LMFDB - Embedded newform 1575.4.a.bg.1.4

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			1.4
Root			$5.87199$ 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)

$f(q)$	$=$	$q+4.87199 q^{2} +15.7363 q^{4} +7.00000 q^{7} +37.6910 q^{8} +O(q^{10})$	$q+4.87199 q^{2} +15.7363 q^{4} +7.00000 q^{7} +37.6910 q^{8} -36.9922 q^{11} -61.3165 q^{13} +34.1039 q^{14} +57.7401 q^{16} +44.8345 q^{17} -139.701 q^{19} -180.226 q^{22} -217.580 q^{23} -298.733 q^{26} +110.154 q^{28} +33.8226 q^{29} +124.437 q^{31} -20.2192 q^{32} +218.433 q^{34} +237.270 q^{37} -680.620 q^{38} -195.117 q^{41} -343.725 q^{43} -582.119 q^{44} -1060.05 q^{46} -16.8224 q^{47} +49.0000 q^{49} -964.894 q^{52} -346.965 q^{53} +263.837 q^{56} +164.783 q^{58} -135.340 q^{59} +490.414 q^{61} +606.255 q^{62} -560.428 q^{64} +477.969 q^{67} +705.529 q^{68} -45.2557 q^{71} +100.781 q^{73} +1155.98 q^{74} -2198.37 q^{76} -258.945 q^{77} +880.534 q^{79} -950.609 q^{82} +1155.03 q^{83} -1674.62 q^{86} -1394.27 q^{88} -619.374 q^{89} -429.216 q^{91} -3423.90 q^{92} -81.9585 q^{94} -231.195 q^{97} +238.727 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.87199	1.72251	0.861254	−	0.508175i	$-0.169680\pi$
$2$	4.87199	1.72251	0.861254	+	0.508175i	$0.169680\pi$
$3$	0	0
$3$	0	0
$4$	15.7363	1.96703
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	37.6910	1.66572
$9$	0	0
$10$	0	0
$11$	−36.9922	−1.01396	−0.506980	−	0.861958i	$-0.669238\pi$
$11$	−36.9922	−1.01396	−0.506980	+	0.861958i	$0.669238\pi$
$12$	0	0
$13$	−61.3165	−1.30817	−0.654083	−	0.756423i	$-0.726945\pi$
$13$	−61.3165	−1.30817	−0.654083	+	0.756423i	$0.726945\pi$
$14$	34.1039	0.651047
$15$	0	0
$16$	57.7401	0.902189
$17$	44.8345	0.639646	0.319823	−	0.947477i	$-0.396377\pi$
$17$	44.8345	0.639646	0.319823	+	0.947477i	$0.396377\pi$
$18$	0	0
$19$	−139.701	−1.68682	−0.843408	−	0.537273i	$-0.819455\pi$
$19$	−139.701	−1.68682	−0.843408	+	0.537273i	$0.819455\pi$
$20$	0	0
$21$	0	0
$22$	−180.226	−1.74656
$23$	−217.580	−1.97255	−0.986275	−	0.165110i	$-0.947202\pi$
$23$	−217.580	−1.97255	−0.986275	+	0.165110i	$0.947202\pi$
$24$	0	0
$25$	0	0
$26$	−298.733	−2.25333
$27$	0	0
$28$	110.154	0.743469
$29$	33.8226	0.216576	0.108288	−	0.994120i	$-0.465463\pi$
$29$	33.8226	0.216576	0.108288	+	0.994120i	$0.465463\pi$
$30$	0	0
$31$	124.437	0.720952	0.360476	−	0.932769i	$-0.382614\pi$
$31$	124.437	0.720952	0.360476	+	0.932769i	$0.382614\pi$
$32$	−20.2192	−0.111697
$33$	0	0
$34$	218.433	1.10179
$35$	0	0
$36$	0	0
$37$	237.270	1.05424	0.527121	−	0.849790i	$-0.323271\pi$
$37$	237.270	1.05424	0.527121	+	0.849790i	$0.323271\pi$
$38$	−680.620	−2.90556
$39$	0	0
$40$	0	0
$41$	−195.117	−0.743224	−0.371612	−	0.928388i	$-0.621195\pi$
$41$	−195.117	−0.743224	−0.371612	+	0.928388i	$0.621195\pi$
$42$	0	0
$43$	−343.725	−1.21901	−0.609506	−	0.792781i	$-0.708632\pi$
$43$	−343.725	−1.21901	−0.609506	+	0.792781i	$0.708632\pi$
$44$	−582.119	−1.99450
$45$	0	0
$46$	−1060.05	−3.39773
$47$	−16.8224	−0.0522085	−0.0261042	−	0.999659i	$-0.508310\pi$
$47$	−16.8224	−0.0522085	−0.0261042	+	0.999659i	$0.508310\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−964.894	−2.57321
$53$	−346.965	−0.899231	−0.449616	−	0.893222i	$-0.648439\pi$
$53$	−346.965	−0.899231	−0.449616	+	0.893222i	$0.648439\pi$
$54$	0	0
$55$	0	0
$56$	263.837	0.629584
$57$	0	0
$58$	164.783	0.373054
$59$	−135.340	−0.298640	−0.149320	−	0.988789i	$-0.547708\pi$
$59$	−135.340	−0.298640	−0.149320	+	0.988789i	$0.547708\pi$
$60$	0	0
$61$	490.414	1.02936	0.514681	−	0.857382i	$-0.327911\pi$
$61$	490.414	1.02936	0.514681	+	0.857382i	$0.327911\pi$
$62$	606.255	1.24185
$63$	0	0
$64$	−560.428	−1.09459
$65$	0	0
$66$	0	0
$67$	477.969	0.871540	0.435770	−	0.900058i	$-0.356476\pi$
$67$	477.969	0.871540	0.435770	+	0.900058i	$0.356476\pi$
$68$	705.529	1.25820
$69$	0	0
$70$	0	0
$71$	−45.2557	−0.0756460	−0.0378230	−	0.999284i	$-0.512042\pi$
$71$	−45.2557	−0.0756460	−0.0378230	+	0.999284i	$0.512042\pi$
$72$	0	0
$73$	100.781	0.161583	0.0807913	−	0.996731i	$-0.474255\pi$
$73$	100.781	0.161583	0.0807913	+	0.996731i	$0.474255\pi$
$74$	1155.98	1.81594
$75$	0	0
$76$	−2198.37	−3.31803
$77$	−258.945	−0.383241
$78$	0	0
$79$	880.534	1.25402	0.627012	−	0.779010i	$-0.284278\pi$
$79$	880.534	1.25402	0.627012	+	0.779010i	$0.284278\pi$
$80$	0	0
$81$	0	0
$82$	−950.609	−1.28021
$83$	1155.03	1.52748	0.763739	−	0.645525i	$-0.223362\pi$
$83$	1155.03	1.52748	0.763739	+	0.645525i	$0.223362\pi$
$84$	0	0
$85$	0	0
$86$	−1674.62	−2.09976
$87$	0	0
$88$	−1394.27	−1.68898
$89$	−619.374	−0.737680	−0.368840	−	0.929493i	$-0.620245\pi$
