214 q^{26} - 63 q^{28} - 352 q^{29} + 66 q^{31} - 309 q^{32} - 186 q^{34} + 188 q^{37} - 224 q^{38} - 16 q^{41} + 396 q^{43} - 156 q^{44} - 960 q^{46} - 188 q^{47} + 98 q^{49} + 642 q^{52} + 982 q^{53} - 357 q^{56} - 774 q^{58} - 516 q^{59} - 880 q^{61} - 680 q^{62} - 479 q^{64} + 356 q^{67} - 558 q^{68} - 310 q^{71} - 326 q^{73} - 1358 q^{74} - 672 q^{76} + 434 q^{77} + 1832 q^{79} + 2190 q^{82} - 680 q^{83} - 1456 q^{86} - 1540 q^{88} - 796 q^{89} - 42 q^{91} - 2880 q^{92} + 2588 q^{94} + 670 q^{97} + 147 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 + 9 * q^4 - 14 * q^7 + 51 * q^8 - 62 * q^11 + 6 * q^13 - 21 * q^14 + 25 * q^16 + 40 * q^17 - 122 * q^19 - 52 * q^22 + 16 * q^23 + 214 * q^26 - 63 * q^28 - 352 * q^29 + 66 * q^31 - 309 * q^32 - 186 * q^34 + 188 * q^37 - 224 * q^38 - 16 * q^41 + 396 * q^43 - 156 * q^44 - 960 * q^46 - 188 * q^47 + 98 * q^49 + 642 * q^52 + 982 * q^53 - 357 * q^56 - 774 * q^58 - 516 * q^59 - 880 * q^61 - 680 * q^62 - 479 * q^64 + 356 * q^67 - 558 * q^68 - 310 * q^71 - 326 * q^73 - 1358 * q^74 - 672 * q^76 + 434 * q^77 + 1832 * q^79 + 2190 * q^82 - 680 * q^83 - 1456 * q^86 - 1540 * q^88 - 796 * q^89 - 42 * q^91 - 2880 * q^92 + 2588 * q^94 + 670 * q^97 + 147 * 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 $

$2$	−1.70156	−0.601593	−0.300797	−	0.953688i	$-0.597253\pi$
$2$	−1.70156	−0.601593	−0.300797	+	0.953688i	$0.597253\pi$
$3$	0	0
$3$	0	0
$4$	−5.10469	−0.638086
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	22.2984	0.985461
$9$	0	0
$10$	0	0
$11$	−37.4031	−1.02522	−0.512612	−	0.858620i	$-0.671322\pi$
$11$	−37.4031	−1.02522	−0.512612	+	0.858620i	$0.671322\pi$
$12$	0	0
$13$	−29.0156	−0.619037	−0.309519	−	0.950893i	$-0.600168\pi$
$13$	−29.0156	−0.619037	−0.309519	+	0.950893i	$0.600168\pi$
$14$	11.9109	0.227381
$15$	0	0
$16$	2.89531	0.0452393
$17$	58.4187	0.833449	0.416724	−	0.909033i	$-0.363178\pi$
$17$	58.4187	0.833449	0.416724	+	0.909033i	$0.363178\pi$
$18$	0	0
$19$	−54.5969	−0.659231	−0.329615	−	0.944115i	$-0.606919\pi$
$19$	−54.5969	−0.659231	−0.329615	+	0.944115i	$0.606919\pi$
$20$	0	0
$21$	0	0
$22$	63.6437	0.616768
$23$	161.675	1.46572	0.732860	−	0.680379i	$-0.238185\pi$
$23$	161.675	1.46572	0.732860	+	0.680379i	$0.238185\pi$
$24$	0	0
$25$	0	0
$26$	49.3719	0.372409
$27$	0	0
$28$	35.7328	0.241174
$29$	−137.581	−0.880972	−0.440486	−	0.897759i	$-0.645194\pi$
$29$	−137.581	−0.880972	−0.440486	+	0.897759i	$0.645194\pi$
$30$	0	0
$31$	154.659	0.896053	0.448026	−	0.894020i	$-0.352127\pi$
$31$	154.659	0.896053	0.448026	+	0.894020i	$0.352127\pi$
$32$	−183.314	−1.01268
$33$	0	0
$34$	−99.4031	−0.501397
$35$	0	0
$36$	0	0
$37$	350.125	1.55568	0.777840	−	0.628462i	$-0.216315\pi$
$37$	350.125	1.55568	0.777840	+	0.628462i	$0.216315\pi$
$38$	92.9000	0.396589
$39$	0	0
$40$	0	0
$41$	−353.769	−1.34755	−0.673773	−	0.738938i	$-0.735327\pi$
$41$	−353.769	−1.34755	−0.673773	+	0.738938i	$0.735327\pi$
$42$	0	0
$43$	518.156	1.83763	0.918815	−	0.394689i	$-0.129148\pi$
$43$	518.156	1.83763	0.918815	+	0.394689i	$0.129148\pi$
$44$	190.931	0.654181
$45$	0	0
$46$	−275.100	−0.881767
$47$	−542.219	−1.68278	−0.841391	−	0.540427i	$-0.818263\pi$
$47$	−542.219	−1.68278	−0.841391	+	0.540427i	$0.818263\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	148.116	0.394999
$53$	305.309	0.791273	0.395637	−	0.918407i	$-0.370524\pi$
$53$	305.309	0.791273	0.395637	+	0.918407i	$0.370524\pi$
$54$	0	0
$55$	0	0
$56$	−156.089	−0.372469
$57$	0	0
$58$	234.103	0.529987
$59$	−14.6813	−0.0323956	−0.0161978	−	0.999869i	$-0.505156\pi$
$59$	−14.6813	−0.0323956	−0.0161978	+	0.999869i	$0.505156\pi$
$60$	0	0
$61$	−171.069	−0.359067	−0.179534	−	0.983752i	$-0.557459\pi$
$61$	−171.069	−0.359067	−0.179534	+	0.983752i	$0.557459\pi$
$62$	−263.163	−0.539059
$63$	0	0
$64$	288.758	0.563980
$65$	0	0
$66$	0	0
$67$	−551.956	−1.00645	−0.503225	−	0.864155i	$-0.667853\pi$
$67$	−551.956	−1.00645	−0.503225	+	0.864155i	$0.667853\pi$
$68$	−298.209	−0.531812
$69$	0	0
$70$	0	0
$71$	120.334	0.201142	0.100571	−	0.994930i	$-0.467933\pi$
$71$	120.334	0.201142	0.100571	+	0.994930i	$0.467933\pi$
$72$	0	0
$73$	−284.659	−0.456395	−0.228198	−	0.973615i	$-0.573283\pi$
$73$	−284.659	−0.456395	−0.228198	+	0.973615i	$0.573283\pi$
$74$	−595.759	−0.935887
$75$	0	0
$76$	278.700	0.420646
$77$	261.822	0.387498
$78$	0	0
$79$	941.612	1.34101	0.670504	−	0.741906i	$-0.266078\pi$
$79$	941.612	1.34101	0.670504	+	0.741906i	$0.266078\pi$
$80$	0	0
$81$	0	0
$82$	601.959	0.810674
$83$	377.150	0.498766	0.249383	−	0.968405i	$-0.419772\pi$
$83$	377.150	0.498766	0.249383	+	0.968405i	$0.419772\pi$
$84$	0	0
$85$	0	0
$86$	−881.675	−1.10551
$87$	0	0
$88$	−834.031	−1.01032
$89$	677.725	0.807176	0.403588	−	0.914941i	$-0.367763\pi$
$89$	677.725	0.807176	0.403588	+	0.914941i	$0.367763\pi$
