LMFDB - Embedded newform 1575.4.a.bg.1.2

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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 32x^{2} - 35x + 120 $ x^4 - 32x^2 - 35x + 120
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.53510$ 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)

$f(q)$	$=$	$q-4.53510 q^{2} +12.5671 q^{4} +7.00000 q^{7} -20.7124 q^{8} +O(q^{10})$	$q-4.53510 q^{2} +12.5671 q^{4} +7.00000 q^{7} -20.7124 q^{8} +54.0684 q^{11} +75.2159 q^{13} -31.7457 q^{14} -6.60441 q^{16} +71.2538 q^{17} -65.5100 q^{19} -245.206 q^{22} -125.688 q^{23} -341.111 q^{26} +87.9699 q^{28} -190.405 q^{29} -193.105 q^{31} +195.650 q^{32} -323.143 q^{34} +114.673 q^{37} +297.094 q^{38} -216.896 q^{41} -413.032 q^{43} +679.484 q^{44} +570.007 q^{46} +113.555 q^{47} +49.0000 q^{49} +945.247 q^{52} -584.366 q^{53} -144.986 q^{56} +863.504 q^{58} -203.748 q^{59} -162.539 q^{61} +875.749 q^{62} -834.459 q^{64} +477.534 q^{67} +895.456 q^{68} -822.294 q^{71} -798.993 q^{73} -520.052 q^{74} -823.273 q^{76} +378.479 q^{77} -468.087 q^{79} +983.643 q^{82} +310.333 q^{83} +1873.14 q^{86} -1119.88 q^{88} -1314.90 q^{89} +526.511 q^{91} -1579.53 q^{92} -514.985 q^{94} -1314.66 q^{97} -222.220 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 36 q^{4} + 28 q^{7} - 27 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 36 * q^4 + 28 * q^7 - 27 * q^8	$ 4 q - 4 q^{2} + 36 q^{4} + 28 q^{7} - 27 q^{8} - 100 q^{11} + 44 q^{13} - 28 q^{14} + 160 q^{16} + 53 q^{17} - 29 q^{19} - 152 q^{22} - 295 q^{23} - 700 q^{26} + 252 q^{28} - 129 q^{29} + 114 q^{31} + 310 q^{32} + 203 q^{34} + 403 q^{37} - 555 q^{38} - 671 q^{41} - 411 q^{43} - 438 q^{44} - 997 q^{46} + 8 q^{47} + 196 q^{49} + 74 q^{52} - 90 q^{53} - 189 q^{56} + 673 q^{58} - 1018 q^{59} + 50 q^{61} + 1626 q^{62} - 2421 q^{64} + 424 q^{67} + 617 q^{68} - 215 q^{71} - 1207 q^{73} - 623 q^{74} - 3257 q^{76} - 700 q^{77} - 951 q^{79} + 1695 q^{82} + 3035 q^{83} + 99 q^{86} + 163 q^{88} - 2819 q^{89} + 308 q^{91} - 3073 q^{92} - 3056 q^{94} - 1100 q^{97} - 196 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 36 * q^4 + 28 * q^7 - 27 * q^8 - 100 * q^11 + 44 * q^13 - 28 * q^14 + 160 * q^16 + 53 * q^17 - 29 * q^19 - 152 * q^22 - 295 * q^23 - 700 * q^26 + 252 * q^28 - 129 * q^29 + 114 * q^31 + 310 * q^32 + 203 * q^34 + 403 * q^37 - 555 * q^38 - 671 * q^41 - 411 * q^43 - 438 * q^44 - 997 * q^46 + 8 * q^47 + 196 * q^49 + 74 * q^52 - 90 * q^53 - 189 * q^56 + 673 * q^58 - 1018 * q^59 + 50 * q^61 + 1626 * q^62 - 2421 * q^64 + 424 * q^67 + 617 * q^68 - 215 * q^71 - 1207 * q^73 - 623 * q^74 - 3257 * q^76 - 700 * q^77 - 951 * q^79 + 1695 * q^82 + 3035 * q^83 + 99 * q^86 + 163 * q^88 - 2819 * q^89 + 308 * q^91 - 3073 * q^92 - 3056 * q^94 - 1100 * q^97 - 196 * 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 $

$2$	−4.53510	−1.60340	−0.801700	−	0.597727i	$-0.796071\pi$
$2$	−4.53510	−1.60340	−0.801700	+	0.597727i	$0.796071\pi$
$3$	0	0
$3$	0	0
$4$	12.5671	1.57089
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−20.7124	−0.915365
$9$	0	0
$10$	0	0
$11$	54.0684	1.48202	0.741011	−	0.671493i	$-0.234347\pi$
$11$	54.0684	1.48202	0.741011	+	0.671493i	$0.234347\pi$
$12$	0	0
$13$	75.2159	1.60470	0.802351	−	0.596853i	$-0.203582\pi$
$13$	75.2159	1.60470	0.802351	+	0.596853i	$0.203582\pi$
$14$	−31.7457	−0.606028
$15$	0	0
$16$	−6.60441	−0.103194
$17$	71.2538	1.01656	0.508282	−	0.861191i	$-0.330281\pi$
$17$	71.2538	1.01656	0.508282	+	0.861191i	$0.330281\pi$
$18$	0	0
$19$	−65.5100	−0.791002	−0.395501	−	0.918466i	$-0.629429\pi$
$19$	−65.5100	−0.791002	−0.395501	+	0.918466i	$0.629429\pi$
$20$	0	0
$21$	0	0
$22$	−245.206	−2.37627
$23$	−125.688	−1.13947	−0.569733	−	0.821830i	$-0.692953\pi$
$23$	−125.688	−1.13947	−0.569733	+	0.821830i	$0.692953\pi$
$24$	0	0
$25$	0	0
$26$	−341.111	−2.57298
$27$	0	0
$28$	87.9699	0.593741
$29$	−190.405	−1.21922	−0.609608	−	0.792703i	$-0.708673\pi$
$29$	−190.405	−1.21922	−0.609608	+	0.792703i	$0.708673\pi$
$30$	0	0
$31$	−193.105	−1.11879	−0.559397	−	0.828900i	$-0.688967\pi$
$31$	−193.105	−1.11879	−0.559397	+	0.828900i	$0.688967\pi$
$32$	195.650	1.08083
$33$	0	0
$34$	−323.143	−1.62996
$35$	0	0
$36$	0	0
$37$	114.673	0.509516	0.254758	−	0.967005i	$-0.418004\pi$
$37$	114.673	0.509516	0.254758	+	0.967005i	$0.418004\pi$
$38$	297.094	1.26829
$39$	0	0
$40$	0	0
$41$	−216.896	−0.826180	−0.413090	−	0.910690i	$-0.635551\pi$
$41$	−216.896	−0.826180	−0.413090	+	0.910690i	$0.635551\pi$
$42$	0	0
$43$	−413.032	−1.46481	−0.732405	−	0.680870i	$-0.761602\pi$
$43$	−413.032	−1.46481	−0.732405	+	0.680870i	$0.761602\pi$
$44$	679.484	2.32809
$45$	0	0
$46$	570.007	1.82702
$47$	113.555	0.352421	0.176210	−	0.984353i	$-0.443616\pi$
$47$	113.555	0.352421	0.176210	+	0.984353i	$0.443616\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	945.247	2.52081
$53$	−584.366	−1.51451	−0.757253	−	0.653122i	$-0.773459\pi$
