LMFDB - Embedded newform 1575.4.a.bo.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:	$5$
Coefficient field:	5.5.78066700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 18x^{3} + 7x^{2} + 30x - 10 $ x^5 - x^4 - 18x^3 + 7x^2 + 30*x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 5 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.35311$ 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)

$f(q)$	$=$	$q+2.20666 q^{2} -3.13065 q^{4} +7.00000 q^{7} -24.5616 q^{8} +O(q^{10})$	$q+2.20666 q^{2} -3.13065 q^{4} +7.00000 q^{7} -24.5616 q^{8} -56.2010 q^{11} +38.9026 q^{13} +15.4466 q^{14} -29.1538 q^{16} +119.322 q^{17} -13.0045 q^{19} -124.016 q^{22} +130.565 q^{23} +85.8448 q^{26} -21.9146 q^{28} -77.9925 q^{29} +61.0660 q^{31} +132.160 q^{32} +263.303 q^{34} +167.391 q^{37} -28.6964 q^{38} -436.142 q^{41} -393.030 q^{43} +175.946 q^{44} +288.112 q^{46} -365.271 q^{47} +49.0000 q^{49} -121.791 q^{52} +282.048 q^{53} -171.931 q^{56} -172.103 q^{58} -414.842 q^{59} -563.802 q^{61} +134.752 q^{62} +524.862 q^{64} +395.230 q^{67} -373.556 q^{68} -103.990 q^{71} -128.026 q^{73} +369.376 q^{74} +40.7125 q^{76} -393.407 q^{77} -641.999 q^{79} -962.417 q^{82} -512.010 q^{83} -867.283 q^{86} +1380.38 q^{88} -1225.10 q^{89} +272.318 q^{91} -408.753 q^{92} -806.028 q^{94} +186.760 q^{97} +108.126 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8}+O(q^{10}) $ 5 * q - q^2 + 27 * q^4 + 35 * q^7 - 33 * q^8	$ 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8} - 66 q^{11} - 2 q^{13} - 7 q^{14} + 155 q^{16} - 108 q^{17} + 174 q^{19} - 506 q^{22} + 116 q^{23} - 446 q^{26} + 189 q^{28} - 370 q^{29} + 342 q^{31} + 55 q^{32} + 112 q^{34} - 408 q^{37} + 34 q^{38} - 802 q^{41} - 584 q^{43} - 290 q^{44} + 640 q^{46} - 716 q^{47} + 245 q^{49} + 338 q^{52} - 98 q^{53} - 231 q^{56} + 482 q^{58} - 704 q^{59} + 650 q^{61} - 2070 q^{62} + 75 q^{64} + 180 q^{67} - 4520 q^{68} - 1470 q^{71} - 534 q^{73} + 1312 q^{74} + 4370 q^{76} - 462 q^{77} - 820 q^{79} - 1338 q^{82} - 1520 q^{83} - 832 q^{86} - 3258 q^{88} - 286 q^{89} - 14 q^{91} - 1288 q^{92} + 2540 q^{94} + 278 q^{97} - 49 q^{98}+O(q^{100}) $ 5 * q - q^2 + 27 * q^4 + 35 * q^7 - 33 * q^8 - 66 * q^11 - 2 * q^13 - 7 * q^14 + 155 * q^16 - 108 * q^17 + 174 * q^19 - 506 * q^22 + 116 * q^23 - 446 * q^26 + 189 * q^28 - 370 * q^29 + 342 * q^31 + 55 * q^32 + 112 * q^34 - 408 * q^37 + 34 * q^38 - 802 * q^41 - 584 * q^43 - 290 * q^44 + 640 * q^46 - 716 * q^47 + 245 * q^49 + 338 * q^52 - 98 * q^53 - 231 * q^56 + 482 * q^58 - 704 * q^59 + 650 * q^61 - 2070 * q^62 + 75 * q^64 + 180 * q^67 - 4520 * q^68 - 1470 * q^71 - 534 * q^73 + 1312 * q^74 + 4370 * q^76 - 462 * q^77 - 820 * q^79 - 1338 * q^82 - 1520 * q^83 - 832 * q^86 - 3258 * q^88 - 286 * q^89 - 14 * q^91 - 1288 * q^92 + 2540 * q^94 + 278 * q^97 - 49 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.20666	0.780172	0.390086	−	0.920778i	$-0.372445\pi$
$2$	2.20666	0.780172	0.390086	+	0.920778i	$0.372445\pi$
$3$	0	0
$3$	0	0
$4$	−3.13065	−0.391332
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−24.5616	−1.08548
$9$	0	0
$10$	0	0
$11$	−56.2010	−1.54048	−0.770238	−	0.637757i	$-0.779862\pi$
$11$	−56.2010	−1.54048	−0.770238	+	0.637757i	$0.779862\pi$
$12$	0	0
$13$	38.9026	0.829972	0.414986	−	0.909828i	$-0.363786\pi$
$13$	38.9026	0.829972	0.414986	+	0.909828i	$0.363786\pi$
$14$	15.4466	0.294877
$15$	0	0
$16$	−29.1538	−0.455528
$17$	119.322	1.70235	0.851173	−	0.524886i	$-0.175892\pi$
$17$	119.322	1.70235	0.851173	+	0.524886i	$0.175892\pi$
$18$	0	0
$19$	−13.0045	−0.157023	−0.0785113	−	0.996913i	$-0.525017\pi$
$19$	−13.0045	−0.157023	−0.0785113	+	0.996913i	$0.525017\pi$
$20$	0	0
$21$	0	0
$22$	−124.016	−1.20184
$23$	130.565	1.18368	0.591840	−	0.806055i	$-0.298402\pi$
$23$	130.565	1.18368	0.591840	+	0.806055i	$0.298402\pi$
$24$	0	0
$25$	0	0
$26$	85.8448	0.647521
$27$	0	0
$28$	−21.9146	−0.147910
$29$	−77.9925	−0.499408	−0.249704	−	0.968322i	$-0.580333\pi$
$29$	−77.9925	−0.499408	−0.249704	+	0.968322i	$0.580333\pi$
$30$	0	0
$31$	61.0660	0.353799	0.176900	−	0.984229i	$-0.443393\pi$
$31$	61.0660	0.353799	0.176900	+	0.984229i	$0.443393\pi$
$32$	132.160	0.730088
$33$	0	0
$34$	263.303	1.32812
$35$	0	0
$36$	0	0
$37$	167.391	0.743757	0.371878	−	0.928282i	$-0.378714\pi$
$37$	167.391	0.743757	0.371878	+	0.928282i	$0.378714\pi$
$38$	−28.6964	−0.122505
$39$	0	0
$40$	0	0
$41$	−436.142	−1.66132	−0.830658	−	0.556783i	$-0.812036\pi$
$41$	−436.142	−1.66132	−0.830658	+	0.556783i	$0.812036\pi$
$42$	0	0
$43$	−393.030	−1.39387	−0.696936	−	0.717134i	$-0.745454\pi$
$43$	−393.030	−1.39387	−0.696936	+	0.717134i	$0.745454\pi$
$44$	175.946	0.602837
$45$	0	0
$46$	288.112	0.923474
$47$	−365.271	−1.13362	−0.566811	−	0.823848i	$-0.691823\pi$
$47$	−365.271	−1.13362	−0.566811	+	0.823848i	$0.691823\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−121.791	−0.324794
$53$	282.048	0.730987	0.365494	−	0.930814i	$-0.380900\pi$