$89$	−619.374	−0.737680	−0.368840	+	0.929493i	$0.620245\pi$
$90$	0	0
$91$	−429.216	−0.494440
$92$	−3423.90	−3.88007
$93$	0	0
$94$	−81.9585	−0.0899295
$95$	0	0
$96$	0	0
$97$	−231.195	−0.242003	−0.121001	−	0.992652i	$-0.538611\pi$
$97$	−231.195	−0.242003	−0.121001	+	0.992652i	$0.538611\pi$
$98$	238.727	0.246073
$99$	0	0
$100$	0	0
$101$	−1875.99	−1.84820	−0.924101	−	0.382149i	$-0.875184\pi$
$101$	−1875.99	−1.84820	−0.924101	+	0.382149i	$0.875184\pi$
$102$	0	0
$103$	1855.94	1.77545	0.887723	−	0.460377i	$-0.152286\pi$
$103$	1855.94	1.77545	0.887723	+	0.460377i	$0.152286\pi$
$104$	−2311.08	−2.17904
$105$	0	0
$106$	−1690.41	−1.54893
$107$	218.016	0.196976	0.0984879	−	0.995138i	$-0.468599\pi$
$107$	218.016	0.196976	0.0984879	+	0.995138i	$0.468599\pi$
$108$	0	0
$109$	847.217	0.744484	0.372242	−	0.928136i	$-0.378589\pi$
$109$	847.217	0.744484	0.372242	+	0.928136i	$0.378589\pi$
$110$	0	0
$111$	0	0
$112$	404.181	0.340995
$113$	−1430.05	−1.19051	−0.595254	−	0.803537i	$-0.702949\pi$
$113$	−1430.05	−1.19051	−0.595254	+	0.803537i	$0.702949\pi$
$114$	0	0
$115$	0	0
$116$	532.242	0.426012
$117$	0	0
$118$	−659.374	−0.514409
$119$	313.842	0.241763
$120$	0	0
$121$	37.4229	0.0281164
$122$	2389.29	1.77308
$123$	0	0
$124$	1958.17	1.41814
$125$	0	0
$126$	0	0
$127$	−1732.84	−1.21075	−0.605374	−	0.795941i	$-0.706976\pi$
$127$	−1732.84	−1.21075	−0.605374	+	0.795941i	$0.706976\pi$
$128$	−2568.65	−1.77374
$129$	0	0
$130$	0	0
$131$	2429.76	1.62052	0.810262	−	0.586068i	$-0.199325\pi$
$131$	2429.76	1.62052	0.810262	+	0.586068i	$0.199325\pi$
$132$	0	0
$133$	−977.904	−0.637557
$134$	2328.66	1.50123
$135$	0	0
$136$	1689.86	1.06547
$137$	−1452.84	−0.906018	−0.453009	−	0.891506i	$-0.649649\pi$
$137$	−1452.84	−0.906018	−0.453009	+	0.891506i	$0.649649\pi$
$138$	0	0
$139$	2927.47	1.78637	0.893183	−	0.449693i	$-0.148467\pi$
$139$	2927.47	1.78637	0.893183	+	0.449693i	$0.148467\pi$
$140$	0	0
$141$	0	0
$142$	−220.485	−0.130301
$143$	2268.23	1.32643
$144$	0	0
$145$	0	0
$146$	491.004	0.278327
$147$	0	0
$148$	3733.75	2.07373
$149$	−901.724	−0.495786	−0.247893	−	0.968787i	$-0.579738\pi$
$149$	−901.724	−0.495786	−0.247893	+	0.968787i	$0.579738\pi$
$150$	0	0
$151$	−1357.63	−0.731669	−0.365834	−	0.930680i	$-0.619216\pi$
$151$	−1357.63	−0.731669	−0.365834	+	0.930680i	$0.619216\pi$
$152$	−5265.46	−2.80977
$153$	0	0
$154$	−1261.58	−0.660136
$155$	0	0
$156$	0	0
$157$	−2318.72	−1.17869	−0.589343	−	0.807883i	$-0.700613\pi$
$157$	−2318.72	−1.17869	−0.589343	+	0.807883i	$0.700613\pi$
$158$	4289.95	2.16006
$159$	0	0
$160$	0	0
$161$	−1523.06	−0.745554
$162$	0	0
$163$	−1577.42	−0.757996	−0.378998	−	0.925397i	$-0.623731\pi$
$163$	−1577.42	−0.757996	−0.378998	+	0.925397i	$0.623731\pi$
$164$	−3070.42	−1.46195
$165$	0	0
$166$	5627.28	2.63109
$167$	−1038.02	−0.480982	−0.240491	−	0.970651i	$-0.577309\pi$
$167$	−1038.02	−0.480982	−0.240491	+	0.970651i	$0.577309\pi$
$168$	0	0
$169$	1562.72	0.711296
$170$	0	0
$171$	0	0
$172$	−5408.95	−2.39784
$173$	482.641	0.212107	0.106054	−	0.994360i	$-0.466178\pi$
$173$	482.641	0.212107	0.106054	+	0.994360i	$0.466178\pi$
$174$	0	0
$175$	0	0
$176$	−2135.93	−0.914784
$177$	0	0
$178$	−3017.58	−1.27066
$179$	−2407.19	−1.00515	−0.502575	−	0.864534i	$-0.667614\pi$
$179$	−2407.19	−1.00515	−0.502575	+	0.864534i	$0.667614\pi$
$180$	0	0
$181$	532.600	0.218717	0.109359	−	0.994002i	$-0.465120\pi$
$181$	532.600	0.218717	0.109359	+	0.994002i	$0.465120\pi$
$182$	−2091.13	−0.851677
$183$	0	0
$184$	−8200.83	−3.28572
$185$	0	0
$186$	0	0
$187$	−1658.53	−0.648575
$188$	−264.722	−0.102696
$189$	0	0
$190$	0	0
$191$	−336.675	−0.127544	−0.0637721	−	0.997964i	$-0.520313\pi$
$191$	−336.675	−0.127544	−0.0637721	+	0.997964i	$0.520313\pi$
$192$	0	0
$193$	22.7366	0.00847988	0.00423994	−	0.999991i	$-0.498650\pi$
$193$	22.7366	0.00847988	0.00423994	+	0.999991i	$0.498650\pi$
$194$	−1126.38	−0.416852
$195$	0	0
$196$	771.077	0.281005
$197$	1085.33	0.392520	0.196260	−	0.980552i	$-0.437120\pi$
$197$	1085.33	0.392520	0.196260	+	0.980552i	$0.437120\pi$
$198$	0	0
$199$	−2630.28	−0.936963	−0.468482	−	0.883473i	$-0.655199\pi$
$199$	−2630.28	−0.936963	−0.468482	+	0.883473i	$0.655199\pi$
$200$	0	0
$201$	0	0
$202$	−9139.82	−3.18354
$203$	236.758	0.0818580
$204$	0	0
$205$	0	0
$206$	9042.11	3.05822
$207$	0	0
$208$	−3540.42	−1.18021
$209$	5167.83	1.71037
$210$	0	0
$211$	614.178	0.200388	0.100194	−	0.994968i	$-0.468054\pi$
$211$	614.178	0.200388	0.100194	+	0.994968i	$0.468054\pi$
$212$	−5459.93	−1.76882
$213$	0	0
$214$	1062.17	0.339292
$215$	0	0
$216$	0	0
$217$	871.057	0.272494