$90$	0	0
$91$	203.109	0.233974
$92$	−825.300	−0.935255
$93$	0	0
$94$	922.619	1.01235
$95$	0	0
$96$	0	0
$97$	1225.03	1.28230	0.641151	−	0.767414i	$-0.278457\pi$
$97$	1225.03	1.28230	0.641151	+	0.767414i	$0.278457\pi$
$98$	−83.3765	−0.0859419
$99$	0	0
$100$	0	0
$101$	338.144	0.333134	0.166567	−	0.986030i	$-0.446732\pi$
$101$	338.144	0.333134	0.166567	+	0.986030i	$0.446732\pi$
$102$	0	0
$103$	566.700	0.542122	0.271061	−	0.962562i	$-0.412625\pi$
$103$	566.700	0.542122	0.271061	+	0.962562i	$0.412625\pi$
$104$	−647.003	−0.610037
$105$	0	0
$106$	−519.503	−0.476024
$107$	−562.531	−0.508242	−0.254121	−	0.967172i	$-0.581786\pi$
$107$	−562.531	−0.508242	−0.254121	+	0.967172i	$0.581786\pi$
$108$	0	0
$109$	1830.79	1.60879	0.804396	−	0.594094i	$-0.202489\pi$
$109$	1830.79	1.60879	0.804396	+	0.594094i	$0.202489\pi$
$110$	0	0
$111$	0	0
$112$	−20.2672	−0.0170988
$113$	−31.8032	−0.0264761	−0.0132380	−	0.999912i	$-0.504214\pi$
$113$	−31.8032	−0.0264761	−0.0132380	+	0.999912i	$0.504214\pi$
$114$	0	0
$115$	0	0
$116$	702.309	0.562136
$117$	0	0
$118$	24.9811	0.0194890
$119$	−408.931	−0.315014
$120$	0	0
$121$	67.9937	0.0510847
$122$	291.084	0.216012
$123$	0	0
$124$	−789.488	−0.571759
$125$	0	0
$126$	0	0
$127$	−2220.81	−1.55169	−0.775847	−	0.630921i	$-0.782677\pi$
$127$	−2220.81	−1.55169	−0.775847	+	0.630921i	$0.782677\pi$
$128$	975.173	0.673390
$129$	0	0
$130$	0	0
$131$	−646.512	−0.431191	−0.215596	−	0.976483i	$-0.569169\pi$
$131$	−646.512	−0.431191	−0.215596	+	0.976483i	$0.569169\pi$
$132$	0	0
$133$	382.178	0.249166
$134$	939.188	0.605474
$135$	0	0
$136$	1302.65	0.821331
$137$	−896.009	−0.558768	−0.279384	−	0.960179i	$-0.590130\pi$
$137$	−896.009	−0.558768	−0.279384	+	0.960179i	$0.590130\pi$
$138$	0	0
$139$	−2313.61	−1.41178	−0.705891	−	0.708320i	$-0.749453\pi$
$139$	−2313.61	−1.41178	−0.705891	+	0.708320i	$0.749453\pi$
$140$	0	0
$141$	0	0
$142$	−204.756	−0.121005
$143$	1085.27	0.634652
$144$	0	0
$145$	0	0
$146$	484.366	0.274564
$147$	0	0
$148$	−1787.28	−0.992658
$149$	−819.337	−0.450488	−0.225244	−	0.974302i	$-0.572318\pi$
$149$	−819.337	−0.450488	−0.225244	+	0.974302i	$0.572318\pi$
$150$	0	0
$151$	534.744	0.288191	0.144095	−	0.989564i	$-0.453973\pi$
$151$	534.744	0.288191	0.144095	+	0.989564i	$0.453973\pi$
$152$	−1217.43	−0.649646
$153$	0	0
$154$	−445.506	−0.233116
$155$	0	0
$156$	0	0
$157$	−1564.76	−0.795423	−0.397711	−	0.917511i	$-0.630195\pi$
$157$	−1564.76	−0.795423	−0.397711	+	0.917511i	$0.630195\pi$
$158$	−1602.21	−0.806741
$159$	0	0
$160$	0	0
$161$	−1131.72	−0.553990
$162$	0	0
$163$	1114.31	0.535455	0.267728	−	0.963495i	$-0.413727\pi$
$163$	1114.31	0.535455	0.267728	+	0.963495i	$0.413727\pi$
$164$	1805.88	0.859850
$165$	0	0
$166$	−641.744	−0.300054
$167$	−1774.47	−0.822231	−0.411115	−	0.911583i	$-0.634861\pi$
$167$	−1774.47	−0.822231	−0.411115	+	0.911583i	$0.634861\pi$
$168$	0	0
$169$	−1355.09	−0.616793
$170$	0	0
$171$	0	0
$172$	−2645.02	−1.17257
$173$	−4215.88	−1.85276	−0.926380	−	0.376590i	$-0.877097\pi$
$173$	−4215.88	−1.85276	−0.926380	+	0.376590i	$0.877097\pi$
$174$	0	0
$175$	0	0
$176$	−108.294	−0.0463804
$177$	0	0
$178$	−1153.19	−0.485592
$179$	2430.70	1.01497	0.507483	−	0.861662i	$-0.330576\pi$
$179$	2430.70	1.01497	0.507483	+	0.861662i	$0.330576\pi$
$180$	0	0
$181$	−2700.91	−1.10916	−0.554578	−	0.832132i	$-0.687120\pi$
$181$	−2700.91	−1.10916	−0.554578	+	0.832132i	$0.687120\pi$
$182$	−345.603	−0.140757
$183$	0	0
$184$	3605.10	1.44441
$185$	0	0
$186$	0	0
$187$	−2185.04	−0.854472
$188$	2767.86	1.07376
$189$	0	0
$190$	0	0
$191$	−3611.10	−1.36801	−0.684005	−	0.729478i	$-0.739763\pi$
$191$	−3611.10	−1.36801	−0.684005	+	0.729478i	$0.739763\pi$
$192$	0	0
$193$	4468.33	1.66651	0.833257	−	0.552886i	$-0.186474\pi$
$193$	4468.33	1.66651	0.833257	+	0.552886i	$0.186474\pi$
$194$	−2084.47	−0.771425
$195$	0	0
$196$	−250.130	−0.0911551
$197$	−434.422	−0.157113	−0.0785566	−	0.996910i	$-0.525031\pi$
$197$	−434.422	−0.157113	−0.0785566	+	0.996910i	$0.525031\pi$
$198$	0	0
$199$	−468.915	−0.167038	−0.0835189	−	0.996506i	$-0.526616\pi$
$199$	−468.915	−0.167038	−0.0835189	+	0.996506i	$0.526616\pi$
$200$	0	0
$201$	0	0
$202$	−575.372	−0.200411
$203$	963.069	0.332976
$204$	0	0
$205$	0	0
$206$	−964.275	−0.326137
$207$	0	0
$208$	−84.0093	−0.0280048
$209$	2042.09	0.675859
$210$	0	0
$211$	3735.51	1.21878	0.609392	−	0.792869i	$-0.291414\pi$
$211$	3735.51	1.21878	0.609392	+	0.792869i	$0.291414\pi$
$212$	−1558.51	−0.504900
$213$	0	0
$214$	957.182	0.305755
$215$	0	0
$216$	0	0
$217$	−1082.62	−0.338676
$218$	−3115.21	−0.967838
$219$	0	0
$220$	0	0
$221$	−1695.06	−0.515936
$222$	0	0
$223$	−842.806	−0.253087	−0.126544	−	0.991961i	$-0.540388\pi$
$223$	−842.806	−0.253087	−0.126544	+	0.991961i	$0.540388\pi$