$53$	−584.366	−1.51451	−0.757253	+	0.653122i	$0.773459\pi$
$54$	0	0
$55$	0	0
$56$	−144.986	−0.345976
$57$	0	0
$58$	863.504	1.95489
$59$	−203.748	−0.449589	−0.224795	−	0.974406i	$-0.572171\pi$
$59$	−203.748	−0.449589	−0.224795	+	0.974406i	$0.572171\pi$
$60$	0	0
$61$	−162.539	−0.341163	−0.170581	−	0.985344i	$-0.554565\pi$
$61$	−162.539	−0.341163	−0.170581	+	0.985344i	$0.554565\pi$
$62$	875.749	1.79387
$63$	0	0
$64$	−834.459	−1.62980
$65$	0	0
$66$	0	0
$67$	477.534	0.870747	0.435374	−	0.900250i	$-0.356616\pi$
$67$	477.534	0.870747	0.435374	+	0.900250i	$0.356616\pi$
$68$	895.456	1.59691
$69$	0	0
$70$	0	0
$71$	−822.294	−1.37448	−0.687242	−	0.726429i	$-0.741179\pi$
$71$	−822.294	−1.37448	−0.687242	+	0.726429i	$0.741179\pi$
$72$	0	0
$73$	−798.993	−1.28103	−0.640514	−	0.767947i	$-0.721279\pi$
$73$	−798.993	−1.28103	−0.640514	+	0.767947i	$0.721279\pi$
$74$	−520.052	−0.816957
$75$	0	0
$76$	−823.273	−1.24258
$77$	378.479	0.560152
$78$	0	0
$79$	−468.087	−0.666632	−0.333316	−	0.942815i	$-0.608168\pi$
$79$	−468.087	−0.666632	−0.333316	+	0.942815i	$0.608168\pi$
$80$	0	0
$81$	0	0
$82$	983.643	1.32470
$83$	310.333	0.410403	0.205202	−	0.978720i	$-0.434215\pi$
$83$	310.333	0.410403	0.205202	+	0.978720i	$0.434215\pi$
$84$	0	0
$85$	0	0
$86$	1873.14	2.34867
$87$	0	0
$88$	−1119.88	−1.35659
$89$	−1314.90	−1.56606	−0.783029	−	0.621985i	$-0.786326\pi$
$89$	−1314.90	−1.56606	−0.783029	+	0.621985i	$0.786326\pi$
$90$	0	0
$91$	526.511	0.606520
$92$	−1579.53	−1.78998
$93$	0	0
$94$	−514.985	−0.565071
$95$	0	0
$96$	0	0
$97$	−1314.66	−1.37612	−0.688058	−	0.725656i	$-0.741537\pi$
$97$	−1314.66	−1.37612	−0.688058	+	0.725656i	$0.741537\pi$
$98$	−222.220	−0.229057
$99$	0	0
$100$	0	0
$101$	401.240	0.395296	0.197648	−	0.980273i	$-0.436670\pi$
$101$	401.240	0.395296	0.197648	+	0.980273i	$0.436670\pi$
$102$	0	0
$103$	291.844	0.279187	0.139594	−	0.990209i	$-0.455420\pi$
$103$	291.844	0.279187	0.139594	+	0.990209i	$0.455420\pi$
$104$	−1557.90	−1.46889
$105$	0	0
$106$	2650.16	2.42836
$107$	21.9594	0.0198402	0.00992009	−	0.999951i	$-0.496842\pi$
$107$	21.9594	0.0198402	0.00992009	+	0.999951i	$0.496842\pi$
$108$	0	0
$109$	−17.3019	−0.0152039	−0.00760194	−	0.999971i	$-0.502420\pi$
$109$	−17.3019	−0.0152039	−0.00760194	+	0.999971i	$0.502420\pi$
$110$	0	0
$111$	0	0
$112$	−46.2308	−0.0390036
$113$	−58.8182	−0.0489660	−0.0244830	−	0.999700i	$-0.507794\pi$
$113$	−58.8182	−0.0489660	−0.0244830	+	0.999700i	$0.507794\pi$
$114$	0	0
$115$	0	0
$116$	−2392.84	−1.91526
$117$	0	0
$118$	924.018	0.720871
$119$	498.777	0.384225
$120$	0	0
$121$	1592.39	1.19639
$122$	737.129	0.547020
$123$	0	0
$124$	−2426.77	−1.75750
$125$	0	0
$126$	0	0
$127$	1862.03	1.30101	0.650507	−	0.759501i	$-0.274557\pi$
$127$	1862.03	1.30101	0.650507	+	0.759501i	$0.274557\pi$
$128$	2219.15	1.53240
$129$	0	0
$130$	0	0
$131$	−775.161	−0.516994	−0.258497	−	0.966012i	$-0.583227\pi$
$131$	−775.161	−0.516994	−0.258497	+	0.966012i	$0.583227\pi$
$132$	0	0
$133$	−458.570	−0.298971
$134$	−2165.66	−1.39616
$135$	0	0
$136$	−1475.83	−0.930528
$137$	1027.95	0.641051	0.320525	−	0.947240i	$-0.396141\pi$
$137$	1027.95	0.641051	0.320525	+	0.947240i	$0.396141\pi$
$138$	0	0
$139$	−1029.66	−0.628309	−0.314154	−	0.949372i	$-0.601721\pi$
$139$	−1029.66	−0.628309	−0.314154	+	0.949372i	$0.601721\pi$
$140$	0	0
$141$	0	0
$142$	3729.18	2.20385
$143$	4066.80	2.37820
$144$	0	0
$145$	0	0
$146$	3623.51	2.05400
$147$	0	0
$148$	1441.11	0.800393
$149$	−1414.67	−0.777814	−0.388907	−	0.921277i	$-0.627147\pi$
$149$	−1414.67	−0.777814	−0.388907	+	0.921277i	$0.627147\pi$
$150$	0	0
$151$	2094.96	1.12904	0.564522	−	0.825418i	$-0.309060\pi$
$151$	2094.96	1.12904	0.564522	+	0.825418i	$0.309060\pi$
$152$	1356.87	0.724056
$153$	0	0
$154$	−1716.44	−0.898147
$155$	0	0
$156$	0	0
$157$	−709.495	−0.360662	−0.180331	−	0.983606i	$-0.557717\pi$
$157$	−709.495	−0.360662	−0.180331	+	0.983606i	$0.557717\pi$
$158$	2122.82	1.06888
$159$	0	0
$160$	0	0
$161$	−879.815	−0.430678
$162$	0	0
$163$	3276.23	1.57432	0.787160	−	0.616749i	$-0.211551\pi$
$163$	3276.23	1.57432	0.787160	+	0.616749i	$0.211551\pi$
$164$	−2725.75	−1.29784
$165$	0	0
$166$	−1407.39	−0.658040
$167$	−2860.69	−1.32555	−0.662775	−	0.748819i	$-0.730621\pi$
$167$	−2860.69	−1.32555	−0.662775	+	0.748819i	$0.730621\pi$
$168$	0	0
$169$	3460.42	1.57507
$170$	0	0
$171$	0	0
$172$	−5190.62	−2.30105
$173$	−2803.44	−1.23203	−0.616015	−	0.787734i	$-0.711254\pi$
$173$	−2803.44	−1.23203	−0.616015	+	0.787734i	$0.711254\pi$
$174$	0	0
$175$	0	0
$176$	−357.090	−0.152936
$177$	0	0
$178$	5963.20	2.51102
$179$	−248.617	−0.103813	−0.0519064	−	0.998652i	$-0.516530\pi$
$179$	−248.617	−0.103813	−0.0519064	+	0.998652i	$0.516530\pi$
$180$	0	0
$181$	2075.56	0.852347	0.426174	−	0.904641i	$-0.359861\pi$
$181$	2075.56	0.852347	0.426174	+	0.904641i	$0.359861\pi$
$182$	−2387.78	−0.972494
$183$	0	0
$184$	2603.29	1.04303
$185$	0	0
$186$	0	0
$187$	3852.58	1.50657
$188$	1427.07	0.553614
$189$	0	0
$190$	0	0