$218$	4127.63	1.28238
$219$	0	0
$220$	0	0
$221$	−2749.10	−0.836762
$222$	0	0
$223$	1930.30	0.579651	0.289826	−	0.957079i	$-0.406403\pi$
$223$	1930.30	0.579651	0.289826	+	0.957079i	$0.406403\pi$
$224$	−141.535	−0.0422173
$225$	0	0
$226$	−6967.17	−2.05066
$227$	−765.365	−0.223784	−0.111892	−	0.993720i	$-0.535691\pi$
$227$	−765.365	−0.223784	−0.111892	+	0.993720i	$0.535691\pi$
$228$	0	0
$229$	−5712.66	−1.64849	−0.824243	−	0.566237i	$-0.808399\pi$
$229$	−5712.66	−1.64849	−0.824243	+	0.566237i	$0.808399\pi$
$230$	0	0
$231$	0	0
$232$	1274.81	0.360756
$233$	864.418	0.243047	0.121523	−	0.992589i	$-0.461222\pi$
$233$	864.418	0.243047	0.121523	+	0.992589i	$0.461222\pi$
$234$	0	0
$235$	0	0
$236$	−2129.74	−0.587435
$237$	0	0
$238$	1529.03	0.416439
$239$	1816.64	0.491669	0.245834	−	0.969312i	$-0.420938\pi$
$239$	1816.64	0.491669	0.245834	+	0.969312i	$0.420938\pi$
$240$	0	0
$241$	−2354.53	−0.629330	−0.314665	−	0.949203i	$-0.601892\pi$
$241$	−2354.53	−0.629330	−0.314665	+	0.949203i	$0.601892\pi$
$242$	182.324	0.0484307
$243$	0	0
$244$	7717.28	2.02479
$245$	0	0
$246$	0	0
$247$	8565.96	2.20664
$248$	4690.15	1.20091
$249$	0	0
$250$	0	0
$251$	2983.66	0.750306	0.375153	−	0.926963i	$-0.377590\pi$
$251$	2983.66	0.750306	0.375153	+	0.926963i	$0.377590\pi$
$252$	0	0
$253$	8048.78	2.00009
$254$	−8442.39	−2.08552
$255$	0	0
$256$	−8030.99	−1.96069
$257$	5121.50	1.24307	0.621537	−	0.783385i	$-0.286508\pi$
$257$	5121.50	1.24307	0.621537	+	0.783385i	$0.286508\pi$
$258$	0	0
$259$	1660.89	0.398466
$260$	0	0
$261$	0	0
$262$	11837.7	2.79137
$263$	−4663.66	−1.09344	−0.546718	−	0.837317i	$-0.684123\pi$
$263$	−4663.66	−1.09344	−0.546718	+	0.837317i	$0.684123\pi$
$264$	0	0
$265$	0	0
$266$	−4764.34	−1.09820
$267$	0	0
$268$	7521.45	1.71435
$269$	−5018.86	−1.13757	−0.568783	−	0.822488i	$-0.692586\pi$
$269$	−5018.86	−1.13757	−0.568783	+	0.822488i	$0.692586\pi$
$270$	0	0
$271$	−2512.66	−0.563222	−0.281611	−	0.959529i	$-0.590869\pi$
$271$	−2512.66	−0.563222	−0.281611	+	0.959529i	$0.590869\pi$
$272$	2588.75	0.577081
$273$	0	0
$274$	−7078.21	−1.56062
$275$	0	0
$276$	0	0
$277$	6286.82	1.36368	0.681839	−	0.731503i	$-0.261181\pi$
$277$	6286.82	1.36368	0.681839	+	0.731503i	$0.261181\pi$
$278$	14262.6	3.07703
$279$	0	0
$280$	0	0
$281$	−8804.33	−1.86912	−0.934560	−	0.355807i	$-0.884206\pi$
$281$	−8804.33	−1.86912	−0.934560	+	0.355807i	$0.884206\pi$
$282$	0	0
$283$	485.298	0.101936	0.0509681	−	0.998700i	$-0.483769\pi$
$283$	485.298	0.101936	0.0509681	+	0.998700i	$0.483769\pi$
$284$	−712.157	−0.148798
$285$	0	0
$286$	11050.8	2.28478
$287$	−1365.82	−0.280912
$288$	0	0
$289$	−2902.86	−0.590854
$290$	0	0
$291$	0	0
$292$	1585.92	0.317838
$293$	−4004.48	−0.798444	−0.399222	−	0.916854i	$-0.630720\pi$
$293$	−4004.48	−0.798444	−0.399222	+	0.916854i	$0.630720\pi$
$294$	0	0
$295$	0	0
$296$	8942.96	1.75608
$297$	0	0
$298$	−4393.19	−0.853995
$299$	13341.3	2.58042
$300$	0	0
$301$	−2406.07	−0.460743
$302$	−6614.33	−1.26031
$303$	0	0
$304$	−8066.32	−1.52183
$305$	0	0
$306$	0	0
$307$	3154.13	0.586370	0.293185	−	0.956056i	$-0.405285\pi$
$307$	3154.13	0.586370	0.293185	+	0.956056i	$0.405285\pi$
$308$	−4074.84	−0.753848
$309$	0	0
$310$	0	0
$311$	1738.29	0.316944	0.158472	−	0.987363i	$-0.449343\pi$
$311$	1738.29	0.316944	0.158472	+	0.987363i	$0.449343\pi$
$312$	0	0
$313$	5092.30	0.919597	0.459799	−	0.888023i	$-0.347922\pi$
$313$	5092.30	0.919597	0.459799	+	0.888023i	$0.347922\pi$
$314$	−11296.8	−2.03030
$315$	0	0
$316$	13856.3	2.46671
$317$	4874.67	0.863686	0.431843	−	0.901949i	$-0.357863\pi$
$317$	4874.67	0.863686	0.431843	+	0.901949i	$0.357863\pi$
$318$	0	0
$319$	−1251.17	−0.219600
$320$	0	0
$321$	0	0
$322$	−7420.35	−1.28422
$323$	−6263.41	−1.07896
$324$	0	0
$325$	0	0
$326$	−7685.19	−1.30565
$327$	0	0
$328$	−7354.17	−1.23801
$329$	−117.757	−0.0197329
$330$	0	0
$331$	−2448.46	−0.406585	−0.203292	−	0.979118i	$-0.565164\pi$
$331$	−2448.46	−0.406585	−0.203292	+	0.979118i	$0.565164\pi$
$332$	18175.8	3.00460
$333$	0	0
$334$	−5057.20	−0.828496
$335$	0	0
$336$	0	0
$337$	−6271.50	−1.01374	−0.506870	−	0.862023i	$-0.669197\pi$
$337$	−6271.50	−1.01374	−0.506870	+	0.862023i	$0.669197\pi$
$338$	7613.54	1.22521
$339$	0	0
$340$	0	0
$341$	−4603.19	−0.731017
$342$	0	0
$343$	343.000	0.0539949
$344$	−12955.3	−2.03054
$345$	0	0
$346$	2351.42	0.365356
$347$	−417.338	−0.0645645	−0.0322822	−	0.999479i	$-0.510278\pi$
$347$	−417.338	−0.0645645	−0.0322822	+	0.999479i	$0.510278\pi$
$348$	0	0
$349$	5971.08	0.915829	0.457915	−	0.888996i	$-0.348597\pi$
$349$	5971.08	0.915829	0.457915	+	0.888996i	$0.348597\pi$