$224$	1283.20	0.382756
$225$	0	0
$226$	54.1152	0.0159278
$227$	992.150	0.290094	0.145047	−	0.989425i	$-0.453667\pi$
$227$	992.150	0.290094	0.145047	+	0.989425i	$0.453667\pi$
$228$	0	0
$229$	−6411.39	−1.85012	−0.925059	−	0.379825i	$-0.875984\pi$
$229$	−6411.39	−1.85012	−0.925059	+	0.379825i	$0.875984\pi$
$230$	0	0
$231$	0	0
$232$	−3067.85	−0.868164
$233$	−2274.35	−0.639476	−0.319738	−	0.947506i	$-0.603595\pi$
$233$	−2274.35	−0.639476	−0.319738	+	0.947506i	$0.603595\pi$
$234$	0	0
$235$	0	0
$236$	74.9433	0.0206712
$237$	0	0
$238$	695.822	0.189510
$239$	−2863.12	−0.774893	−0.387447	−	0.921892i	$-0.626643\pi$
$239$	−2863.12	−0.774893	−0.387447	+	0.921892i	$0.626643\pi$
$240$	0	0
$241$	−5364.23	−1.43378	−0.716889	−	0.697187i	$-0.754435\pi$
$241$	−5364.23	−1.43378	−0.716889	+	0.697187i	$0.754435\pi$
$242$	−115.696	−0.0307322
$243$	0	0
$244$	873.252	0.229116
$245$	0	0
$246$	0	0
$247$	1584.16	0.408088
$248$	3448.66	0.883025
$249$	0	0
$250$	0	0
$251$	−5569.81	−1.40065	−0.700325	−	0.713824i	$-0.746961\pi$
$251$	−5569.81	−1.40065	−0.700325	+	0.713824i	$0.746961\pi$
$252$	0	0
$253$	−6047.15	−1.50269
$254$	3778.85	0.933489
$255$	0	0
$256$	−3969.38	−0.969087
$257$	2095.36	0.508580	0.254290	−	0.967128i	$-0.418158\pi$
$257$	2095.36	0.508580	0.254290	+	0.967128i	$0.418158\pi$
$258$	0	0
$259$	−2450.87	−0.587992
$260$	0	0
$261$	0	0
$262$	1100.08	0.259402
$263$	−7465.88	−1.75044	−0.875220	−	0.483724i	$-0.839284\pi$
$263$	−7465.88	−1.75044	−0.875220	+	0.483724i	$0.839284\pi$
$264$	0	0
$265$	0	0
$266$	−650.300	−0.149896
$267$	0	0
$268$	2817.56	0.642202
$269$	6521.38	1.47812	0.739062	−	0.673637i	$-0.235269\pi$
$269$	6521.38	1.47812	0.739062	+	0.673637i	$0.235269\pi$
$270$	0	0
$271$	2409.70	0.540144	0.270072	−	0.962840i	$-0.412952\pi$
$271$	2409.70	0.540144	0.270072	+	0.962840i	$0.412952\pi$
$272$	169.141	0.0377046
$273$	0	0
$274$	1524.62	0.336151
$275$	0	0
$276$	0	0
$277$	2219.83	0.481503	0.240752	−	0.970587i	$-0.422606\pi$
$277$	2219.83	0.481503	0.240752	+	0.970587i	$0.422606\pi$
$278$	3936.75	0.849319
$279$	0	0
$280$	0	0
$281$	−5838.56	−1.23950	−0.619749	−	0.784800i	$-0.712766\pi$
$281$	−5838.56	−1.23950	−0.619749	+	0.784800i	$0.712766\pi$
$282$	0	0
$283$	3645.04	0.765636	0.382818	−	0.923824i	$-0.374954\pi$
$283$	3645.04	0.765636	0.382818	+	0.923824i	$0.374954\pi$
$284$	−614.269	−0.128346
$285$	0	0
$286$	−1846.66	−0.381802
$287$	2476.38	0.509325
$288$	0	0
$289$	−1500.25	−0.305363
$290$	0	0
$291$	0	0
$292$	1453.10	0.291219
$293$	3777.91	0.753268	0.376634	−	0.926362i	$-0.377081\pi$
$293$	3777.91	0.753268	0.376634	+	0.926362i	$0.377081\pi$
$294$	0	0
$295$	0	0
$296$	7807.24	1.53306
$297$	0	0
$298$	1394.15	0.271011
$299$	−4691.10	−0.907336
$300$	0	0
$301$	−3627.09	−0.694559
$302$	−909.900	−0.173374
$303$	0	0
$304$	−158.075	−0.0298231
$305$	0	0
$306$	0	0
$307$	4799.64	0.892281	0.446140	−	0.894963i	$-0.352798\pi$
$307$	4799.64	0.892281	0.446140	+	0.894963i	$0.352798\pi$
$308$	−1336.52	−0.247257
$309$	0	0
$310$	0	0
$311$	−580.113	−0.105772	−0.0528861	−	0.998601i	$-0.516842\pi$
$311$	−580.113	−0.105772	−0.0528861	+	0.998601i	$0.516842\pi$
$312$	0	0
$313$	6114.78	1.10424	0.552121	−	0.833764i	$-0.313818\pi$
$313$	6114.78	1.10424	0.552121	+	0.833764i	$0.313818\pi$
$314$	2662.53	0.478521
$315$	0	0
$316$	−4806.64	−0.855679
$317$	−4300.63	−0.761979	−0.380989	−	0.924579i	$-0.624417\pi$
$317$	−4300.63	−0.761979	−0.380989	+	0.924579i	$0.624417\pi$
$318$	0	0
$319$	5145.97	0.903194
$320$	0	0
$321$	0	0
$322$	1925.70	0.333277
$323$	−3189.48	−0.549435
$324$	0	0
$325$	0	0
$326$	−1896.06	−0.322126
$327$	0	0
$328$	−7888.49	−1.32795
$329$	3795.53	0.636032
$330$	0	0
$331$	−6687.54	−1.11051	−0.555257	−	0.831679i	$-0.687380\pi$
$331$	−6687.54	−1.11051	−0.555257	+	0.831679i	$0.687380\pi$
$332$	−1925.23	−0.318256
$333$	0	0
$334$	3019.37	0.494648
$335$	0	0
$336$	0	0
$337$	−5869.28	−0.948723	−0.474362	−	0.880330i	$-0.657321\pi$
$337$	−5869.28	−0.948723	−0.474362	+	0.880330i	$0.657321\pi$
$338$	2305.78	0.371058
$339$	0	0
$340$	0	0
$341$	−5784.74	−0.918655
$342$	0	0
$343$	−343.000	−0.0539949
$344$	11554.1	1.81091
$345$	0	0
$346$	7173.58	1.11461
$347$	1937.22	0.299699	0.149850	−	0.988709i	$-0.452121\pi$
$347$	1937.22	0.299699	0.149850	+	0.988709i	$0.452121\pi$
$348$	0	0
$349$	−9748.82	−1.49525	−0.747625	−	0.664121i	$-0.768806\pi$
$349$	−9748.82	−1.49525	−0.747625	+	0.664121i	$0.768806\pi$
$350$	0	0
$351$	0	0
$352$	6856.52	1.03822
$353$	−4576.61	−0.690052	−0.345026	−	0.938593i	$-0.612130\pi$
$353$	−4576.61	−0.690052	−0.345026	+	0.938593i	$0.612130\pi$
$354$	0	0
$355$	0	0
$356$	−3459.57	−0.515048
$357$	0	0
$358$	−4135.98	−0.610596
$359$	−10849.9	−1.59509	−0.797546	−	0.603258i	$-0.793869\pi$
$359$	−10849.9	−1.59509	−0.797546	+	0.603258i	$0.793869\pi$
$360$	0	0