$191$	1252.95	0.474663	0.237331	−	0.971429i	$-0.423727\pi$
$191$	1252.95	0.474663	0.237331	+	0.971429i	$0.423727\pi$
$192$	0	0
$193$	1602.12	0.597527	0.298764	−	0.954327i	$-0.403426\pi$
$193$	1602.12	0.597527	0.298764	+	0.954327i	$0.403426\pi$
$194$	5962.10	2.20646
$195$	0	0
$196$	615.789	0.224413
$197$	−2346.38	−0.848593	−0.424296	−	0.905523i	$-0.639479\pi$
$197$	−2346.38	−0.848593	−0.424296	+	0.905523i	$0.639479\pi$
$198$	0	0
$199$	−1993.53	−0.710140	−0.355070	−	0.934840i	$-0.615543\pi$
$199$	−1993.53	−0.710140	−0.355070	+	0.934840i	$0.615543\pi$
$200$	0	0
$201$	0	0
$202$	−1819.67	−0.633818
$203$	−1332.83	−0.460820
$204$	0	0
$205$	0	0
$206$	−1323.54	−0.447649
$207$	0	0
$208$	−496.756	−0.165595
$209$	−3542.02	−1.17228
$210$	0	0
$211$	−5852.55	−1.90951	−0.954754	−	0.297396i	$-0.903882\pi$
$211$	−5852.55	−1.90951	−0.954754	+	0.297396i	$0.903882\pi$
$212$	−7343.80	−2.37912
$213$	0	0
$214$	−99.5883	−0.0318117
$215$	0	0
$216$	0	0
$217$	−1351.73	−0.422864
$218$	78.4659	0.0243779
$219$	0	0
$220$	0	0
$221$	5359.42	1.63128
$222$	0	0
$223$	−1719.58	−0.516374	−0.258187	−	0.966095i	$-0.583125\pi$
$223$	−1719.58	−0.516374	−0.258187	+	0.966095i	$0.583125\pi$
$224$	1369.55	0.408514
$225$	0	0
$226$	266.747	0.0785120
$227$	5686.00	1.66252	0.831262	−	0.555881i	$-0.187619\pi$
$227$	5686.00	1.66252	0.831262	+	0.555881i	$0.187619\pi$
$228$	0	0
$229$	−3087.40	−0.890923	−0.445461	−	0.895301i	$-0.646960\pi$
$229$	−3087.40	−0.890923	−0.445461	+	0.895301i	$0.646960\pi$
$230$	0	0
$231$	0	0
$232$	3943.73	1.11603
$233$	−3997.55	−1.12398	−0.561991	−	0.827143i	$-0.689965\pi$
$233$	−3997.55	−1.12398	−0.561991	+	0.827143i	$0.689965\pi$
$234$	0	0
$235$	0	0
$236$	−2560.53	−0.706255
$237$	0	0
$238$	−2262.00	−0.616067
$239$	4499.79	1.21786	0.608928	−	0.793226i	$-0.291600\pi$
$239$	4499.79	1.21786	0.608928	+	0.793226i	$0.291600\pi$
$240$	0	0
$241$	3633.26	0.971115	0.485557	−	0.874205i	$-0.338617\pi$
$241$	3633.26	0.971115	0.485557	+	0.874205i	$0.338617\pi$
$242$	−7221.66	−1.91829
$243$	0	0
$244$	−2042.64	−0.535929
$245$	0	0
$246$	0	0
$247$	−4927.39	−1.26932
$248$	3999.65	1.02411
$249$	0	0
$250$	0	0
$251$	1211.21	0.304585	0.152292	−	0.988335i	$-0.451334\pi$
$251$	1211.21	0.304585	0.152292	+	0.988335i	$0.451334\pi$
$252$	0	0
$253$	−6795.74	−1.68871
$254$	−8444.50	−2.08604
$255$	0	0
$256$	−3388.39	−0.827245
$257$	6225.81	1.51111	0.755556	−	0.655085i	$-0.227367\pi$
$257$	6225.81	1.51111	0.755556	+	0.655085i	$0.227367\pi$
$258$	0	0
$259$	802.709	0.192579
$260$	0	0
$261$	0	0
$262$	3515.43	0.828947
$263$	757.377	0.177574	0.0887869	−	0.996051i	$-0.471701\pi$
$263$	757.377	0.177574	0.0887869	+	0.996051i	$0.471701\pi$
$264$	0	0
$265$	0	0
$266$	2079.66	0.479369
$267$	0	0
$268$	6001.23	1.36785
$269$	−1750.59	−0.396786	−0.198393	−	0.980123i	$-0.563572\pi$
$269$	−1750.59	−0.396786	−0.198393	+	0.980123i	$0.563572\pi$
$270$	0	0
$271$	4451.86	0.997902	0.498951	−	0.866630i	$-0.333719\pi$
$271$	4451.86	0.997902	0.498951	+	0.866630i	$0.333719\pi$
$272$	−470.589	−0.104903
$273$	0	0
$274$	−4661.87	−1.02786
$275$	0	0
$276$	0	0
$277$	−7691.69	−1.66841	−0.834204	−	0.551456i	$-0.814072\pi$
$277$	−7691.69	−1.66841	−0.834204	+	0.551456i	$0.814072\pi$
$278$	4669.62	1.00743
$279$	0	0
$280$	0	0
$281$	199.034	0.0422540	0.0211270	−	0.999777i	$-0.493275\pi$
$281$	199.034	0.0422540	0.0211270	+	0.999777i	$0.493275\pi$
$282$	0	0
$283$	−799.307	−0.167893	−0.0839467	−	0.996470i	$-0.526753\pi$
$283$	−799.307	−0.167893	−0.0839467	+	0.996470i	$0.526753\pi$
$284$	−10333.9	−2.15916
$285$	0	0
$286$	−18443.3	−3.81321
$287$	−1518.27	−0.312267
$288$	0	0
$289$	164.110	0.0334031
$290$	0	0
$291$	0	0
$292$	−10041.0	−2.01235
$293$	−3302.75	−0.658528	−0.329264	−	0.944238i	$-0.606801\pi$
$293$	−3302.75	−0.658528	−0.329264	+	0.944238i	$0.606801\pi$
$294$	0	0
$295$	0	0
$296$	−2375.14	−0.466393
$297$	0	0
$298$	6415.67	1.24715
$299$	−9453.72	−1.82850
$300$	0	0
$301$	−2891.22	−0.553646
$302$	−9500.87	−1.81031
$303$	0	0
$304$	432.655	0.0816265
$305$	0	0
$306$	0	0
$307$	−8628.17	−1.60402	−0.802012	−	0.597308i	$-0.796237\pi$
$307$	−8628.17	−1.60402	−0.802012	+	0.597308i	$0.796237\pi$
$308$	4756.39	0.879937
$309$	0	0
$310$	0	0
$311$	4900.60	0.893530	0.446765	−	0.894651i	$-0.352576\pi$
$311$	4900.60	0.893530	0.446765	+	0.894651i	$0.352576\pi$
$312$	0	0
$313$	−9114.26	−1.64590	−0.822952	−	0.568110i	$-0.807675\pi$
$313$	−9114.26	−1.64590	−0.822952	+	0.568110i	$0.807675\pi$
$314$	3217.63	0.578285
$315$	0	0
$316$	−5882.51	−1.04721
$317$	8022.82	1.42147	0.710736	−	0.703459i	$-0.248362\pi$
$317$	8022.82	1.42147	0.710736	+	0.703459i	$0.248362\pi$
$318$	0	0
$319$	−10294.9	−1.80691
$320$	0	0
$321$	0	0
$322$	3990.05	0.690549
$323$	−4667.84	−0.804104
$324$	0	0
$325$	0	0
$326$	−14858.0	−2.52426
$327$	0	0
$328$	4492.42	0.756257
$329$	794.888	0.133202
$330$	0	0
$331$	−1333.58	−0.221451	−0.110725	−	0.993851i	$-0.535317\pi$
$331$	−1333.58	−0.221451	−0.110725	+	0.993851i	$0.535317\pi$
$332$	3899.99	0.644698