$350$	0	0
$351$	0	0
$352$	747.954	0.113256
$353$	12012.6	1.81124	0.905620	−	0.424089i	$-0.139406\pi$
$353$	12012.6	1.81124	0.905620	+	0.424089i	$0.139406\pi$
$354$	0	0
$355$	0	0
$356$	−9746.64	−1.45104
$357$	0	0
$358$	−11727.8	−1.73138
$359$	−6312.46	−0.928020	−0.464010	−	0.885830i	$-0.653590\pi$
$359$	−6312.46	−0.928020	−0.464010	+	0.885830i	$0.653590\pi$
$360$	0	0
$361$	12657.3	1.84535
$362$	2594.82	0.376743
$363$	0	0
$364$	−6754.26	−0.972580
$365$	0	0
$366$	0	0
$367$	12228.6	1.73932	0.869659	−	0.493652i	$-0.164338\pi$
$367$	12228.6	1.73932	0.869659	+	0.493652i	$0.164338\pi$
$368$	−12563.1	−1.77961
$369$	0	0
$370$	0	0
$371$	−2428.75	−0.339877
$372$	0	0
$373$	7876.24	1.09334	0.546671	−	0.837348i	$-0.315895\pi$
$373$	7876.24	1.09334	0.546671	+	0.837348i	$0.315895\pi$
$374$	−8080.33	−1.11718
$375$	0	0
$376$	−634.053	−0.0869649
$377$	−2073.89	−0.283317
$378$	0	0
$379$	5992.79	0.812214	0.406107	−	0.913826i	$-0.366886\pi$
$379$	5992.79	0.812214	0.406107	+	0.913826i	$0.366886\pi$
$380$	0	0
$381$	0	0
$382$	−1640.28	−0.219696
$383$	−7373.61	−0.983743	−0.491872	−	0.870668i	$-0.663687\pi$
$383$	−7373.61	−0.983743	−0.491872	+	0.870668i	$0.663687\pi$
$384$	0	0
$385$	0	0
$386$	110.772	0.0146067
$387$	0	0
$388$	−3638.14	−0.476028
$389$	−1896.88	−0.247238	−0.123619	−	0.992330i	$-0.539450\pi$
$389$	−1896.88	−0.247238	−0.123619	+	0.992330i	$0.539450\pi$
$390$	0	0
$391$	−9755.12	−1.26173
$392$	1846.86	0.237961
$393$	0	0
$394$	5287.71	0.676119
$395$	0	0
$396$	0	0
$397$	−791.156	−0.100018	−0.0500088	−	0.998749i	$-0.515925\pi$
$397$	−791.156	−0.100018	−0.0500088	+	0.998749i	$0.515925\pi$
$398$	−12814.7	−1.61393
$399$	0	0
$400$	0	0
$401$	−4410.55	−0.549258	−0.274629	−	0.961550i	$-0.588555\pi$
$401$	−4410.55	−0.549258	−0.274629	+	0.961550i	$0.588555\pi$
$402$	0	0
$403$	−7630.03	−0.943124
$404$	−29521.1	−3.63548
$405$	0	0
$406$	1153.48	0.141001
$407$	−8777.14	−1.06896
$408$	0	0
$409$	−4403.73	−0.532397	−0.266199	−	0.963918i	$-0.585768\pi$
$409$	−4403.73	−0.532397	−0.266199	+	0.963918i	$0.585768\pi$
$410$	0	0
$411$	0	0
$412$	29205.5	3.49236
$413$	−947.379	−0.112875
$414$	0	0
$415$	0	0
$416$	1239.77	0.146117
$417$	0	0
$418$	25177.6	2.94612
$419$	14272.6	1.66411	0.832057	−	0.554691i	$-0.187163\pi$
$419$	14272.6	1.66411	0.832057	+	0.554691i	$0.187163\pi$
$420$	0	0
$421$	−15530.4	−1.79788	−0.898939	−	0.438074i	$-0.855661\pi$
$421$	−15530.4	−1.79788	−0.898939	+	0.438074i	$0.855661\pi$
$422$	2992.27	0.345169
$423$	0	0
$424$	−13077.5	−1.49787
$425$	0	0
$426$	0	0
$427$	3432.90	0.389062
$428$	3430.76	0.387458
$429$	0	0
$430$	0	0
$431$	5260.75	0.587938	0.293969	−	0.955815i	$-0.405024\pi$
$431$	5260.75	0.587938	0.293969	+	0.955815i	$0.405024\pi$
$432$	0	0
$433$	6610.65	0.733690	0.366845	−	0.930282i	$-0.380438\pi$
$433$	6610.65	0.733690	0.366845	+	0.930282i	$0.380438\pi$
$434$	4243.78	0.469373
$435$	0	0
$436$	13332.0	1.46442
$437$	30396.1	3.32733
$438$	0	0
$439$	−8870.46	−0.964383	−0.482191	−	0.876066i	$-0.660159\pi$
$439$	−8870.46	−0.964383	−0.482191	+	0.876066i	$0.660159\pi$
$440$	0	0
$441$	0	0
$442$	−13393.6	−1.44133
$443$	13221.8	1.41802	0.709012	−	0.705196i	$-0.249141\pi$
$443$	13221.8	1.41802	0.709012	+	0.705196i	$0.249141\pi$
$444$	0	0
$445$	0	0
$446$	9404.38	0.998454
$447$	0	0
$448$	−3923.00	−0.413715
$449$	−4192.70	−0.440681	−0.220341	−	0.975423i	$-0.570717\pi$
$449$	−4192.70	−0.440681	−0.220341	+	0.975423i	$0.570717\pi$
$450$	0	0
$451$	7217.82	0.753600
$452$	−22503.6	−2.34177
$453$	0	0
$454$	−3728.85	−0.385470
$455$	0	0
$456$	0	0
$457$	−4701.78	−0.481270	−0.240635	−	0.970616i	$-0.577356\pi$
$457$	−4701.78	−0.481270	−0.240635	+	0.970616i	$0.577356\pi$
$458$	−27832.0	−2.83953
$459$	0	0
$460$	0	0
$461$	1949.82	0.196989	0.0984946	−	0.995138i	$-0.468597\pi$
$461$	1949.82	0.196989	0.0984946	+	0.995138i	$0.468597\pi$
$462$	0	0
$463$	7315.85	0.734333	0.367166	−	0.930155i	$-0.380328\pi$
$463$	7315.85	0.734333	0.367166	+	0.930155i	$0.380328\pi$
$464$	1952.92	0.195392
$465$	0	0
$466$	4211.43	0.418650
$467$	1897.54	0.188025	0.0940126	−	0.995571i	$-0.470031\pi$
$467$	1897.54	0.188025	0.0940126	+	0.995571i	$0.470031\pi$
$468$	0	0
$469$	3345.78	0.329411
$470$	0	0
$471$	0	0
$472$	−5101.10	−0.497451
$473$	12715.1	1.23603
$474$	0	0
$475$	0	0
$476$	4938.70	0.475557
$477$	0	0
$478$	8850.66	0.846903
$479$	−10534.4	−1.00486	−0.502429	−	0.864619i	$-0.667560\pi$
$479$	−10534.4	−1.00486	−0.502429	+	0.864619i	$0.667560\pi$
$480$	0	0
$481$	−14548.6	−1.37912
$482$	−11471.2	−1.08403
$483$	0	0
$484$	588.897	0.0553058
$485$	0	0
$486$	0	0