$361$	−3878.18	−0.565415
$362$	4595.77	0.667261
$363$	0	0
$364$	−1036.81	−0.149296
$365$	0	0
$366$	0	0
$367$	−11467.7	−1.63108	−0.815541	−	0.578699i	$-0.803561\pi$
$367$	−11467.7	−1.63108	−0.815541	+	0.578699i	$0.803561\pi$
$368$	468.100	0.0663081
$369$	0	0
$370$	0	0
$371$	−2137.17	−0.299073
$372$	0	0
$373$	−539.982	−0.0749576	−0.0374788	−	0.999297i	$-0.511933\pi$
$373$	−539.982	−0.0749576	−0.0374788	+	0.999297i	$0.511933\pi$
$374$	3717.99	0.514044
$375$	0	0
$376$	−12090.6	−1.65832
$377$	3992.01	0.545355
$378$	0	0
$379$	8577.57	1.16253	0.581267	−	0.813713i	$-0.302557\pi$
$379$	8577.57	1.16253	0.581267	+	0.813713i	$0.302557\pi$
$380$	0	0
$381$	0	0
$382$	6144.51	0.822985
$383$	8627.96	1.15109	0.575546	−	0.817770i	$-0.304790\pi$
$383$	8627.96	1.15109	0.575546	+	0.817770i	$0.304790\pi$
$384$	0	0
$385$	0	0
$386$	−7603.13	−1.00256
$387$	0	0
$388$	−6253.42	−0.818219
$389$	−9234.06	−1.20356	−0.601781	−	0.798661i	$-0.705542\pi$
$389$	−9234.06	−1.20356	−0.601781	+	0.798661i	$0.705542\pi$
$390$	0	0
$391$	9444.85	1.22160
$392$	1092.62	0.140780
$393$	0	0
$394$	739.196	0.0945182
$395$	0	0
$396$	0	0
$397$	−11618.0	−1.46874	−0.734372	−	0.678747i	$-0.762523\pi$
$397$	−11618.0	−1.46874	−0.734372	+	0.678747i	$0.762523\pi$
$398$	797.889	0.100489
$399$	0	0
$400$	0	0
$401$	−11157.1	−1.38942	−0.694711	−	0.719289i	$-0.744468\pi$
$401$	−11157.1	−1.38942	−0.694711	+	0.719289i	$0.744468\pi$
$402$	0	0
$403$	−4487.54	−0.554690
$404$	−1726.12	−0.212568
$405$	0	0
$406$	−1638.72	−0.200316
$407$	−13095.8	−1.59492
$408$	0	0
$409$	−7428.08	−0.898031	−0.449015	−	0.893524i	$-0.648225\pi$
$409$	−7428.08	−0.898031	−0.449015	+	0.893524i	$0.648225\pi$
$410$	0	0
$411$	0	0
$412$	−2892.83	−0.345921
$413$	102.769	0.0122444
$414$	0	0
$415$	0	0
$416$	5318.97	0.626885
$417$	0	0
$418$	−3474.75	−0.406592
$419$	−9644.74	−1.12453	−0.562263	−	0.826959i	$-0.690069\pi$
$419$	−9644.74	−1.12453	−0.562263	+	0.826959i	$0.690069\pi$
$420$	0	0
$421$	9918.18	1.14818	0.574088	−	0.818793i	$-0.305357\pi$
$421$	9918.18	1.14818	0.574088	+	0.818793i	$0.305357\pi$
$422$	−6356.21	−0.733212
$423$	0	0
$424$	6807.92	0.779769
$425$	0	0
$426$	0	0
$427$	1197.48	0.135715
$428$	2871.54	0.324302
$429$	0	0
$430$	0	0
$431$	16324.1	1.82437	0.912185	−	0.409779i	$-0.134394\pi$
$431$	16324.1	1.82437	0.912185	+	0.409779i	$0.134394\pi$
$432$	0	0
$433$	5168.75	0.573659	0.286829	−	0.957982i	$-0.407399\pi$
$433$	5168.75	0.573659	0.286829	+	0.957982i	$0.407399\pi$
$434$	1842.14	0.203745
$435$	0	0
$436$	−9345.63	−1.02655
$437$	−8826.95	−0.966248
$438$	0	0
$439$	18339.0	1.99378	0.996892	−	0.0787782i	$-0.0251019\pi$
$439$	18339.0	1.99378	0.996892	+	0.0787782i	$0.0251019\pi$
$440$	0	0
$441$	0	0
$442$	2884.24	0.310383
$443$	−1613.28	−0.173023	−0.0865113	−	0.996251i	$-0.527572\pi$
$443$	−1613.28	−0.173023	−0.0865113	+	0.996251i	$0.527572\pi$
$444$	0	0
$445$	0	0
$446$	1434.09	0.152256
$447$	0	0
$448$	−2021.30	−0.213164
$449$	886.750	0.0932034	0.0466017	−	0.998914i	$-0.485161\pi$
$449$	886.750	0.0932034	0.0466017	+	0.998914i	$0.485161\pi$
$450$	0	0
$451$	13232.1	1.38154
$452$	162.345	0.0168940
$453$	0	0
$454$	−1688.21	−0.174518
$455$	0	0
$456$	0	0
$457$	−7391.22	−0.756557	−0.378279	−	0.925692i	$-0.623484\pi$
$457$	−7391.22	−0.756557	−0.378279	+	0.925692i	$0.623484\pi$
$458$	10909.4	1.11302
$459$	0	0
$460$	0	0
$461$	7133.35	0.720679	0.360340	−	0.932821i	$-0.382661\pi$
$461$	7133.35	0.720679	0.360340	+	0.932821i	$0.382661\pi$
$462$	0	0
$463$	−14461.8	−1.45162	−0.725808	−	0.687897i	$-0.758534\pi$
$463$	−14461.8	−1.45162	−0.725808	+	0.687897i	$0.758534\pi$
$464$	−398.341	−0.0398546
$465$	0	0
$466$	3869.95	0.384704
$467$	−16393.5	−1.62441	−0.812206	−	0.583370i	$-0.801734\pi$
$467$	−16393.5	−1.62441	−0.812206	+	0.583370i	$0.801734\pi$
$468$	0	0
$469$	3863.69	0.380403
$470$	0	0
$471$	0	0
$472$	−327.370	−0.0319246
$473$	−19380.7	−1.88398
$474$	0	0
$475$	0	0
$476$	2087.47	0.201006
$477$	0	0
$478$	4871.77	0.466171
$479$	12991.4	1.23923	0.619617	−	0.784904i	$-0.287288\pi$
$479$	12991.4	1.23923	0.619617	+	0.784904i	$0.287288\pi$
$480$	0	0
$481$	−10159.1	−0.963025
$482$	9127.57	0.862551
$483$	0	0
$484$	−347.087	−0.0325964
$485$	0	0
$486$	0	0
$487$	12863.5	1.19692	0.598461	−	0.801152i	$-0.295779\pi$
$487$	12863.5	1.19692	0.598461	+	0.801152i	$0.295779\pi$
$488$	−3814.57	−0.353847
$489$	0	0
$490$	0	0
$491$	4898.10	0.450200	0.225100	−	0.974336i	$-0.427729\pi$
$491$	4898.10	0.450200	0.225100	+	0.974336i	$0.427729\pi$
$492$	0	0
$493$	−8037.32	−0.734245
$494$	−2695.55	−0.245503
$495$	0	0
$496$	447.787	0.0405368
$497$	−842.340	−0.0760244
$498$	0	0
$499$	10308.0	0.924746	0.462373	−	0.886686i	$-0.346998\pi$
$499$	10308.0	0.924746	0.462373	+	0.886686i	$0.346998\pi$
$500$	0	0
$501$	0	0
$502$	9477.37	0.842621