$53$	282.048	0.730987	0.365494	+	0.930814i	$0.380900\pi$
$54$	0	0
$55$	0	0
$56$	−171.931	−0.410272
$57$	0	0
$58$	−172.103	−0.389624
$59$	−414.842	−0.915388	−0.457694	−	0.889110i	$-0.651324\pi$
$59$	−414.842	−0.915388	−0.457694	+	0.889110i	$0.651324\pi$
$60$	0	0
$61$	−563.802	−1.18340	−0.591701	−	0.806158i	$-0.701543\pi$
$61$	−563.802	−1.18340	−0.591701	+	0.806158i	$0.701543\pi$
$62$	134.752	0.276024
$63$	0	0
$64$	524.862	1.02512
$65$	0	0
$66$	0	0
$67$	395.230	0.720673	0.360336	−	0.932822i	$-0.382662\pi$
$67$	395.230	0.720673	0.360336	+	0.932822i	$0.382662\pi$
$68$	−373.556	−0.666182
$69$	0	0
$70$	0	0
$71$	−103.990	−0.173821	−0.0869107	−	0.996216i	$-0.527699\pi$
$71$	−103.990	−0.173821	−0.0869107	+	0.996216i	$0.527699\pi$
$72$	0	0
$73$	−128.026	−0.205264	−0.102632	−	0.994719i	$-0.532726\pi$
$73$	−128.026	−0.205264	−0.102632	+	0.994719i	$0.532726\pi$
$74$	369.376	0.580258
$75$	0	0
$76$	40.7125	0.0614479
$77$	−393.407	−0.582245
$78$	0	0
$79$	−641.999	−0.914310	−0.457155	−	0.889387i	$-0.651132\pi$
$79$	−641.999	−0.914310	−0.457155	+	0.889387i	$0.651132\pi$
$80$	0	0
$81$	0	0
$82$	−962.417	−1.29611
$83$	−512.010	−0.677113	−0.338557	−	0.940946i	$-0.609939\pi$
$83$	−512.010	−0.677113	−0.338557	+	0.940946i	$0.609939\pi$
$84$	0	0
$85$	0	0
$86$	−867.283	−1.08746
$87$	0	0
$88$	1380.38	1.67215
$89$	−1225.10	−1.45911	−0.729554	−	0.683923i	$-0.760272\pi$
$89$	−1225.10	−1.45911	−0.729554	+	0.683923i	$0.760272\pi$
$90$	0	0
$91$	272.318	0.313700
$92$	−408.753	−0.463212
$93$	0	0
$94$	−806.028	−0.884420
$95$	0	0
$96$	0	0
$97$	186.760	0.195491	0.0977454	−	0.995211i	$-0.468837\pi$
$97$	186.760	0.195491	0.0977454	+	0.995211i	$0.468837\pi$
$98$	108.126	0.111453
$99$	0	0
$100$	0	0
$101$	−1650.68	−1.62623	−0.813114	−	0.582104i	$-0.802230\pi$
$101$	−1650.68	−1.62623	−0.813114	+	0.582104i	$0.802230\pi$
$102$	0	0
$103$	−72.1876	−0.0690568	−0.0345284	−	0.999404i	$-0.510993\pi$
$103$	−72.1876	−0.0690568	−0.0345284	+	0.999404i	$0.510993\pi$
$104$	−955.508	−0.900916
$105$	0	0
$106$	622.385	0.570296
$107$	1202.55	1.08649	0.543246	−	0.839574i	$-0.317195\pi$
$107$	1202.55	1.08649	0.543246	+	0.839574i	$0.317195\pi$
$108$	0	0
$109$	−1551.36	−1.36324	−0.681622	−	0.731704i	$-0.738725\pi$
$109$	−1551.36	−1.36324	−0.681622	+	0.731704i	$0.738725\pi$
$110$	0	0
$111$	0	0
$112$	−204.076	−0.172173
$113$	−2080.90	−1.73234	−0.866170	−	0.499749i	$-0.833426\pi$
$113$	−2080.90	−1.73234	−0.866170	+	0.499749i	$0.833426\pi$