$487$	11937.0	1.11071	0.555355	−	0.831614i	$-0.312583\pi$
$487$	11937.0	1.11071	0.555355	+	0.831614i	$0.312583\pi$
$488$	18484.2	1.71463
$489$	0	0
$490$	0	0
$491$	−100.060	−0.00919683	−0.00459841	−	0.999989i	$-0.501464\pi$
$491$	−100.060	−0.00919683	−0.00459841	+	0.999989i	$0.501464\pi$
$492$	0	0
$493$	1516.42	0.138532
$494$	41733.3	3.80095
$495$	0	0
$496$	7184.99	0.650435
$497$	−316.790	−0.0285915
$498$	0	0
$499$	−15611.7	−1.40055	−0.700275	−	0.713874i	$-0.746939\pi$
$499$	−15611.7	−1.40055	−0.700275	+	0.713874i	$0.746939\pi$
$500$	0	0
$501$	0	0
$502$	14536.3	1.29241
$503$	−2270.39	−0.201256	−0.100628	−	0.994924i	$-0.532085\pi$
$503$	−2270.39	−0.201256	−0.100628	+	0.994924i	$0.532085\pi$
$504$	0	0
$505$	0	0
$506$	39213.6	3.44517
$507$	0	0
$508$	−27268.5	−2.38158
$509$	2579.79	0.224651	0.112325	−	0.993671i	$-0.464170\pi$
$509$	2579.79	0.224651	0.112325	+	0.993671i	$0.464170\pi$
$510$	0	0
$511$	705.467	0.0610725
$512$	−18577.7	−1.60357
$513$	0	0
$514$	24951.9	2.14121
$515$	0	0
$516$	0	0
$517$	622.297	0.0529373
$518$	8091.84	0.686361
$519$	0	0
$520$	0	0
$521$	−12793.9	−1.07584	−0.537918	−	0.842997i	$-0.680789\pi$
$521$	−12793.9	−1.07584	−0.537918	+	0.842997i	$0.680789\pi$
$522$	0	0
$523$	−11273.4	−0.942548	−0.471274	−	0.881987i	$-0.656206\pi$
$523$	−11273.4	−0.942548	−0.471274	+	0.881987i	$0.656206\pi$
$524$	38235.3	3.18763
$525$	0	0
$526$	−22721.3	−1.88345
$527$	5579.07	0.461154
$528$	0	0
$529$	35174.2	2.89095
$530$	0	0
$531$	0	0
$532$	−15388.6	−1.25410
$533$	11963.9	0.972260
$534$	0	0
$535$	0	0
$536$	18015.1	1.45175
$537$	0	0
$538$	−24451.8	−1.95947
$539$	−1812.62	−0.144852
$540$	0	0
$541$	10282.9	0.817183	0.408591	−	0.912717i	$-0.366020\pi$
$541$	10282.9	0.817183	0.408591	+	0.912717i	$0.366020\pi$
$542$	−12241.6	−0.970154
$543$	0	0
$544$	−906.520	−0.0714462
$545$	0	0
$546$	0	0
$547$	−2160.94	−0.168912	−0.0844562	−	0.996427i	$-0.526915\pi$
$547$	−2160.94	−0.168912	−0.0844562	+	0.996427i	$0.526915\pi$
$548$	−22862.3	−1.78217
$549$	0	0
$550$	0	0
$551$	−4725.04	−0.365324
$552$	0	0
$553$	6163.74	0.473976
$554$	30629.3	2.34895
$555$	0	0
$556$	46067.5	3.51384
$557$	−15648.4	−1.19038	−0.595190	−	0.803585i	$-0.702923\pi$
$557$	−15648.4	−1.19038	−0.595190	+	0.803585i	$0.702923\pi$
$558$	0	0
$559$	21076.0	1.59467
$560$	0	0
$561$	0	0
$562$	−42894.6	−3.21957
$563$	21951.6	1.64325	0.821623	−	0.570031i	$-0.193069\pi$
$563$	21951.6	1.64325	0.821623	+	0.570031i	$0.193069\pi$
$564$	0	0
$565$	0	0
$566$	2364.36	0.175586
$567$	0	0
$568$	−1705.74	−0.126005
$569$	−14019.2	−1.03289	−0.516445	−	0.856321i	$-0.672745\pi$
$569$	−14019.2	−1.03289	−0.516445	+	0.856321i	$0.672745\pi$
$570$	0	0
$571$	−21706.3	−1.59086	−0.795430	−	0.606045i	$-0.792755\pi$
$571$	−21706.3	−1.59086	−0.795430	+	0.606045i	$0.792755\pi$
$572$	35693.5	2.60913
$573$	0	0
$574$	−6654.26	−0.483874
$575$	0	0
$576$	0	0
$577$	12569.0	0.906853	0.453427	−	0.891294i	$-0.350201\pi$
$577$	12569.0	0.906853	0.453427	+	0.891294i	$0.350201\pi$
$578$	−14142.7	−1.01775
$579$	0	0
$580$	0	0
$581$	8085.19	0.577332
$582$	0	0
$583$	12835.0	0.911785
$584$	3798.54	0.269152
$585$	0	0
$586$	−19509.8	−1.37533
$587$	1330.20	0.0935322	0.0467661	−	0.998906i	$-0.485108\pi$
$587$	1330.20	0.0935322	0.0467661	+	0.998906i	$0.485108\pi$
$588$	0	0
$589$	−17383.9	−1.21611
$590$	0	0
$591$	0	0
$592$	13700.0	0.951126
$593$	−9866.91	−0.683281	−0.341640	−	0.939831i	$-0.610982\pi$
$593$	−9866.91	−0.683281	−0.341640	+	0.939831i	$0.610982\pi$
$594$	0	0
$595$	0	0
$596$	−14189.8	−0.975228
$597$	0	0
$598$	64998.6	4.44480
$599$	−4185.17	−0.285478	−0.142739	−	0.989760i	$-0.545591\pi$
$599$	−4185.17	−0.285478	−0.142739	+	0.989760i	$0.545591\pi$
$600$	0	0
$601$	10599.8	0.719426	0.359713	−	0.933063i	$-0.382875\pi$
$601$	10599.8	0.719426	0.359713	+	0.933063i	$0.382875\pi$
$602$	−11722.4	−0.793634
$603$	0	0
$604$	−21364.0	−1.43922
$605$	0	0
$606$	0	0
$607$	−13290.1	−0.888679	−0.444340	−	0.895858i	$-0.646562\pi$
$607$	−13290.1	−0.888679	−0.444340	+	0.895858i	$0.646562\pi$
$608$	2824.64	0.188412
$609$	0	0
$610$	0	0
$611$	1031.49	0.0682973
$612$	0	0
$613$	21327.1	1.40521	0.702606	−	0.711579i	$-0.252020\pi$
$613$	21327.1	1.40521	0.702606	+	0.711579i	$0.252020\pi$
$614$	15366.9	1.01003
$615$	0	0
$616$	−9759.92	−0.638374
$617$	−13741.6	−0.896622	−0.448311	−	0.893878i	$-0.647974\pi$
$617$	−13741.6	−0.896622	−0.448311	+	0.893878i	$0.647974\pi$
$618$	0	0
$619$	19444.3	1.26257	0.631285	−	0.775551i	$-0.282528\pi$
$619$	19444.3	1.26257	0.631285	+	0.775551i	$0.282528\pi$
$620$	0	0
$621$	0	0
$622$	8468.94	0.545938
$623$	−4335.62	−0.278817
$624$	0	0
$625$	0	0