$503$	−15119.6	−1.34026	−0.670130	−	0.742244i	$-0.733762\pi$
$503$	−15119.6	−1.34026	−0.670130	+	0.742244i	$0.733762\pi$
$504$	0	0
$505$	0	0
$506$	10289.6	0.904009
$507$	0	0
$508$	11336.6	0.990114
$509$	−14183.8	−1.23514	−0.617571	−	0.786515i	$-0.711883\pi$
$509$	−14183.8	−1.23514	−0.617571	+	0.786515i	$0.711883\pi$
$510$	0	0
$511$	1992.62	0.172501
$512$	−1047.24	−0.0903943
$513$	0	0
$514$	−3565.39	−0.305958
$515$	0	0
$516$	0	0
$517$	20280.7	1.72523
$518$	4170.32	0.353732
$519$	0	0
$520$	0	0
$521$	7464.08	0.627653	0.313827	−	0.949480i	$-0.398389\pi$
$521$	7464.08	0.627653	0.313827	+	0.949480i	$0.398389\pi$
$522$	0	0
$523$	16642.9	1.39148	0.695739	−	0.718295i	$-0.255077\pi$
$523$	16642.9	1.39148	0.695739	+	0.718295i	$0.255077\pi$
$524$	3300.24	0.275137
$525$	0	0
$526$	12703.7	1.05305
$527$	9035.01	0.746814
$528$	0	0
$529$	13971.8	1.14834
$530$	0	0
$531$	0	0
$532$	−1950.90	−0.158989
$533$	10264.8	0.834181
$534$	0	0
$535$	0	0
$536$	−12307.8	−0.991818
$537$	0	0
$538$	−11096.5	−0.889230
$539$	−1832.75	−0.146461
$540$	0	0
$541$	67.8755	0.00539408	0.00269704	−	0.999996i	$-0.499142\pi$
$541$	67.8755	0.00539408	0.00269704	+	0.999996i	$0.499142\pi$
$542$	−4100.26	−0.324947
$543$	0	0
$544$	−10709.0	−0.844014
$545$	0	0
$546$	0	0
$547$	9212.91	0.720138	0.360069	−	0.932926i	$-0.382753\pi$
$547$	9212.91	0.720138	0.360069	+	0.932926i	$0.382753\pi$
$548$	4573.85	0.356542
$549$	0	0
$550$	0	0
$551$	7511.51	0.580764
$552$	0	0
$553$	−6591.29	−0.506854
$554$	−3777.17	−0.289669
$555$	0	0
$556$	11810.2	0.900838
$557$	16699.6	1.27035	0.635173	−	0.772370i	$-0.280929\pi$
$557$	16699.6	1.27035	0.635173	+	0.772370i	$0.280929\pi$
$558$	0	0
$559$	−15034.6	−1.13756
$560$	0	0
$561$	0	0
$562$	9934.67	0.745674
$563$	14772.8	1.10586	0.552931	−	0.833227i	$-0.313509\pi$
$563$	14772.8	1.10586	0.552931	+	0.833227i	$0.313509\pi$
$564$	0	0
$565$	0	0
$566$	−6202.26	−0.460601
$567$	0	0
$568$	2683.27	0.198217
$569$	−5663.76	−0.417289	−0.208644	−	0.977992i	$-0.566905\pi$
$569$	−5663.76	−0.417289	−0.208644	+	0.977992i	$0.566905\pi$
$570$	0	0
$571$	5579.58	0.408929	0.204464	−	0.978874i	$-0.434455\pi$
$571$	5579.58	0.408929	0.204464	+	0.978874i	$0.434455\pi$
$572$	−5539.99	−0.404962
$573$	0	0
$574$	−4213.72	−0.306406
$575$	0	0
$576$	0	0
$577$	2301.23	0.166034	0.0830170	−	0.996548i	$-0.473544\pi$
$577$	2301.23	0.166034	0.0830170	+	0.996548i	$0.473544\pi$
$578$	2552.77	0.183704
$579$	0	0
$580$	0	0
$581$	−2640.05	−0.188516
$582$	0	0
$583$	−11419.5	−0.811232
$584$	−6347.46	−0.449760
$585$	0	0
$586$	−6428.34	−0.453161
$587$	−16470.4	−1.15810	−0.579052	−	0.815291i	$-0.696577\pi$
$587$	−16470.4	−1.15810	−0.579052	+	0.815291i	$0.696577\pi$
$588$	0	0
$589$	−8443.92	−0.590706
$590$	0	0
$591$	0	0
$592$	1013.72	0.0703779
$593$	−13570.0	−0.939715	−0.469858	−	0.882742i	$-0.655695\pi$
$593$	−13570.0	−0.939715	−0.469858	+	0.882742i	$0.655695\pi$
$594$	0	0
$595$	0	0
$596$	4182.46	0.287450
$597$	0	0
$598$	7982.20	0.545847
$599$	−27814.1	−1.89725	−0.948625	−	0.316403i	$-0.897525\pi$
$599$	−27814.1	−1.89725	−0.948625	+	0.316403i	$0.897525\pi$
$600$	0	0
$601$	20646.1	1.40128	0.700641	−	0.713514i	$-0.252898\pi$
$601$	20646.1	1.40128	0.700641	+	0.713514i	$0.252898\pi$
$602$	6171.72	0.417842
$603$	0	0
$604$	−2729.70	−0.183891
$605$	0	0
$606$	0	0
$607$	−3315.28	−0.221686	−0.110843	−	0.993838i	$-0.535355\pi$
$607$	−3315.28	−0.221686	−0.110843	+	0.993838i	$0.535355\pi$
$608$	10008.4	0.667588
$609$	0	0
$610$	0	0
$611$	15732.8	1.04170
$612$	0	0
$613$	11113.9	0.732278	0.366139	−	0.930560i	$-0.380680\pi$
$613$	11113.9	0.732278	0.366139	+	0.930560i	$0.380680\pi$
$614$	−8166.89	−0.536790
$615$	0	0
$616$	5838.22	0.381865
$617$	7871.34	0.513595	0.256797	−	0.966465i	$-0.417333\pi$
$617$	7871.34	0.513595	0.256797	+	0.966465i	$0.417333\pi$
$618$	0	0
$619$	−19107.1	−1.24068	−0.620339	−	0.784334i	$-0.713005\pi$
$619$	−19107.1	−1.24068	−0.620339	+	0.784334i	$0.713005\pi$
$620$	0	0
$621$	0	0
$622$	987.098	0.0636319
$623$	−4744.07	−0.305084
$624$	0	0
$625$	0	0
$626$	−10404.7	−0.664305
$627$	0	0
$628$	7987.60	0.507548
$629$	20453.9	1.29658
$630$	0	0
$631$	−25769.9	−1.62580	−0.812902	−	0.582401i	$-0.802113\pi$
$631$	−25769.9	−1.62580	−0.812902	+	0.582401i	$0.802113\pi$
$632$	20996.5	1.32151
$633$	0	0
$634$	7317.79	0.458401
$635$	0	0
$636$	0	0
$637$	−1421.77	−0.0884339
$638$	−8756.19	−0.543355
$639$	0	0
$640$	0	0
$641$	1954.61	0.120440	0.0602202	−	0.998185i	$-0.480820\pi$
$641$	1954.61	0.120440	0.0602202	+	0.998185i	$0.480820\pi$
$642$	0	0
$643$	19396.5	1.18961	0.594807	−	0.803868i	$-0.297228\pi$
$643$	19396.5	1.18961	0.594807	+	0.803868i	$0.297228\pi$
$644$	5777.10	0.353493
$645$	0	0
$646$	5427.10	0.330536
$647$	31264.3	1.89973	0.949865	−	0.312661i	$-0.101220\pi$
$647$	31264.3	1.89973	0.949865	+	0.312661i	$0.101220\pi$