$333$	0	0
$334$	12973.5	2.12539
$335$	0	0
$336$	0	0
$337$	−1687.33	−0.272744	−0.136372	−	0.990658i	$-0.543544\pi$
$337$	−1687.33	−0.272744	−0.136372	+	0.990658i	$0.543544\pi$
$338$	−15693.4	−2.52546
$339$	0	0
$340$	0	0
$341$	−10440.9	−1.65808
$342$	0	0
$343$	343.000	0.0539949
$344$	8554.87	1.34084
$345$	0	0
$346$	12713.9	1.97544
$347$	−5955.18	−0.921299	−0.460649	−	0.887582i	$-0.652383\pi$
$347$	−5955.18	−0.921299	−0.460649	+	0.887582i	$0.652383\pi$
$348$	0	0
$349$	6876.59	1.05471	0.527357	−	0.849644i	$-0.323183\pi$
$349$	6876.59	1.05471	0.527357	+	0.849644i	$0.323183\pi$
$350$	0	0
$351$	0	0
$352$	10578.5	1.60181
$353$	−1395.95	−0.210479	−0.105239	−	0.994447i	$-0.533561\pi$
$353$	−1395.95	−0.210479	−0.105239	+	0.994447i	$0.533561\pi$
$354$	0	0
$355$	0	0
$356$	−16524.5	−2.46011
$357$	0	0
$358$	1127.50	0.166453
$359$	382.988	0.0563046	0.0281523	−	0.999604i	$-0.491038\pi$
$359$	382.988	0.0563046	0.0281523	+	0.999604i	$0.491038\pi$
$360$	0	0
$361$	−2567.44	−0.374316
$362$	−9412.85	−1.36665
$363$	0	0
$364$	6616.73	0.952777
$365$	0	0
$366$	0	0
$367$	−10178.4	−1.44771	−0.723856	−	0.689951i	$-0.757632\pi$
$367$	−10178.4	−1.44771	−0.723856	+	0.689951i	$0.757632\pi$
$368$	830.094	0.117586
$369$	0	0
$370$	0	0
$371$	−4090.56	−0.572430
$372$	0	0
$373$	−2991.00	−0.415196	−0.207598	−	0.978214i	$-0.566565\pi$
$373$	−2991.00	−0.415196	−0.207598	+	0.978214i	$0.566565\pi$
$374$	−17471.8	−2.41563
$375$	0	0
$376$	−2352.00	−0.322594
$377$	−14321.5	−1.95648
$378$	0	0
$379$	7400.66	1.00302	0.501512	−	0.865150i	$-0.332777\pi$
$379$	7400.66	1.00302	0.501512	+	0.865150i	$0.332777\pi$
$380$	0	0
$381$	0	0
$382$	−5682.27	−0.761074
$383$	10769.6	1.43682	0.718408	−	0.695622i	$-0.244871\pi$
$383$	10769.6	1.43682	0.718408	+	0.695622i	$0.244871\pi$
$384$	0	0
$385$	0	0
$386$	−7265.75	−0.958075
$387$	0	0
$388$	−16521.5	−2.16173
$389$	11193.3	1.45893	0.729466	−	0.684017i	$-0.239769\pi$
$389$	11193.3	1.45893	0.729466	+	0.684017i	$0.239769\pi$
$390$	0	0
$391$	−8955.74	−1.15834
$392$	−1014.91	−0.130766
$393$	0	0
$394$	10641.1	1.36063
$395$	0	0
$396$	0	0
$397$	10371.9	1.31121	0.655605	−	0.755104i	$-0.272414\pi$
$397$	10371.9	1.31121	0.655605	+	0.755104i	$0.272414\pi$
$398$	9040.88	1.13864
$399$	0	0
$400$	0	0
$401$	−5402.20	−0.672751	−0.336375	−	0.941728i	$-0.609201\pi$
$401$	−5402.20	−0.672751	−0.336375	+	0.941728i	$0.609201\pi$
$402$	0	0
$403$	−14524.5	−1.79533
$404$	5042.44	0.620967
$405$	0	0
$406$	6044.53	0.738879
$407$	6200.17	0.755113
$408$	0	0
$409$	−12829.9	−1.55110	−0.775549	−	0.631288i	$-0.782527\pi$
$409$	−12829.9	−1.55110	−0.775549	+	0.631288i	$0.782527\pi$
$410$	0	0
$411$	0	0
$412$	3667.64	0.438572
$413$	−1426.24	−0.169929
$414$	0	0
$415$	0	0
$416$	14716.0	1.73440
$417$	0	0
$418$	16063.4	1.87964
$419$	−5620.85	−0.655362	−0.327681	−	0.944788i	$-0.606267\pi$
$419$	−5620.85	−0.655362	−0.327681	+	0.944788i	$0.606267\pi$
$420$	0	0
$421$	−2177.11	−0.252033	−0.126017	−	0.992028i	$-0.540219\pi$
$421$	−2177.11	−0.252033	−0.126017	+	0.992028i	$0.540219\pi$
$422$	26541.9	3.06171
$423$	0	0
$424$	12103.6	1.38633
$425$	0	0
$426$	0	0
$427$	−1137.77	−0.128947
$428$	275.967	0.0311668
$429$	0	0
$430$	0	0
$431$	−10396.4	−1.16190	−0.580950	−	0.813939i	$-0.697319\pi$
$431$	−10396.4	−1.16190	−0.580950	+	0.813939i	$0.697319\pi$
$432$	0	0
$433$	11875.6	1.31803	0.659015	−	0.752129i	$-0.270973\pi$
$433$	11875.6	1.31803	0.659015	+	0.752129i	$0.270973\pi$
$434$	6130.24	0.678021
$435$	0	0
$436$	−217.435	−0.0238836
$437$	8233.81	0.901320
$438$	0	0
$439$	640.369	0.0696200	0.0348100	−	0.999394i	$-0.488917\pi$
$439$	640.369	0.0696200	0.0348100	+	0.999394i	$0.488917\pi$
$440$	0	0
$441$	0	0
$442$	−24305.5	−2.61560
$443$	854.852	0.0916823	0.0458411	−	0.998949i	$-0.485403\pi$
$443$	854.852	0.0916823	0.0458411	+	0.998949i	$0.485403\pi$
$444$	0	0
$445$	0	0
$446$	7798.46	0.827954
$447$	0	0
$448$	−5841.21	−0.616008
$449$	−3427.44	−0.360247	−0.180123	−	0.983644i	$-0.557650\pi$
$449$	−3427.44	−0.360247	−0.180123	+	0.983644i	$0.557650\pi$
$450$	0	0
$451$	−11727.2	−1.22442
$452$	−739.176	−0.0769202
$453$	0	0
$454$	−25786.6	−2.66569
$455$	0	0
$456$	0	0
$457$	1709.31	0.174963	0.0874814	−	0.996166i	$-0.472118\pi$
$457$	1709.31	0.174963	0.0874814	+	0.996166i	$0.472118\pi$
$458$	14001.7	1.42851
$459$	0	0
$460$	0	0
$461$	−2251.18	−0.227436	−0.113718	−	0.993513i	$-0.536276\pi$
$461$	−2251.18	−0.227436	−0.113718	+	0.993513i	$0.536276\pi$
$462$	0	0
$463$	−3200.80	−0.321282	−0.160641	−	0.987013i	$-0.551356\pi$
$463$	−3200.80	−0.321282	−0.160641	+	0.987013i	$0.551356\pi$
$464$	1257.51	0.125816
$465$	0	0
$466$	18129.3	1.80219
$467$	−5477.39	−0.542748	−0.271374	−	0.962474i	$-0.587478\pi$
$467$	−5477.39	−0.542748	−0.271374	+	0.962474i	$0.587478\pi$
$468$	0	0
$469$	3342.74	0.329112
$470$	0	0
$471$	0	0
$472$	4220.11	0.411538
$473$	−22332.0	−2.17088
$474$	0	0
$475$	0	0
$476$	6268.19	0.603576
$477$	0	0
$478$	−20407.0	−1.95271
$479$	14182.0	1.35280	0.676399	−	0.736535i	$-0.263539\pi$