$626$	24809.6	1.58401
$627$	0	0
$628$	−36488.0	−2.31852
$629$	10637.9	0.674341
$630$	0	0
$631$	−27425.6	−1.73026	−0.865130	−	0.501547i	$-0.832764\pi$
$631$	−27425.6	−1.73026	−0.865130	+	0.501547i	$0.832764\pi$
$632$	33188.2	2.08886
$633$	0	0
$634$	23749.3	1.48771
$635$	0	0
$636$	0	0
$637$	−3004.51	−0.186881
$638$	−6095.70	−0.378262
$639$	0	0
$640$	0	0
$641$	10053.8	0.619502	0.309751	−	0.950818i	$-0.399754\pi$
$641$	10053.8	0.619502	0.309751	+	0.950818i	$0.399754\pi$
$642$	0	0
$643$	−4044.18	−0.248036	−0.124018	−	0.992280i	$-0.539578\pi$
$643$	−4044.18	−0.248036	−0.124018	+	0.992280i	$0.539578\pi$
$644$	−23967.3	−1.46653
$645$	0	0
$646$	−30515.3	−1.85853
$647$	−1486.90	−0.0903491	−0.0451746	−	0.998979i	$-0.514384\pi$
$647$	−1486.90	−0.0903491	−0.0451746	+	0.998979i	$0.514384\pi$
$648$	0	0
$649$	5006.52	0.302809
$650$	0	0
$651$	0	0
$652$	−24822.8	−1.49100
$653$	−4269.30	−0.255851	−0.127925	−	0.991784i	$-0.540832\pi$
$653$	−4269.30	−0.255851	−0.127925	+	0.991784i	$0.540832\pi$
$654$	0	0
$655$	0	0
$656$	−11266.1	−0.670528
$657$	0	0
$658$	−573.710	−0.0339902
$659$	−1607.29	−0.0950094	−0.0475047	−	0.998871i	$-0.515127\pi$
$659$	−1607.29	−0.0950094	−0.0475047	+	0.998871i	$0.515127\pi$
$660$	0	0
$661$	8001.31	0.470824	0.235412	−	0.971896i	$-0.424356\pi$
$661$	8001.31	0.470824	0.235412	+	0.971896i	$0.424356\pi$
$662$	−11928.9	−0.700345
$663$	0	0
$664$	43534.1	2.54436
$665$	0	0
$666$	0	0
$667$	−7359.14	−0.427207
$668$	−16334.5	−0.946109
$669$	0	0
$670$	0	0
$671$	−18141.5	−1.04373
$672$	0	0
$673$	3347.96	0.191760	0.0958800	−	0.995393i	$-0.469433\pi$
$673$	3347.96	0.191760	0.0958800	+	0.995393i	$0.469433\pi$
$674$	−30554.7	−1.74617
$675$	0	0
$676$	24591.4	1.39914
$677$	7.91810	0.000449508 0	0.000224754	−	1.00000i	$-0.499928\pi$
$677$	7.91810	0.000449508 0	0.000224754		1.00000i	$0.499928\pi$
$678$	0	0
$679$	−1618.36	−0.0914684
$680$	0	0
$681$	0	0
$682$	−22426.7	−1.25918
$683$	−32136.0	−1.80036	−0.900182	−	0.435513i	$-0.856567\pi$
$683$	−32136.0	−1.80036	−0.900182	+	0.435513i	$0.856567\pi$
$684$	0	0
$685$	0	0
$686$	1671.09	0.0930067
$687$	0	0
$688$	−19846.7	−1.09978
$689$	21274.7	1.17634
$690$	0	0
$691$	−6196.84	−0.341156	−0.170578	−	0.985344i	$-0.554563\pi$
$691$	−6196.84	−0.341156	−0.170578	+	0.985344i	$0.554563\pi$
$692$	7594.98	0.417222
$693$	0	0
$694$	−2033.27	−0.111213
$695$	0	0
$696$	0	0
$697$	−8747.99	−0.475400
$698$	29091.0	1.57752
$699$	0	0
$700$	0	0
$701$	−1651.38	−0.0889755	−0.0444878	−	0.999010i	$-0.514166\pi$
$701$	−1651.38	−0.0889755	−0.0444878	+	0.999010i	$0.514166\pi$
$702$	0	0
$703$	−33146.8	−1.77831
$704$	20731.5	1.10987
$705$	0	0
$706$	58525.4	3.11988
$707$	−13132.0	−0.698555
$708$	0	0
$709$	15399.8	0.815728	0.407864	−	0.913043i	$-0.366274\pi$
$709$	15399.8	0.815728	0.407864	+	0.913043i	$0.366274\pi$
$710$	0	0
$711$	0	0
$712$	−23344.8	−1.22877
$713$	−27075.0	−1.42211
$714$	0	0
$715$	0	0
$716$	−37880.2	−1.97716
$717$	0	0
$718$	−30754.3	−1.59852
$719$	−3507.31	−0.181920	−0.0909600	−	0.995855i	$-0.528994\pi$
$719$	−3507.31	−0.181920	−0.0909600	+	0.995855i	$0.528994\pi$
$720$	0	0
$721$	12991.6	0.671056
$722$	61666.0	3.17863
$723$	0	0
$724$	8381.14	0.430225
$725$	0	0
$726$	0	0
$727$	22516.7	1.14869	0.574346	−	0.818613i	$-0.305256\pi$
$727$	22516.7	1.14869	0.574346	+	0.818613i	$0.305256\pi$
$728$	−16177.6	−0.823600
$729$	0	0
$730$	0	0
$731$	−15410.7	−0.779736
$732$	0	0
$733$	−10964.7	−0.552509	−0.276255	−	0.961084i	$-0.589093\pi$
$733$	−10964.7	−0.552509	−0.276255	+	0.961084i	$0.589093\pi$
$734$	59577.8	2.99599
$735$	0	0
$736$	4399.31	0.220327
$737$	−17681.1	−0.883707
$738$	0	0
$739$	3029.33	0.150793	0.0753963	−	0.997154i	$-0.475978\pi$
$739$	3029.33	0.150793	0.0753963	+	0.997154i	$0.475978\pi$
$740$	0	0
$741$	0	0
$742$	−11832.9	−0.585442
$743$	8921.04	0.440486	0.220243	−	0.975445i	$-0.429315\pi$
$743$	8921.04	0.440486	0.220243	+	0.975445i	$0.429315\pi$
$744$	0	0
$745$	0	0
$746$	38373.0	1.88329
$747$	0	0
$748$	−26099.1	−1.27577
$749$	1526.11	0.0744498
$750$	0	0
$751$	−6202.64	−0.301382	−0.150691	−	0.988581i	$-0.548150\pi$
$751$	−6202.64	−0.301382	−0.150691	+	0.988581i	$0.548150\pi$
$752$	−971.326	−0.0471019
$753$	0	0
$754$	−10104.0	−0.488016
$755$	0	0
$756$	0	0
$757$	−12099.3	−0.580922	−0.290461	−	0.956887i	$-0.593809\pi$
$757$	−12099.3	−0.580922	−0.290461	+	0.956887i	$0.593809\pi$
$758$	29196.8	1.39905
$759$	0	0
$760$	0	0
$761$	−29332.7	−1.39725	−0.698626	−	0.715487i	$-0.746205\pi$
$761$	−29332.7	−1.39725	−0.698626	+	0.715487i	$0.746205\pi$
$762$	0	0
$763$	5930.52	0.281388
$764$	−5298.01	−0.250884
$765$	0	0