$648$	0	0
$649$	549.126	0.0332127
$650$	0	0
$651$	0	0
$652$	−5688.18	−0.341666
$653$	6442.75	0.386102	0.193051	−	0.981189i	$-0.438162\pi$
$653$	6442.75	0.386102	0.193051	+	0.981189i	$0.438162\pi$
$654$	0	0
$655$	0	0
$656$	−1024.27	−0.0609620
$657$	0	0
$658$	−6458.33	−0.382632
$659$	3584.70	0.211897	0.105949	−	0.994372i	$-0.466212\pi$
$659$	3584.70	0.211897	0.105949	+	0.994372i	$0.466212\pi$
$660$	0	0
$661$	6294.43	0.370386	0.185193	−	0.982702i	$-0.440709\pi$
$661$	6294.43	0.370386	0.185193	+	0.982702i	$0.440709\pi$
$662$	11379.3	0.668078
$663$	0	0
$664$	8409.85	0.491514
$665$	0	0
$666$	0	0
$667$	−22243.4	−1.29126
$668$	9058.11	0.524654
$669$	0	0
$670$	0	0
$671$	6398.51	0.368125
$672$	0	0
$673$	10233.9	0.586162	0.293081	−	0.956088i	$-0.405319\pi$
$673$	10233.9	0.586162	0.293081	+	0.956088i	$0.405319\pi$
$674$	9986.94	0.570745
$675$	0	0
$676$	6917.33	0.393567
$677$	−7100.75	−0.403108	−0.201554	−	0.979477i	$-0.564599\pi$
$677$	−7100.75	−0.403108	−0.201554	+	0.979477i	$0.564599\pi$
$678$	0	0
$679$	−8575.24	−0.484665
$680$	0	0
$681$	0	0
$682$	9843.10	0.552657
$683$	−35274.6	−1.97620	−0.988100	−	0.153813i	$-0.950845\pi$
$683$	−35274.6	−1.97620	−0.988100	+	0.153813i	$0.950845\pi$
$684$	0	0
$685$	0	0
$686$	583.636	0.0324830
$687$	0	0
$688$	1500.22	0.0831330
$689$	−8858.74	−0.489828
$690$	0	0
$691$	4945.12	0.272245	0.136122	−	0.990692i	$-0.456536\pi$
$691$	4945.12	0.272245	0.136122	+	0.990692i	$0.456536\pi$
$692$	21520.8	1.18222
$693$	0	0
$694$	−3296.31	−0.180297
$695$	0	0
$696$	0	0
$697$	−20666.7	−1.12311
$698$	16588.2	0.899532
$699$	0	0
$700$	0	0
$701$	15300.4	0.824379	0.412190	−	0.911098i	$-0.364764\pi$
$701$	15300.4	0.824379	0.412190	+	0.911098i	$0.364764\pi$
$702$	0	0
$703$	−19115.7	−1.02555
$704$	−10800.4	−0.578206
$705$	0	0
$706$	7787.38	0.415130
$707$	−2367.01	−0.125913
$708$	0	0
$709$	−28297.4	−1.49892	−0.749458	−	0.662052i	$-0.769686\pi$
$709$	−28297.4	−1.49892	−0.749458	+	0.662052i	$0.769686\pi$
$710$	0	0
$711$	0	0
$712$	15112.2	0.795441
$713$	25004.5	1.31336
$714$	0	0
$715$	0	0
$716$	−12407.9	−0.647635
$717$	0	0
$718$	18461.9	0.959597
$719$	−8548.96	−0.443425	−0.221712	−	0.975112i	$-0.571165\pi$
$719$	−8548.96	−0.443425	−0.221712	+	0.975112i	$0.571165\pi$
$720$	0	0
$721$	−3966.90	−0.204903
$722$	6598.97	0.340150
$723$	0	0
$724$	13787.3	0.707737
$725$	0	0
$726$	0	0
$727$	−14345.3	−0.731827	−0.365913	−	0.930649i	$-0.619243\pi$
$727$	−14345.3	−0.731827	−0.365913	+	0.930649i	$0.619243\pi$
$728$	4529.02	0.230572
$729$	0	0
$730$	0	0
$731$	30270.0	1.53157
$732$	0	0
$733$	22624.0	1.14002	0.570012	−	0.821637i	$-0.306939\pi$
$733$	22624.0	1.14002	0.570012	+	0.821637i	$0.306939\pi$
$734$	19512.9	0.981248
$735$	0	0
$736$	−29637.3	−1.48430
$737$	20644.9	1.03184
$738$	0	0
$739$	14837.3	0.738566	0.369283	−	0.929317i	$-0.379603\pi$
$739$	14837.3	0.738566	0.369283	+	0.929317i	$0.379603\pi$
$740$	0	0
$741$	0	0
$742$	3636.52	0.179920
$743$	13073.1	0.645497	0.322749	−	0.946485i	$-0.395393\pi$
$743$	13073.1	0.645497	0.322749	+	0.946485i	$0.395393\pi$
$744$	0	0
$745$	0	0
$746$	918.813	0.0450940
$747$	0	0
$748$	11154.0	0.545226
$749$	3937.72	0.192098
$750$	0	0
$751$	16213.3	0.787791	0.393895	−	0.919155i	$-0.371127\pi$
$751$	16213.3	0.787791	0.393895	+	0.919155i	$0.371127\pi$
$752$	−1569.89	−0.0761278
$753$	0	0
$754$	−6792.65	−0.328082
$755$	0	0
$756$	0	0
$757$	−19903.9	−0.955642	−0.477821	−	0.878457i	$-0.658573\pi$
$757$	−19903.9	−0.955642	−0.477821	+	0.878457i	$0.658573\pi$
$758$	−14595.3	−0.699372
$759$	0	0
$760$	0	0
$761$	30125.5	1.43502	0.717509	−	0.696549i	$-0.245282\pi$
$761$	30125.5	1.43502	0.717509	+	0.696549i	$0.245282\pi$
$762$	0	0
$763$	−12815.6	−0.608066
$764$	18433.5	0.872907
$765$	0	0
$766$	−14681.0	−0.692489
$767$	425.986	0.0200541
$768$	0	0
$769$	−36049.1	−1.69046	−0.845230	−	0.534402i	$-0.820537\pi$
$769$	−36049.1	−1.69046	−0.845230	+	0.534402i	$0.820537\pi$
$770$	0	0
$771$	0	0
$772$	−22809.4	−1.06338
$773$	−9644.77	−0.448769	−0.224384	−	0.974501i	$-0.572037\pi$
$773$	−9644.77	−0.448769	−0.224384	+	0.974501i	$0.572037\pi$
$774$	0	0
$775$	0	0
$776$	27316.4	1.26366
$777$	0	0
$778$	15712.3	0.724054
$779$	19314.7	0.888344
$780$	0	0
$781$	−4500.88	−0.206215
$782$	−16071.0	−0.734908
$783$	0	0
$784$	141.870	0.00646275
$785$	0	0
$786$	0	0
$787$	25218.6	1.14224	0.571122	−	0.820865i	$-0.306508\pi$
$787$	25218.6	1.14224	0.571122	+	0.820865i	$0.306508\pi$
$788$	2217.59	0.100252
$789$	0	0
$790$	0	0
$791$	222.623	0.0100070
$792$	0	0
$793$	4963.67	0.222276
$794$	19768.8	0.883586
$795$	0	0
$796$	2393.67	0.106584
$797$	32042.3	1.42409	0.712044	−	0.702135i	$-0.247770\pi$
$797$	32042.3	1.42409	0.712044	+	0.702135i	$0.247770\pi$
$798$	0	0
$799$	−31675.7	−1.40251
$800$	0	0
$801$	0	0
$802$	18984.5	0.835867