$479$	14182.0	1.35280	0.676399	+	0.736535i	$0.263539\pi$
$480$	0	0
$481$	8625.20	0.817620
$482$	−16477.2	−1.55709
$483$	0	0
$484$	20011.8	1.87940
$485$	0	0
$486$	0	0
$487$	−5320.83	−0.495092	−0.247546	−	0.968876i	$-0.579624\pi$
$487$	−5320.83	−0.495092	−0.247546	+	0.968876i	$0.579624\pi$
$488$	3366.56	0.312289
$489$	0	0
$490$	0	0
$491$	6574.80	0.604311	0.302155	−	0.953259i	$-0.402294\pi$
$491$	6574.80	0.604311	0.302155	+	0.953259i	$0.402294\pi$
$492$	0	0
$493$	−13567.1	−1.23941
$494$	22346.2	2.03523
$495$	0	0
$496$	1275.34	0.115453
$497$	−5756.06	−0.519506
$498$	0	0
$499$	−1507.68	−0.135256	−0.0676282	−	0.997711i	$-0.521543\pi$
$499$	−1507.68	−0.135256	−0.0676282	+	0.997711i	$0.521543\pi$
$500$	0	0
$501$	0	0
$502$	−5492.95	−0.488371
$503$	8782.24	0.778491	0.389245	−	0.921134i	$-0.372736\pi$
$503$	8782.24	0.778491	0.389245	+	0.921134i	$0.372736\pi$
$504$	0	0
$505$	0	0
$506$	30819.4	2.70768
$507$	0	0
$508$	23400.4	2.04375
$509$	−12696.6	−1.10563	−0.552815	−	0.833304i	$-0.686447\pi$
$509$	−12696.6	−1.10563	−0.552815	+	0.833304i	$0.686447\pi$
$510$	0	0
$511$	−5592.95	−0.484183
$512$	−2386.50	−0.205995
$513$	0	0
$514$	−28234.7	−2.42291
$515$	0	0
$516$	0	0
$517$	6139.76	0.522295
$518$	−3640.36	−0.308781
$519$	0	0
$520$	0	0
$521$	10434.7	0.877455	0.438727	−	0.898620i	$-0.355429\pi$
$521$	10434.7	0.877455	0.438727	+	0.898620i	$0.355429\pi$
$522$	0	0
$523$	−8403.03	−0.702560	−0.351280	−	0.936270i	$-0.614254\pi$
$523$	−8403.03	−0.702560	−0.351280	+	0.936270i	$0.614254\pi$
$524$	−9741.55	−0.812140
$525$	0	0
$526$	−3434.78	−0.284722
$527$	−13759.4	−1.13733
$528$	0	0
$529$	3630.43	0.298384
$530$	0	0
$531$	0	0
$532$	−5762.91	−0.469650
$533$	−16314.0	−1.32577
$534$	0	0
$535$	0	0
$536$	−9890.85	−0.797052
$537$	0	0
$538$	7939.11	0.636207
$539$	2649.35	0.211717
$540$	0	0
$541$	8349.66	0.663549	0.331775	−	0.943359i	$-0.392353\pi$
$541$	8349.66	0.663549	0.331775	+	0.943359i	$0.392353\pi$
$542$	−20189.6	−1.60004
$543$	0	0
$544$	13940.8	1.09873
$545$	0	0
$546$	0	0
$547$	12185.7	0.952509	0.476254	−	0.879308i	$-0.341994\pi$
$547$	12185.7	0.952509	0.476254	+	0.879308i	$0.341994\pi$
$548$	12918.4	1.00702
$549$	0	0
$550$	0	0
$551$	12473.4	0.964402
$552$	0	0
$553$	−3276.61	−0.251963
$554$	34882.6	2.67512
$555$	0	0
$556$	−12939.9	−0.987004
$557$	540.209	0.0410940	0.0205470	−	0.999789i	$-0.493459\pi$
$557$	540.209	0.0410940	0.0205470	+	0.999789i	$0.493459\pi$
$558$	0	0
$559$	−31066.6	−2.35058
$560$	0	0
$561$	0	0
$562$	−902.639	−0.0677501
$563$	1949.26	0.145917	0.0729587	−	0.997335i	$-0.476756\pi$
$563$	1949.26	0.145917	0.0729587	+	0.997335i	$0.476756\pi$
$564$	0	0
$565$	0	0
$566$	3624.94	0.269200
$567$	0	0
$568$	17031.6	1.25815
$569$	−9487.28	−0.698994	−0.349497	−	0.936938i	$-0.613647\pi$
$569$	−9487.28	−0.698994	−0.349497	+	0.936938i	$0.613647\pi$
$570$	0	0
$571$	−20172.3	−1.47843	−0.739215	−	0.673470i	$-0.764803\pi$
$571$	−20172.3	−1.47843	−0.739215	+	0.673470i	$0.764803\pi$
$572$	51108.0	3.73590
$573$	0	0
$574$	6885.50	0.500689
$575$	0	0
$576$	0	0
$577$	12937.3	0.933423	0.466712	−	0.884410i	$-0.345439\pi$
$577$	12937.3	0.933423	0.466712	+	0.884410i	$0.345439\pi$
$578$	−744.254	−0.0535586
$579$	0	0
$580$	0	0
$581$	2172.33	0.155118
$582$	0	0
$583$	−31595.7	−2.24453
$584$	16549.0	1.17261
$585$	0	0
$586$	14978.3	1.05588
$587$	6489.43	0.456299	0.228149	−	0.973626i	$-0.426733\pi$
$587$	6489.43	0.456299	0.228149	+	0.973626i	$0.426733\pi$
$588$	0	0
$589$	12650.3	0.884968
$590$	0	0
$591$	0	0
$592$	−757.345	−0.0525789
$593$	16803.6	1.16364	0.581821	−	0.813317i	$-0.302340\pi$
$593$	16803.6	1.16364	0.581821	+	0.813317i	$0.302340\pi$
$594$	0	0
$595$	0	0
$596$	−17778.3	−1.22186
$597$	0	0
$598$	42873.5	2.93182
$599$	−19139.6	−1.30554	−0.652772	−	0.757554i	$-0.726394\pi$
$599$	−19139.6	−1.30554	−0.652772	+	0.757554i	$0.726394\pi$
$600$	0	0
$601$	14304.0	0.970838	0.485419	−	0.874282i	$-0.338667\pi$
$601$	14304.0	0.970838	0.485419	+	0.874282i	$0.338667\pi$
$602$	13112.0	0.887715
$603$	0	0
$604$	26327.7	1.77361
$605$	0	0
$606$	0	0
$607$	−2018.10	−0.134946	−0.0674730	−	0.997721i	$-0.521494\pi$
$607$	−2018.10	−0.134946	−0.0674730	+	0.997721i	$0.521494\pi$
$608$	−12817.1	−0.854935
$609$	0	0
$610$	0	0
$611$	8541.17	0.565530
$612$	0	0
$613$	21479.4	1.41524	0.707621	−	0.706592i	$-0.249768\pi$
$613$	21479.4	1.41524	0.707621	+	0.706592i	$0.249768\pi$
$614$	39129.6	2.57189
$615$	0	0
$616$	−7839.19	−0.512743
$617$	5856.83	0.382151	0.191075	−	0.981575i	$-0.438803\pi$
$617$	5856.83	0.382151	0.191075	+	0.981575i	$0.438803\pi$
$618$	0	0
$619$	−17619.3	−1.14407	−0.572035	−	0.820229i	$-0.693846\pi$
$619$	−17619.3	−1.14407	−0.572035	+	0.820229i	$0.693846\pi$
$620$	0	0
$621$	0	0
$622$	−22224.7	−1.43269
$623$	−9204.30	−0.591914
$624$	0	0
$625$	0	0
$626$	41334.1	2.63904
$627$	0	0
$628$	−8916.31	−0.566560
$629$	8170.87	0.517955
$630$	0	0
$631$	26395.7	1.66529	0.832643	−	0.553810i	$-0.186827\pi$