$766$	−35924.1	−1.69451
$767$	8298.57	0.390670
$768$	0	0
$769$	−13280.1	−0.622748	−0.311374	−	0.950287i	$-0.600789\pi$
$769$	−13280.1	−0.622748	−0.311374	+	0.950287i	$0.600789\pi$
$770$	0	0
$771$	0	0
$772$	357.789	0.0166802
$773$	−11248.1	−0.523370	−0.261685	−	0.965153i	$-0.584278\pi$
$773$	−11248.1	−0.523370	−0.261685	+	0.965153i	$0.584278\pi$
$774$	0	0
$775$	0	0
$776$	−8713.96	−0.403110
$777$	0	0
$778$	−9241.57	−0.425869
$779$	27258.0	1.25368
$780$	0	0
$781$	1674.11	0.0767021
$782$	−47526.8	−2.17335
$783$	0	0
$784$	2829.26	0.128884
$785$	0	0
$786$	0	0
$787$	−436.193	−0.0197568	−0.00987839	−	0.999951i	$-0.503144\pi$
$787$	−436.193	−0.0197568	−0.00987839	+	0.999951i	$0.503144\pi$
$788$	17079.0	0.772100
$789$	0	0
$790$	0	0
$791$	−10010.3	−0.449970
$792$	0	0
$793$	−30070.5	−1.34657
$794$	−3854.50	−0.172281
$795$	0	0
$796$	−41390.8	−1.84304
$797$	24034.0	1.06816	0.534082	−	0.845433i	$-0.320657\pi$
$797$	24034.0	1.06816	0.534082	+	0.845433i	$0.320657\pi$
$798$	0	0
$799$	−754.224	−0.0333949
$800$	0	0
$801$	0	0
$802$	−21488.1	−0.946101
$803$	−3728.11	−0.163838
$804$	0	0
$805$	0	0
$806$	−37173.4	−1.62454
$807$	0	0
$808$	−70708.1	−3.07859
$809$	−16188.0	−0.703509	−0.351754	−	0.936092i	$-0.614415\pi$
$809$	−16188.0	−0.703509	−0.351754	+	0.936092i	$0.614415\pi$
$810$	0	0
$811$	3732.42	0.161607	0.0808033	−	0.996730i	$-0.474251\pi$
$811$	3732.42	0.161607	0.0808033	+	0.996730i	$0.474251\pi$
$812$	3725.69	0.161018
$813$	0	0
$814$	−42762.1	−1.84129
$815$	0	0
$816$	0	0
$817$	48018.6	2.05625
$818$	−21454.9	−0.917058
$819$	0	0
$820$	0	0
$821$	−37226.4	−1.58247	−0.791237	−	0.611510i	$-0.790562\pi$
$821$	−37226.4	−1.58247	−0.791237	+	0.611510i	$0.790562\pi$
$822$	0	0
$823$	21355.7	0.904511	0.452256	−	0.891888i	$-0.350620\pi$
$823$	21355.7	0.904511	0.452256	+	0.891888i	$0.350620\pi$
$824$	69952.2	2.95740
$825$	0	0
$826$	−4615.62	−0.194428
$827$	18852.9	0.792718	0.396359	−	0.918096i	$-0.370274\pi$
$827$	18852.9	0.792718	0.396359	+	0.918096i	$0.370274\pi$
$828$	0	0
$829$	−43227.9	−1.81106	−0.905528	−	0.424286i	$-0.860525\pi$
$829$	−43227.9	−1.81106	−0.905528	+	0.424286i	$0.860525\pi$
$830$	0	0
$831$	0	0
$832$	34363.5	1.43190
$833$	2196.89	0.0913779
$834$	0	0
$835$	0	0
$836$	81322.4	3.36435
$837$	0	0
$838$	69536.1	2.86645
$839$	−30311.3	−1.24727	−0.623637	−	0.781714i	$-0.714346\pi$
$839$	−30311.3	−1.24727	−0.623637	+	0.781714i	$0.714346\pi$
$840$	0	0
$841$	−23245.0	−0.953095
$842$	−75664.0	−3.09686
$843$	0	0
$844$	9664.88	0.394169
$845$	0	0
$846$	0	0
$847$	261.960	0.0106270
$848$	−20033.8	−0.811276
$849$	0	0
$850$	0	0
$851$	−51625.3	−2.07955
$852$	0	0
$853$	−22797.0	−0.915071	−0.457535	−	0.889191i	$-0.651268\pi$
$853$	−22797.0	−0.915071	−0.457535	+	0.889191i	$0.651268\pi$
$854$	16725.0	0.670162
$855$	0	0
$856$	8217.25	0.328107
$857$	11280.3	0.449625	0.224813	−	0.974402i	$-0.427823\pi$
$857$	11280.3	0.449625	0.224813	+	0.974402i	$0.427823\pi$
$858$	0	0
$859$	43369.3	1.72263	0.861315	−	0.508071i	$-0.169641\pi$
$859$	43369.3	1.72263	0.861315	+	0.508071i	$0.169641\pi$
$860$	0	0
$861$	0	0
$862$	25630.3	1.01273
$863$	−9846.22	−0.388377	−0.194189	−	0.980964i	$-0.562207\pi$
$863$	−9846.22	−0.388377	−0.194189	+	0.980964i	$0.562207\pi$
$864$	0	0
$865$	0	0
$866$	32207.0	1.26379
$867$	0	0
$868$	13707.2	0.536005
$869$	−32572.9	−1.27153
$870$	0	0
$871$	−29307.4	−1.14012
$872$	31932.5	1.24010
$873$	0	0
$874$	148090.	5.73136
$875$	0	0
$876$	0	0
$877$	−15229.8	−0.586400	−0.293200	−	0.956051i	$-0.594720\pi$
$877$	−15229.8	−0.586400	−0.293200	+	0.956051i	$0.594720\pi$
$878$	−43216.8	−1.66116
$879$	0	0
$880$	0	0
$881$	−27891.7	−1.06662	−0.533311	−	0.845919i	$-0.679052\pi$
$881$	−27891.7	−1.06662	−0.533311	+	0.845919i	$0.679052\pi$
$882$	0	0
$883$	18300.3	0.697456	0.348728	−	0.937224i	$-0.386614\pi$
$883$	18300.3	0.697456	0.348728	+	0.937224i	$0.386614\pi$
$884$	−43260.6	−1.64594
$885$	0	0
$886$	64416.3	2.44256
$887$	19104.8	0.723197	0.361598	−	0.932334i	$-0.382231\pi$
$887$	19104.8	0.723197	0.361598	+	0.932334i	$0.382231\pi$
$888$	0	0
$889$	−12129.9	−0.457620
$890$	0	0
$891$	0	0
$892$	30375.7	1.14019
$893$	2350.10	0.0880661
$894$	0	0
$895$	0	0
$896$	−17980.5	−0.670410
$897$	0	0
$898$	−20426.8	−0.759077
$899$	4208.78	0.156141
$900$	0	0
$901$	−15556.0	−0.575189
$902$	35165.1	1.29808
$903$	0	0
$904$	−53899.9	−1.98306
$905$	0	0
$906$	0	0
$907$	−29518.8	−1.08066	−0.540328	−	0.841455i	$-0.681700\pi$
$907$	−29518.8	−1.08066	−0.540328	+	0.841455i	$0.681700\pi$
$908$	−12044.0	−0.440191
$909$	0	0
$910$	0	0
$911$	30322.2	1.10277	0.551383	−	0.834253i	$-0.314100\pi$