$114$	0	0
$115$	0	0
$116$	244.167	0.195434
$117$	0	0
$118$	−915.416	−0.714160
$119$	835.255	0.643426
$120$	0	0
$121$	1827.55	1.37306
$122$	−1244.12	−0.923257
$123$	0	0
$124$	−191.177	−0.138453
$125$	0	0
$126$	0	0
$127$	1414.70	0.988461	0.494231	−	0.869331i	$-0.335450\pi$
$127$	1414.70	0.988461	0.494231	+	0.869331i	$0.335450\pi$
$128$	100.912	0.0696832
$129$	0	0
$130$	0	0
$131$	2472.51	1.64904	0.824520	−	0.565833i	$-0.191445\pi$
$131$	2472.51	1.64904	0.824520	+	0.565833i	$0.191445\pi$
$132$	0	0
$133$	−91.0313	−0.0593490
$134$	872.139	0.562249
$135$	0	0
$136$	−2930.74	−1.84786
$137$	214.391	0.133698	0.0668491	−	0.997763i	$-0.478705\pi$
$137$	214.391	0.133698	0.0668491	+	0.997763i	$0.478705\pi$
$138$	0	0
$139$	942.774	0.575288	0.287644	−	0.957737i	$-0.407128\pi$
$139$	942.774	0.575288	0.287644	+	0.957737i	$0.407128\pi$
$140$	0	0
$141$	0	0
$142$	−229.470	−0.135611
$143$	−2186.36	−1.27855
$144$	0	0
$145$	0	0
$146$	−282.509	−0.160141
$147$	0	0
$148$	−524.045	−0.291056
$149$	−1693.07	−0.930882	−0.465441	−	0.885079i	$-0.654104\pi$
$149$	−1693.07	−0.930882	−0.465441	+	0.885079i	$0.654104\pi$
$150$	0	0
$151$	2519.69	1.35795	0.678973	−	0.734163i	$-0.262425\pi$
$151$	2519.69	1.35795	0.678973	+	0.734163i	$0.262425\pi$
$152$	319.410	0.170445
$153$	0	0
$154$	−868.115	−0.454251
$155$	0	0
$156$	0	0
$157$	1621.48	0.824258	0.412129	−	0.911125i	$-0.364785\pi$
$157$	1621.48	0.824258	0.412129	+	0.911125i	$0.364785\pi$
$158$	−1416.67	−0.713319
$159$	0	0
$160$	0	0
$161$	913.954	0.447389
$162$	0	0
$163$	925.194	0.444582	0.222291	−	0.974980i	$-0.428647\pi$
$163$	925.194	0.444582	0.222291	+	0.974980i	$0.428647\pi$
$164$	1365.41	0.650126
$165$	0	0
$166$	−1129.83	−0.528265
$167$	−2681.55	−1.24254	−0.621271	−	0.783596i	$-0.713383\pi$
$167$	−2681.55	−1.24254	−0.621271	+	0.783596i	$0.713383\pi$
$168$	0	0
$169$	−683.589	−0.311147
$170$	0	0
$171$	0	0
$172$	1230.44	0.545466
$173$	−287.591	−0.126388	−0.0631940	−	0.998001i	$-0.520129\pi$
$173$	−287.591	−0.126388	−0.0631940	+	0.998001i	$0.520129\pi$
$174$	0	0
$175$	0	0
$176$	1638.47	0.701729
$177$	0	0
$178$	−2703.38	−1.13835
$179$	−3683.47	−1.53808	−0.769038	−	0.639203i	$-0.779265\pi$
$179$	−3683.47	−1.53808	−0.769038	+	0.639203i	$0.779265\pi$
$180$	0	0
$181$	3132.65	1.28645	0.643227	−	0.765676i	$-0.277595\pi$
$181$	3132.65	1.28645	0.643227	+	0.765676i	$0.277595\pi$
$182$	600.913	0.244740
$183$	0	0
$184$	−3206.88	−1.28486
$185$	0	0
$186$	0	0
$187$	−6706.02	−2.62242