$631$	26395.7	1.66529	0.832643	+	0.553810i	$0.186827\pi$
$632$	9695.19	0.610212
$633$	0	0
$634$	−36384.3	−2.27919
$635$	0	0
$636$	0	0
$637$	3685.58	0.229243
$638$	46688.3	2.89719
$639$	0	0
$640$	0	0
$641$	−3209.30	−0.197753	−0.0988764	−	0.995100i	$-0.531525\pi$
$641$	−3209.30	−0.197753	−0.0988764	+	0.995100i	$0.531525\pi$
$642$	0	0
$643$	−1762.48	−0.108096	−0.0540478	−	0.998538i	$-0.517212\pi$
$643$	−1762.48	−0.108096	−0.0540478	+	0.998538i	$0.517212\pi$
$644$	−11056.7	−0.676548
$645$	0	0
$646$	21169.1	1.28930
$647$	24372.7	1.48098	0.740488	−	0.672070i	$-0.234595\pi$
$647$	24372.7	1.48098	0.740488	+	0.672070i	$0.234595\pi$
$648$	0	0
$649$	−11016.3	−0.666301
$650$	0	0
$651$	0	0
$652$	41172.8	2.47308
$653$	−6389.97	−0.382938	−0.191469	−	0.981499i	$-0.561325\pi$
$653$	−6389.97	−0.382938	−0.191469	+	0.981499i	$0.561325\pi$
$654$	0	0
$655$	0	0
$656$	1432.47	0.0852567
$657$	0	0
$658$	−3604.90	−0.213577
$659$	6111.68	0.361270	0.180635	−	0.983550i	$-0.442185\pi$
$659$	6111.68	0.361270	0.180635	+	0.983550i	$0.442185\pi$
$660$	0	0
$661$	16391.1	0.964510	0.482255	−	0.876031i	$-0.339818\pi$
$661$	16391.1	0.964510	0.482255	+	0.876031i	$0.339818\pi$
$662$	6047.92	0.355074
$663$	0	0
$664$	−6427.72	−0.375669
$665$	0	0
$666$	0	0
$667$	23931.6	1.38926
$668$	−35950.6	−2.08229
$669$	0	0
$670$	0	0
$671$	−8788.20	−0.505611
$672$	0	0
$673$	−13055.9	−0.747799	−0.373900	−	0.927469i	$-0.621980\pi$
$673$	−13055.9	−0.747799	−0.373900	+	0.927469i	$0.621980\pi$
$674$	7652.20	0.437317
$675$	0	0
$676$	43487.6	2.47426
$677$	3154.53	0.179082	0.0895410	−	0.995983i	$-0.471460\pi$
$677$	3154.53	0.179082	0.0895410	+	0.995983i	$0.471460\pi$
$678$	0	0
$679$	−9202.60	−0.520123
$680$	0	0
$681$	0	0
$682$	47350.4	2.65856
$683$	−17282.7	−0.968233	−0.484117	−	0.875004i	$-0.660859\pi$
$683$	−17282.7	−0.968233	−0.484117	+	0.875004i	$0.660859\pi$
$684$	0	0
$685$	0	0
$686$	−1555.54	−0.0865754
$687$	0	0
$688$	2727.83	0.151159
$689$	−43953.6	−2.43033
$690$	0	0
$691$	26838.3	1.47753	0.738766	−	0.673961i	$-0.235409\pi$
$691$	26838.3	1.47753	0.738766	+	0.673961i	$0.235409\pi$
$692$	−35231.1	−1.93538
$693$	0	0
$694$	27007.3	1.47721
$695$	0	0
$696$	0	0
$697$	−15454.6	−0.839866
$698$	−31186.0	−1.69113
$699$	0	0
$700$	0	0
$701$	8374.52	0.451214	0.225607	−	0.974218i	$-0.427563\pi$
$701$	8374.52	0.451214	0.225607	+	0.974218i	$0.427563\pi$
$702$	0	0
$703$	−7512.21	−0.403028
$704$	−45117.9	−2.41540
$705$	0	0
$706$	6330.79	0.337482
$707$	2808.68	0.149408
$708$	0	0
$709$	11929.4	0.631902	0.315951	−	0.948775i	$-0.397676\pi$
$709$	11929.4	0.631902	0.315951	+	0.948775i	$0.397676\pi$
$710$	0	0
$711$	0	0
$712$	27234.7	1.43352
$713$	24270.9	1.27483
$714$	0	0
$715$	0	0
$716$	−3124.40	−0.163078
$717$	0	0
$718$	−1736.89	−0.0902788
$719$	23392.3	1.21333	0.606666	−	0.794957i	$-0.292507\pi$
$719$	23392.3	1.21333	0.606666	+	0.794957i	$0.292507\pi$
$720$	0	0
$721$	2042.91	0.105523
$722$	11643.6	0.600179
$723$	0	0
$724$	26083.8	1.33894
$725$	0	0
$726$	0	0
$727$	10659.8	0.543810	0.271905	−	0.962324i	$-0.412346\pi$
$727$	10659.8	0.543810	0.271905	+	0.962324i	$0.412346\pi$
$728$	−10905.3	−0.555188
$729$	0	0
$730$	0	0
$731$	−29430.1	−1.48907
$732$	0	0
$733$	23919.1	1.20528	0.602642	−	0.798011i	$-0.294115\pi$
$733$	23919.1	1.20528	0.602642	+	0.798011i	$0.294115\pi$
$734$	46160.2	2.32126
$735$	0	0
$736$	−24590.9	−1.23157
$737$	25819.5	1.29047
$738$	0	0
$739$	34535.1	1.71907	0.859536	−	0.511074i	$-0.170752\pi$
$739$	34535.1	1.71907	0.859536	+	0.511074i	$0.170752\pi$
$740$	0	0
$741$	0	0
$742$	18551.1	0.917833
$743$	−17526.8	−0.865403	−0.432702	−	0.901537i	$-0.642440\pi$
$743$	−17526.8	−0.865403	−0.432702	+	0.901537i	$0.642440\pi$
$744$	0	0
$745$	0	0
$746$	13564.5	0.665725
$747$	0	0
$748$	48415.9	2.36666
$749$	153.716	0.00749889
$750$	0	0
$751$	18534.3	0.900565	0.450283	−	0.892886i	$-0.351323\pi$
$751$	18534.3	0.900565	0.450283	+	0.892886i	$0.351323\pi$
$752$	−749.966	−0.0363676
$753$	0	0
$754$	64949.2	3.13702
$755$	0	0
$756$	0	0
$757$	−9309.64	−0.446981	−0.223491	−	0.974706i	$-0.571745\pi$
$757$	−9309.64	−0.446981	−0.223491	+	0.974706i	$0.571745\pi$
$758$	−33562.7	−1.60825
$759$	0	0
$760$	0	0
$761$	2968.41	0.141399	0.0706996	−	0.997498i	$-0.477477\pi$
$761$	2968.41	0.141399	0.0706996	+	0.997498i	$0.477477\pi$
$762$	0	0
$763$	−121.113	−0.00574653
$764$	15746.0	0.745643
$765$	0	0
$766$	−48841.1	−2.30379
$767$	−15325.1	−0.721457
$768$	0	0
$769$	−34932.9	−1.63812	−0.819060	−	0.573708i	$-0.805505\pi$
$769$	−34932.9	−1.63812	−0.819060	+	0.573708i	$0.805505\pi$
$770$	0	0
$771$	0	0
$772$	20134.0	0.938650
$773$	13595.4	0.632591	0.316295	−	0.948661i	$-0.397561\pi$
$773$	13595.4	0.632591	0.316295	+	0.948661i	$0.397561\pi$
$774$	0	0
$775$	0	0
$776$	27229.7	1.25965
$777$	0	0
$778$	−50762.9	−2.33925
$779$	14208.8	0.653510
$780$	0	0
$781$	−44460.1	−2.03701
$782$	40615.2	1.85728
$783$	0	0
$784$	−323.616	−0.0147420
$785$	0	0
$786$	0	0
$787$	−14228.1	−0.644444	−0.322222	−	0.946664i	$-0.604430\pi$