$911$	30322.2	1.10277	0.551383	+	0.834253i	$0.314100\pi$
$912$	0	0
$913$	−42727.0	−1.54880
$914$	−22907.0	−0.828991
$915$	0	0
$916$	−89895.9	−3.24263
$917$	17008.3	0.612501
$918$	0	0
$919$	−37780.9	−1.35612	−0.678062	−	0.735005i	$-0.737180\pi$
$919$	−37780.9	−1.35612	−0.678062	+	0.735005i	$0.737180\pi$
$920$	0	0
$921$	0	0
$922$	9499.48	0.339315
$923$	2774.93	0.0989575
$924$	0	0
$925$	0	0
$926$	35642.7	1.26489
$927$	0	0
$928$	−683.867	−0.0241908
$929$	−48493.3	−1.71261	−0.856305	−	0.516470i	$-0.827246\pi$
$929$	−48493.3	−1.71261	−0.856305	+	0.516470i	$0.827246\pi$
$930$	0	0
$931$	−6845.33	−0.240974
$932$	13602.7	0.478081
$933$	0	0
$934$	9244.81	0.323875
$935$	0	0
$936$	0	0
$937$	18661.7	0.650641	0.325320	−	0.945604i	$-0.394528\pi$
$937$	18661.7	0.650641	0.325320	+	0.945604i	$0.394528\pi$
$938$	16300.6	0.567413
$939$	0	0
$940$	0	0
$941$	31270.1	1.08329	0.541644	−	0.840608i	$-0.317802\pi$
$941$	31270.1	1.08329	0.541644	+	0.840608i	$0.317802\pi$
$942$	0	0
$943$	42453.7	1.46605
$944$	−7814.53	−0.269429
$945$	0	0
$946$	61948.0	2.12907
$947$	22405.3	0.768823	0.384412	−	0.923162i	$-0.374404\pi$
$947$	22405.3	0.768823	0.384412	+	0.923162i	$0.374404\pi$
$948$	0	0
$949$	−6179.55	−0.211377
$950$	0	0
$951$	0	0
$952$	11829.0	0.402711
$953$	−21777.9	−0.740247	−0.370123	−	0.928983i	$-0.620685\pi$
$953$	−21777.9	−0.740247	−0.370123	+	0.928983i	$0.620685\pi$
$954$	0	0
$955$	0	0
$956$	28587.2	0.967129
$957$	0	0
$958$	−51323.3	−1.73088
$959$	−10169.9	−0.342442
$960$	0	0
$961$	−14306.5	−0.480229
$962$	−70880.5	−2.37555
$963$	0	0
$964$	−37051.5	−1.23791
$965$	0	0
$966$	0	0
$967$	−14432.2	−0.479945	−0.239973	−	0.970780i	$-0.577138\pi$
$967$	−14432.2	−0.479945	−0.239973	+	0.970780i	$0.577138\pi$
$968$	1410.51	0.0468341
$969$	0	0
$970$	0	0
$971$	−6672.57	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$971$	−6672.57	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$972$	0	0
$973$	20492.3	0.675183
$974$	58156.7	1.91321
$975$	0	0
$976$	28316.5	0.928678
$977$	33300.2	1.09045	0.545224	−	0.838290i	$-0.316444\pi$
$977$	33300.2	1.09045	0.545224	+	0.838290i	$0.316444\pi$
$978$	0	0
$979$	22912.0	0.747979
$980$	0	0
$981$	0	0
$982$	−487.491	−0.0158416
$983$	−17243.3	−0.559486	−0.279743	−	0.960075i	$-0.590249\pi$
$983$	−17243.3	−0.559486	−0.279743	+	0.960075i	$0.590249\pi$
$984$	0	0
$985$	0	0
$986$	7387.99	0.238622
$987$	0	0
$988$	134796.	4.34053
$989$	74787.8	2.40456
$990$	0	0
$991$	−32266.7	−1.03429	−0.517147	−	0.855897i	$-0.673006\pi$
$991$	−32266.7	−1.03429	−0.517147	+	0.855897i	$0.673006\pi$
$992$	−2516.02	−0.0805278
$993$	0	0
$994$	−1543.40	−0.0492491
$995$	0	0
$996$	0	0
$997$	−14339.8	−0.455514	−0.227757	−	0.973718i	$-0.573139\pi$
$997$	−14339.8	−0.455514	−0.227757	+	0.973718i	$0.573139\pi$
$998$	−76059.8	−2.41246
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bg.1.4		4
3.2	odd	2		175.4.a.h.1.1	yes	4
5.4	even	2		1575.4.a.bl.1.1		4
15.2	even	4		175.4.b.f.99.1		8
15.8	even	4		175.4.b.f.99.8		8
15.14	odd	2		175.4.a.g.1.4	✓	4
21.20	even	2		1225.4.a.bd.1.1		4
105.104	even	2		1225.4.a.z.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.g.1.4	✓	4	15.14	odd	2
175.4.a.h.1.1	yes	4	3.2	odd	2
175.4.b.f.99.1		8	15.2	even	4
175.4.b.f.99.8		8	15.8	even	4
1225.4.a.z.1.4		4	105.104	even	2
1225.4.a.bd.1.1		4	21.20	even	2
1575.4.a.bg.1.4		4	1.1	even	1	trivial
1575.4.a.bl.1.1		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)

\(f(q)\)	\(=\)	\(q+4.87199 q^{2} +15.7363 q^{4} +7.00000 q^{7} +37.6910 q^{8} +O(q^{10})\)	\(q+4.87199 q^{2} +15.7363 q^{4} +7.00000 q^{7} +37.6910 q^{8} -36.9922 q^{11} -61.3165 q^{13} +34.1039 q^{14} +57.7401 q^{16} +44.8345 q^{17} -139.701 q^{19} -180.226 q^{22} -217.580 q^{23} -298.733 q^{26} +110.154 q^{28} +33.8226 q^{29} +124.437 q^{31} -20.2192 q^{32} +218.433 q^{34} +237.270 q^{37} -680.620 q^{38} -195.117 q^{41} -343.725 q^{43} -582.119 q^{44} -1060.05 q^{46} -16.8224 q^{47} +49.0000 q^{49} -964.894 q^{52} -346.965 q^{53} +263.837 q^{56} +164.783 q^{58} -135.340 q^{59} +490.414 q^{61} +606.255 q^{62} -560.428 q^{64} +477.969 q^{67} +705.529 q^{68} -45.2557 q^{71} +100.781 q^{73} +1155.98 q^{74} -2198.37 q^{76} -258.945 q^{77} +880.534 q^{79} -950.609 q^{82} +1155.03 q^{83} -1674.62 q^{86} -1394.27 q^{88} -619.374 q^{89} -429.216 q^{91} -3423.90 q^{92} -81.9585 q^{94} -231.195 q^{97} +238.727 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 36 q^{4} + 28 q^{7} - 27 q^{8}+O(q^{10}) \) 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

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