$803$	10647.1	0.467908
$804$	0	0
$805$	0	0
$806$	7635.82	0.333698
$807$	0	0
$808$	7540.07	0.328291
$809$	−3427.90	−0.148972	−0.0744860	−	0.997222i	$-0.523732\pi$
$809$	−3427.90	−0.148972	−0.0744860	+	0.997222i	$0.523732\pi$
$810$	0	0
$811$	23094.4	0.999943	0.499972	−	0.866042i	$-0.333344\pi$
$811$	23094.4	0.999943	0.499972	+	0.866042i	$0.333344\pi$
$812$	−4916.16	−0.212467
$813$	0	0
$814$	22283.3	0.959494
$815$	0	0
$816$	0	0
$817$	−28289.7	−1.21142
$818$	12639.3	0.540249
$819$	0	0
$820$	0	0
$821$	−474.741	−0.0201810	−0.0100905	−	0.999949i	$-0.503212\pi$
$821$	−474.741	−0.0201810	−0.0100905	+	0.999949i	$0.503212\pi$
$822$	0	0
$823$	−24159.8	−1.02328	−0.511638	−	0.859201i	$-0.670961\pi$
$823$	−24159.8	−1.02328	−0.511638	+	0.859201i	$0.670961\pi$
$824$	12636.5	0.534240
$825$	0	0
$826$	−174.868	−0.00736613
$827$	−7566.35	−0.318147	−0.159074	−	0.987267i	$-0.550851\pi$
$827$	−7566.35	−0.318147	−0.159074	+	0.987267i	$0.550851\pi$
$828$	0	0
$829$	−10580.9	−0.443295	−0.221648	−	0.975127i	$-0.571143\pi$
$829$	−10580.9	−0.443295	−0.221648	+	0.975127i	$0.571143\pi$
$830$	0	0
$831$	0	0
$832$	−8378.49	−0.349125
$833$	2862.52	0.119064
$834$	0	0
$835$	0	0
$836$	−10424.2	−0.431256
$837$	0	0
$838$	16411.1	0.676507
$839$	−15315.6	−0.630218	−0.315109	−	0.949055i	$-0.602041\pi$
$839$	−15315.6	−0.630218	−0.315109	+	0.949055i	$0.602041\pi$
$840$	0	0
$841$	−5460.40	−0.223888
$842$	−16876.4	−0.690735
$843$	0	0
$844$	−19068.6	−0.777688
$845$	0	0
$846$	0	0
$847$	−475.956	−0.0193082
$848$	883.966	0.0357966
$849$	0	0
$850$	0	0
$851$	56606.4	2.28019
$852$	0	0
$853$	−18598.2	−0.746528	−0.373264	−	0.927725i	$-0.621761\pi$
$853$	−18598.2	−0.746528	−0.373264	+	0.927725i	$0.621761\pi$
$854$	−2037.59	−0.0816450
$855$	0	0
$856$	−12543.6	−0.500853
$857$	41775.3	1.66513	0.832566	−	0.553926i	$-0.186871\pi$
$857$	41775.3	1.66513	0.832566	+	0.553926i	$0.186871\pi$
$858$	0	0
$859$	32414.7	1.28752	0.643758	−	0.765229i	$-0.277374\pi$
$859$	32414.7	1.28752	0.643758	+	0.765229i	$0.277374\pi$
$860$	0	0
$861$	0	0
$862$	−27776.4	−1.09753
$863$	−31299.7	−1.23459	−0.617297	−	0.786730i	$-0.711772\pi$
$863$	−31299.7	−1.23459	−0.617297	+	0.786730i	$0.711772\pi$
$864$	0	0
$865$	0	0
$866$	−8794.94	−0.345109
$867$	0	0
$868$	5526.41	0.216104
$869$	−35219.2	−1.37483
$870$	0	0
$871$	16015.4	0.623030
$872$	40823.8	1.58540
$873$	0	0
$874$	15019.6	0.581288
$875$	0	0
$876$	0	0
$877$	19973.0	0.769031	0.384515	−	0.923119i	$-0.374369\pi$
$877$	19973.0	0.769031	0.384515	+	0.923119i	$0.374369\pi$
$878$	−31204.9	−1.19945
$879$	0	0
$880$	0	0
$881$	−17367.9	−0.664176	−0.332088	−	0.943248i	$-0.607753\pi$
$881$	−17367.9	−0.664176	−0.332088	+	0.943248i	$0.607753\pi$
$882$	0	0
$883$	14364.2	0.547446	0.273723	−	0.961809i	$-0.411745\pi$
$883$	14364.2	0.547446	0.273723	+	0.961809i	$0.411745\pi$
$884$	8652.73	0.329211
$885$	0	0
$886$	2745.09	0.104089
$887$	−33738.1	−1.27713	−0.638564	−	0.769568i	$-0.720471\pi$
$887$	−33738.1	−1.27713	−0.638564	+	0.769568i	$0.720471\pi$
$888$	0	0
$889$	15545.7	0.586485
$890$	0	0
$891$	0	0
$892$	4302.26	0.161491
$893$	29603.4	1.10934
$894$	0	0
$895$	0	0
$896$	−6826.21	−0.254518
$897$	0	0
$898$	−1508.86	−0.0560705
$899$	−21278.2	−0.789398
$900$	0	0
$901$	17835.8	0.659485
$902$	−22515.2	−0.831123
$903$	0	0
$904$	−709.162	−0.0260911
$905$	0	0
$906$	0	0
$907$	32998.6	1.20805	0.604024	−	0.796966i	$-0.293563\pi$
$907$	32998.6	1.20805	0.604024	+	0.796966i	$0.293563\pi$
$908$	−5064.62	−0.185105
$909$	0	0
$910$	0	0
$911$	−33446.3	−1.21638	−0.608192	−	0.793790i	$-0.708105\pi$
$911$	−33446.3	−1.21638	−0.608192	+	0.793790i	$0.708105\pi$
$912$	0	0
$913$	−14106.6	−0.511347
$914$	12576.6	0.455139
$915$	0	0
$916$	32728.2	1.18053
$917$	4525.59	0.162975
$918$	0	0
$919$	41708.7	1.49711	0.748554	−	0.663074i	$-0.230748\pi$
$919$	41708.7	1.49711	0.748554	+	0.663074i	$0.230748\pi$
$920$	0	0
$921$	0	0
$922$	−12137.8	−0.433556
$923$	−3491.58	−0.124514
$924$	0	0
$925$	0	0
$926$	24607.7	0.873283
$927$	0	0
$928$	25220.6	0.892140
$929$	−49024.6	−1.73137	−0.865686	−	0.500587i	$-0.833117\pi$
$929$	−49024.6	−1.73137	−0.865686	+	0.500587i	$0.833117\pi$
$930$	0	0
$931$	−2675.25	−0.0941758
$932$	11609.9	0.408040
$933$	0	0
$934$	27894.6	0.977235
$935$	0	0
$936$	0	0
$937$	5447.58	0.189930	0.0949651	−	0.995481i	$-0.469726\pi$
$937$	5447.58	0.189930	0.0949651	+	0.995481i	$0.469726\pi$
$938$	−6574.31	−0.228848
$939$	0	0
$940$	0	0
$941$	−4125.75	−0.142928	−0.0714642	−	0.997443i	$-0.522767\pi$
$941$	−4125.75	−0.142928	−0.0714642	+	0.997443i	$0.522767\pi$
$942$	0	0
$943$	−57195.5	−1.97513
$944$	−42.5069	−0.00146555
$945$	0	0
$946$	32977.4	1.13339
$947$	17332.3	0.594747	0.297374	−	0.954761i	$-0.403889\pi$
$947$	17332.3	0.594747	0.297374	+	0.954761i	$0.403889\pi$
$948$	0	0