$787$	−14228.1	−0.644444	−0.322222	+	0.946664i	$0.604430\pi$
$788$	−29487.3	−1.33305
$789$	0	0
$790$	0	0
$791$	−411.728	−0.0185074
$792$	0	0
$793$	−12225.5	−0.547465
$794$	−47037.5	−2.10239
$795$	0	0
$796$	−25053.0	−1.11555
$797$	−11442.3	−0.508539	−0.254270	−	0.967133i	$-0.581835\pi$
$797$	−11442.3	−0.508539	−0.254270	+	0.967133i	$0.581835\pi$
$798$	0	0
$799$	8091.26	0.358258
$800$	0	0
$801$	0	0
$802$	24499.5	1.07869
$803$	−43200.3	−1.89851
$804$	0	0
$805$	0	0
$806$	65870.2	2.87863
$807$	0	0
$808$	−8310.63	−0.361840
$809$	−22732.9	−0.987944	−0.493972	−	0.869478i	$-0.664455\pi$
$809$	−22732.9	−0.987944	−0.493972	+	0.869478i	$0.664455\pi$
$810$	0	0
$811$	−29768.2	−1.28890	−0.644452	−	0.764644i	$-0.722915\pi$
$811$	−29768.2	−1.28890	−0.644452	+	0.764644i	$0.722915\pi$
$812$	−16749.9	−0.723898
$813$	0	0
$814$	−28118.4	−1.21075
$815$	0	0
$816$	0	0
$817$	27057.7	1.15867
$818$	58185.0	2.48703
$819$	0	0
$820$	0	0
$821$	−6408.35	−0.272416	−0.136208	−	0.990680i	$-0.543491\pi$
$821$	−6408.35	−0.272416	−0.136208	+	0.990680i	$0.543491\pi$
$822$	0	0
$823$	−20876.0	−0.884194	−0.442097	−	0.896967i	$-0.645765\pi$
$823$	−20876.0	−0.884194	−0.442097	+	0.896967i	$0.645765\pi$
$824$	−6044.78	−0.255558
$825$	0	0
$826$	6468.13	0.272464
$827$	41650.1	1.75129	0.875645	−	0.482955i	$-0.160437\pi$
$827$	41650.1	1.75129	0.875645	+	0.482955i	$0.160437\pi$
$828$	0	0
$829$	17194.7	0.720380	0.360190	−	0.932879i	$-0.382712\pi$
$829$	17194.7	0.720380	0.360190	+	0.932879i	$0.382712\pi$
$830$	0	0
$831$	0	0
$832$	−62764.5	−2.61535
$833$	3491.44	0.145223
$834$	0	0
$835$	0	0
$836$	−44513.0	−1.84153
$837$	0	0
$838$	25491.1	1.05081
$839$	−40583.7	−1.66997	−0.834985	−	0.550273i	$-0.814524\pi$
$839$	−40583.7	−1.66997	−0.834985	+	0.550273i	$0.814524\pi$
$840$	0	0
$841$	11865.0	0.486489
$842$	9873.42	0.404110
$843$	0	0
$844$	−73549.7	−2.99963
$845$	0	0
$846$	0	0
$847$	11146.8	0.452192
$848$	3859.39	0.156288
$849$	0	0
$850$	0	0
$851$	−14413.0	−0.580576
$852$	0	0
$853$	23020.1	0.924023	0.462012	−	0.886874i	$-0.347128\pi$
$853$	23020.1	0.924023	0.462012	+	0.886874i	$0.347128\pi$
$854$	5159.90	0.206754
$855$	0	0
$856$	−454.832	−0.0181610
$857$	18075.5	0.720476	0.360238	−	0.932861i	$-0.382696\pi$
$857$	18075.5	0.720476	0.360238	+	0.932861i	$0.382696\pi$
$858$	0	0
$859$	20308.2	0.806643	0.403321	−	0.915058i	$-0.367856\pi$
$859$	20308.2	0.806643	0.403321	+	0.915058i	$0.367856\pi$
$860$	0	0
$861$	0	0
$862$	47148.9	1.86299
$863$	31023.5	1.22370	0.611849	−	0.790975i	$-0.290426\pi$
$863$	31023.5	1.22370	0.611849	+	0.790975i	$0.290426\pi$
$864$	0	0
$865$	0	0
$866$	−53857.2	−2.11333
$867$	0	0
$868$	−16987.4	−0.664274
$869$	−25308.7	−0.987963
$870$	0	0
$871$	35918.1	1.39729
$872$	358.363	0.0139171
$873$	0	0
$874$	−37341.2	−1.44518
$875$	0	0
$876$	0	0
$877$	−28344.1	−1.09135	−0.545674	−	0.837997i	$-0.683726\pi$
$877$	−28344.1	−1.09135	−0.545674	+	0.837997i	$0.683726\pi$
$878$	−2904.14	−0.111629
$879$	0	0
$880$	0	0
$881$	−41264.5	−1.57802	−0.789010	−	0.614380i	$-0.789406\pi$
$881$	−41264.5	−1.57802	−0.789010	+	0.614380i	$0.789406\pi$
$882$	0	0
$883$	−7995.59	−0.304726	−0.152363	−	0.988325i	$-0.548688\pi$
$883$	−7995.59	−0.304726	−0.152363	+	0.988325i	$0.548688\pi$
$884$	67352.5	2.56257
$885$	0	0
$886$	−3876.84	−0.147003
$887$	−24438.3	−0.925093	−0.462547	−	0.886595i	$-0.653064\pi$
$887$	−24438.3	−0.925093	−0.462547	+	0.886595i	$0.653064\pi$
$888$	0	0
$889$	13034.2	0.491737
$890$	0	0
$891$	0	0
$892$	−21610.1	−0.811167
$893$	−7439.02	−0.278765
$894$	0	0
$895$	0	0
$896$	15534.1	0.579192
$897$	0	0
$898$	15543.8	0.577620
$899$	36768.0	1.36405
$900$	0	0
$901$	−41638.3	−1.53959
$902$	53184.0	1.96323
$903$	0	0
$904$	1218.26	0.0448218
$905$	0	0
$906$	0	0
$907$	17241.6	0.631201	0.315601	−	0.948892i	$-0.397794\pi$
$907$	17241.6	0.631201	0.315601	+	0.948892i	$0.397794\pi$
$908$	71456.6	2.61164
$909$	0	0
$910$	0	0
$911$	−17407.8	−0.633092	−0.316546	−	0.948577i	$-0.602523\pi$
$911$	−17407.8	−0.633092	−0.316546	+	0.948577i	$0.602523\pi$
$912$	0	0
$913$	16779.2	0.608226
$914$	−7751.87	−0.280535
$915$	0	0
$916$	−38799.8	−1.39954
$917$	−5426.13	−0.195405
$918$	0	0
$919$	7704.64	0.276553	0.138277	−	0.990394i	$-0.455844\pi$
$919$	7704.64	0.276553	0.138277	+	0.990394i	$0.455844\pi$
$920$	0	0
$921$	0	0
$922$	10209.3	0.364671
$923$	−61849.5	−2.20564
$924$	0	0
$925$	0	0
$926$	14515.9	0.515144
$927$	0	0
$928$	−37252.8	−1.31776
$929$	48184.3	1.70170	0.850849	−	0.525411i	$-0.176088\pi$
$929$	48184.3	1.70170	0.850849	+	0.525411i	$0.176088\pi$
$930$	0	0
$931$	−3209.99	−0.113000
$932$	−50237.7	−1.76565
$933$	0	0
$934$	24840.5	0.870242
$935$	0	0
$936$	0	0
$937$	23371.6	0.814853	0.407427	−	0.913238i	$-0.366426\pi$
$937$	23371.6	0.814853	0.407427	+	0.913238i	$0.366426\pi$
$938$	−15159.7	−0.527697
$939$	0	0
$940$	0	0
$941$	−18048.6	−0.625258	−0.312629	−	0.949875i	$-0.601210\pi$
$941$	−18048.6	−0.625258	−0.312629	+	0.949875i	$0.601210\pi$
$942$	0	0
$943$	27261.1	0.941405
$944$	1345.64	0.0463948
$945$	0	0
$946$	101278.	3.48079
$947$	−10346.2	−0.355022	−0.177511	−	0.984119i	$-0.556805\pi$
$947$	−10346.2	−0.355022	−0.177511	+	0.984119i	$0.556805\pi$
$948$	0	0
$949$	−60096.9	−2.05567
$950$	0	0
$951$	0	0
$952$	−10330.8	−0.351706
$953$	−8691.77	−0.295440	−0.147720	−	0.989029i	$-0.547193\pi$
$953$	−8691.77	−0.295440	−0.147720	+	0.989029i	$0.547193\pi$
$954$	0	0
$955$	0	0
$956$	56549.5	1.91312
$957$	0	0
$958$	−64316.6	−2.16908
$959$	7195.67	0.242294
$960$	0	0
$961$	7498.41	0.251701
$962$	−39116.2	−1.31097
$963$	0	0
$964$	45659.6	1.52552
$965$	0	0
$966$	0	0
$967$	8971.58	0.298352	0.149176	−	0.988811i	$-0.452338\pi$
$967$	8971.58	0.298352	0.149176	+	0.988811i	$0.452338\pi$
$968$	−32982.2	−1.09513
$969$	0	0
$970$	0	0
$971$	−29275.1	−0.967541	−0.483771	−	0.875195i	$-0.660733\pi$
$971$	−29275.1	−0.967541	−0.483771	+	0.875195i	$0.660733\pi$
$972$	0	0
$973$	−7207.64	−0.237478
$974$	24130.5	0.793831
$975$	0	0
$976$	1073.47	0.0352059
$977$	16477.5	0.539572	0.269786	−	0.962920i	$-0.413047\pi$
$977$	16477.5	0.539572	0.269786	+	0.962920i	$0.413047\pi$
$978$	0	0
$979$	−71094.6	−2.32093
$980$	0	0
$981$	0	0
$982$	−29817.4	−0.968952
$983$	−45912.9	−1.48972	−0.744859	−	0.667222i	$-0.767483\pi$
$983$	−45912.9	−1.48972	−0.744859	+	0.667222i	$0.767483\pi$
$984$	0	0
$985$	0	0
$986$	61528.0	1.98727
$987$	0	0
$988$	−61923.2	−1.99397
$989$	51913.1	1.66910
$990$	0	0
$991$	46124.8	1.47851	0.739255	−	0.673426i	$-0.235178\pi$
$991$	46124.8	1.47851	0.739255	+	0.673426i	$0.235178\pi$
$992$	−37781.0	−1.20922
$993$	0	0
$994$	26104.3	0.832975
$995$	0	0
$996$	0	0
$997$	−30984.5	−0.984240	−0.492120	−	0.870527i	$-0.663778\pi$
$997$	−30984.5	−0.984240	−0.492120	+	0.870527i	$0.663778\pi$
$998$	6837.47	0.216870
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bg.1.2		4
3.2	odd	2		175.4.a.h.1.3	yes	4
5.4	even	2		1575.4.a.bl.1.3		4
15.2	even	4		175.4.b.f.99.6		8
15.8	even	4		175.4.b.f.99.3		8
15.14	odd	2		175.4.a.g.1.2	✓	4
21.20	even	2		1225.4.a.bd.1.3		4
105.104	even	2		1225.4.a.z.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.g.1.2	✓	4	15.14	odd	2
175.4.a.h.1.3	yes	4	3.2	odd	2
175.4.b.f.99.3		8	15.8	even	4
175.4.b.f.99.6		8	15.2	even	4
1225.4.a.z.1.2		4	105.104	even	2
1225.4.a.bd.1.3		4	21.20	even	2
1575.4.a.bg.1.2		4	1.1	even	1	trivial
1575.4.a.bl.1.3		4	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)

\(f(q)\)	\(=\)	\(q-4.53510 q^{2} +12.5671 q^{4} +7.00000 q^{7} -20.7124 q^{8} +O(q^{10})\)	\(q-4.53510 q^{2} +12.5671 q^{4} +7.00000 q^{7} -20.7124 q^{8} +54.0684 q^{11} +75.2159 q^{13} -31.7457 q^{14} -6.60441 q^{16} +71.2538 q^{17} -65.5100 q^{19} -245.206 q^{22} -125.688 q^{23} -341.111 q^{26} +87.9699 q^{28} -190.405 q^{29} -193.105 q^{31} +195.650 q^{32} -323.143 q^{34} +114.673 q^{37} +297.094 q^{38} -216.896 q^{41} -413.032 q^{43} +679.484 q^{44} +570.007 q^{46} +113.555 q^{47} +49.0000 q^{49} +945.247 q^{52} -584.366 q^{53} -144.986 q^{56} +863.504 q^{58} -203.748 q^{59} -162.539 q^{61} +875.749 q^{62} -834.459 q^{64} +477.534 q^{67} +895.456 q^{68} -822.294 q^{71} -798.993 q^{73} -520.052 q^{74} -823.273 q^{76} +378.479 q^{77} -468.087 q^{79} +983.643 q^{82} +310.333 q^{83} +1873.14 q^{86} -1119.88 q^{88} -1314.90 q^{89} +526.511 q^{91} -1579.53 q^{92} -514.985 q^{94} -1314.66 q^{97} -222.220 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 36 q^{4} + 28 q^{7} - 27 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 36 * q^4 + 28 * q^7 - 27 * q^8	\( 4 q - 4 q^{2} + 36 q^{4} + 28 q^{7} - 27 q^{8} - 100 q^{11} + 44 q^{13} - 28 q^{14} + 160 q^{16} + 53 q^{17} - 29 q^{19} - 152 q^{22} - 295 q^{23} - 700 q^{26} + 252 q^{28} - 129 q^{29} + 114 q^{31} + 310 q^{32} + 203 q^{34} + 403 q^{37} - 555 q^{38} - 671 q^{41} - 411 q^{43} - 438 q^{44} - 997 q^{46} + 8 q^{47} + 196 q^{49} + 74 q^{52} - 90 q^{53} - 189 q^{56} + 673 q^{58} - 1018 q^{59} + 50 q^{61} + 1626 q^{62} - 2421 q^{64} + 424 q^{67} + 617 q^{68} - 215 q^{71} - 1207 q^{73} - 623 q^{74} - 3257 q^{76} - 700 q^{77} - 951 q^{79} + 1695 q^{82} + 3035 q^{83} + 99 q^{86} + 163 q^{88} - 2819 q^{89} + 308 q^{91} - 3073 q^{92} - 3056 q^{94} - 1100 q^{97} - 196 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 36 * q^4 + 28 * q^7 - 27 * q^8 - 100 * q^11 + 44 * q^13 - 28 * q^14 + 160 * q^16 + 53 * q^17 - 29 * q^19 - 152 * q^22 - 295 * q^23 - 700 * q^26 + 252 * q^28 - 129 * q^29 + 114 * q^31 + 310 * q^32 + 203 * q^34 + 403 * q^37 - 555 * q^38 - 671 * q^41 - 411 * q^43 - 438 * q^44 - 997 * q^46 + 8 * q^47 + 196 * q^49 + 74 * q^52 - 90 * q^53 - 189 * q^56 + 673 * q^58 - 1018 * q^59 + 50 * q^61 + 1626 * q^62 - 2421 * q^64 + 424 * q^67 + 617 * q^68 - 215 * q^71 - 1207 * q^73 - 623 * q^74 - 3257 * q^76 - 700 * q^77 - 951 * q^79 + 1695 * q^82 + 3035 * q^83 + 99 * q^86 + 163 * q^88 - 2819 * q^89 + 308 * q^91 - 3073 * q^92 - 3056 * q^94 - 1100 * q^97 - 196 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more