$949$	8259.57	0.282526
$950$	0	0
$951$	0	0
$952$	−9118.53	−0.310434
$953$	56839.4	1.93201	0.966007	−	0.258517i	$-0.0832337\pi$
$953$	56839.4	1.93201	0.966007	+	0.258517i	$0.0832337\pi$
$954$	0	0
$955$	0	0
$956$	14615.3	0.494449
$957$	0	0
$958$	−22105.7	−0.745515
$959$	6272.07	0.211195
$960$	0	0
$961$	−5871.48	−0.197089
$962$	17286.3	0.579349
$963$	0	0
$964$	27382.7	0.914873
$965$	0	0
$966$	0	0
$967$	13284.2	0.441769	0.220884	−	0.975300i	$-0.429106\pi$
$967$	13284.2	0.441769	0.220884	+	0.975300i	$0.429106\pi$
$968$	1516.15	0.0503420
$969$	0	0
$970$	0	0
$971$	−12153.0	−0.401658	−0.200829	−	0.979626i	$-0.564364\pi$
$971$	−12153.0	−0.401658	−0.200829	+	0.979626i	$0.564364\pi$
$972$	0	0
$973$	16195.3	0.533604
$974$	−21888.1	−0.720060
$975$	0	0
$976$	−495.298	−0.0162440
$977$	−37999.3	−1.24433	−0.622163	−	0.782888i	$-0.713746\pi$
$977$	−37999.3	−1.24433	−0.622163	+	0.782888i	$0.713746\pi$
$978$	0	0
$979$	−25349.0	−0.827537
$980$	0	0
$981$	0	0
$982$	−8334.42	−0.270837
$983$	22375.0	0.725993	0.362996	−	0.931791i	$-0.381754\pi$
$983$	22375.0	0.725993	0.362996	+	0.931791i	$0.381754\pi$
$984$	0	0
$985$	0	0
$986$	13676.0	0.441717
$987$	0	0
$988$	−8086.65	−0.260395
$989$	83772.9	2.69345
$990$	0	0
$991$	18985.3	0.608564	0.304282	−	0.952582i	$-0.401583\pi$
$991$	18985.3	0.608564	0.304282	+	0.952582i	$0.401583\pi$
$992$	−28351.2	−0.907412
$993$	0	0
$994$	1433.29	0.0457358
$995$	0	0
$996$	0	0
$997$	56476.5	1.79401	0.897006	−	0.442019i	$-0.145738\pi$
$997$	56476.5	1.79401	0.897006	+	0.442019i	$0.145738\pi$
$998$	−17539.6	−0.556321
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.y.1.1		2
3.2	odd	2		525.4.a.i.1.2		2
5.4	even	2		315.4.a.g.1.2		2
15.2	even	4		525.4.d.j.274.3		4
15.8	even	4		525.4.d.j.274.2		4
15.14	odd	2		105.4.a.g.1.1	✓	2
35.34	odd	2		2205.4.a.v.1.2		2
60.59	even	2		1680.4.a.y.1.1		2
105.104	even	2		735.4.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.g.1.1	✓	2	15.14	odd	2
315.4.a.g.1.2		2	5.4	even	2
525.4.a.i.1.2		2	3.2	odd	2
525.4.d.j.274.2		4	15.8	even	4
525.4.d.j.274.3		4	15.2	even	4
735.4.a.q.1.1		2	105.104	even	2
1575.4.a.y.1.1		2	1.1	even	1	trivial
1680.4.a.y.1.1		2	60.59	even	2
2205.4.a.v.1.2		2	35.34	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70156 q^{2} -5.10469 q^{4} -7.00000 q^{7} +22.2984 q^{8} +O(q^{10})\)	\(q-1.70156 q^{2} -5.10469 q^{4} -7.00000 q^{7} +22.2984 q^{8} -37.4031 q^{11} -29.0156 q^{13} +11.9109 q^{14} +2.89531 q^{16} +58.4187 q^{17} -54.5969 q^{19} +63.6437 q^{22} +161.675 q^{23} +49.3719 q^{26} +35.7328 q^{28} -137.581 q^{29} +154.659 q^{31} -183.314 q^{32} -99.4031 q^{34} +350.125 q^{37} +92.9000 q^{38} -353.769 q^{41} +518.156 q^{43} +190.931 q^{44} -275.100 q^{46} -542.219 q^{47} +49.0000 q^{49} +148.116 q^{52} +305.309 q^{53} -156.089 q^{56} +234.103 q^{58} -14.6813 q^{59} -171.069 q^{61} -263.163 q^{62} +288.758 q^{64} -551.956 q^{67} -298.209 q^{68} +120.334 q^{71} -284.659 q^{73} -595.759 q^{74} +278.700 q^{76} +261.822 q^{77} +941.612 q^{79} +601.959 q^{82} +377.150 q^{83} -881.675 q^{86} -834.031 q^{88} +677.725 q^{89} +203.109 q^{91} -825.300 q^{92} +922.619 q^{94} +1225.03 q^{97} -83.3765 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 9 q^{4} - 14 q^{7} + 51 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 + 9 * q^4 - 14 * q^7 + 51 * q^8	\( 2 q + 3 q^{2} + 9 q^{4} - 14 q^{7} + 51 q^{8} - 62 q^{11} + 6 q^{13} - 21 q^{14} + 25 q^{16} + 40 q^{17} - 122 q^{19} - 52 q^{22} + 16 q^{23} + 214 q^{26} - 63 q^{28} - 352 q^{29} + 66 q^{31} - 309 q^{32} - 186 q^{34} + 188 q^{37} - 224 q^{38} - 16 q^{41} + 396 q^{43} - 156 q^{44} - 960 q^{46} - 188 q^{47} + 98 q^{49} + 642 q^{52} + 982 q^{53} - 357 q^{56} - 774 q^{58} - 516 q^{59} - 880 q^{61} - 680 q^{62} - 479 q^{64} + 356 q^{67} - 558 q^{68} - 310 q^{71} - 326 q^{73} - 1358 q^{74} - 672 q^{76} + 434 q^{77} + 1832 q^{79} + 2190 q^{82} - 680 q^{83} - 1456 q^{86} - 1540 q^{88} - 796 q^{89} - 42 q^{91} - 2880 q^{92} + 2588 q^{94} + 670 q^{97} + 147 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 + 9 * q^4 - 14 * q^7 + 51 * q^8 - 62 * q^11 + 6 * q^13 - 21 * q^14 + 25 * q^16 + 40 * q^17 - 122 * q^19 - 52 * q^22 + 16 * q^23 + 214 * q^26 - 63 * q^28 - 352 * q^29 + 66 * q^31 - 309 * q^32 - 186 * q^34 + 188 * q^37 - 224 * q^38 - 16 * q^41 + 396 * q^43 - 156 * q^44 - 960 * q^46 - 188 * q^47 + 98 * q^49 + 642 * q^52 + 982 * q^53 - 357 * q^56 - 774 * q^58 - 516 * q^59 - 880 * q^61 - 680 * q^62 - 479 * q^64 + 356 * q^67 - 558 * q^68 - 310 * q^71 - 326 * q^73 - 1358 * q^74 - 672 * q^76 + 434 * q^77 + 1832 * q^79 + 2190 * q^82 - 680 * q^83 - 1456 * q^86 - 1540 * q^88 - 796 * q^89 - 42 * q^91 - 2880 * q^92 + 2588 * q^94 + 670 * q^97 + 147 * q^98

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