$188$	1143.54	0.443622
$189$	0	0
$190$	0	0
$191$	−1586.93	−0.601184	−0.300592	−	0.953753i	$-0.597184\pi$
$191$	−1586.93	−0.601184	−0.300592	+	0.953753i	$0.597184\pi$
$192$	0	0
$193$	−5179.00	−1.93157	−0.965783	−	0.259352i	$-0.916491\pi$
$193$	−5179.00	−1.93157	−0.965783	+	0.259352i	$0.916491\pi$
$194$	412.116	0.152516
$195$	0	0
$196$	−153.402	−0.0559045
$197$	−903.798	−0.326868	−0.163434	−	0.986554i	$-0.552257\pi$
$197$	−903.798	−0.326868	−0.163434	+	0.986554i	$0.552257\pi$
$198$	0	0
$199$	−1171.51	−0.417317	−0.208659	−	0.977989i	$-0.566910\pi$
$199$	−1171.51	−0.417317	−0.208659	+	0.977989i	$0.566910\pi$
$200$	0	0
$201$	0	0
$202$	−3642.50	−1.26874
$203$	−545.947	−0.188759
$204$	0	0
$205$	0	0
$206$	−159.293	−0.0538762
$207$	0	0
$208$	−1134.16	−0.378075
$209$	730.864	0.241889
$210$	0	0
$211$	−1103.16	−0.359928	−0.179964	−	0.983673i	$-0.557598\pi$
$211$	−1103.16	−0.359928	−0.179964	+	0.983673i	$0.557598\pi$
$212$	−882.996	−0.286059
$213$	0	0
$214$	2653.61	0.847651
$215$	0	0
$216$	0	0
$217$	427.462	0.133724
$218$	−3423.33	−1.06357
$219$	0	0
$220$	0	0
$221$	4641.94	1.41290
$222$	0	0
$223$	4079.95	1.22517	0.612586	−	0.790404i	$-0.290129\pi$
$223$	4079.95	1.22517	0.612586	+	0.790404i	$0.290129\pi$
$224$	925.120	0.275947
$225$	0	0
$226$	−4591.83	−1.35152
$227$	−931.964	−0.272496	−0.136248	−	0.990675i	$-0.543504\pi$
$227$	−931.964	−0.272496	−0.136248	+	0.990675i	$0.543504\pi$
$228$	0	0
$229$	−1471.55	−0.424641	−0.212321	−	0.977200i	$-0.568102\pi$
$229$	−1471.55	−0.424641	−0.212321	+	0.977200i	$0.568102\pi$
$230$	0	0
$231$	0	0
$232$	1915.62	0.542097
$233$	−2479.06	−0.697034	−0.348517	−	0.937303i	$-0.613315\pi$
$233$	−2479.06	−0.697034	−0.348517	+	0.937303i	$0.613315\pi$
$234$	0	0
$235$	0	0
$236$	1298.73	0.358220
$237$	0	0
$238$	1843.12	0.501983
$239$	954.068	0.258216	0.129108	−	0.991631i	$-0.458789\pi$
$239$	954.068	0.258216	0.129108	+	0.991631i	$0.458789\pi$
$240$	0	0
$241$	−5297.02	−1.41581	−0.707906	−	0.706306i	$-0.750360\pi$
$241$	−5297.02	−1.41581	−0.707906	+	0.706306i	$0.750360\pi$
$242$	4032.77	1.07123
$243$	0	0
$244$	1765.07	0.463103
$245$	0	0
$246$	0	0
$247$	−505.907	−0.130324
$248$	−1499.88	−0.384041
$249$	0	0
$250$	0	0
$251$	−1855.17	−0.466524	−0.233262	−	0.972414i	$-0.574940\pi$
$251$	−1855.17	−0.466524	−0.233262	+	0.972414i	$0.574940\pi$
$252$	0	0
$253$	−7337.87	−1.82343
$254$	3121.77	0.771170
$255$	0	0
$256$	−3976.22	−0.970757
$257$	−6233.09	−1.51288	−0.756438	−	0.654065i	$-0.773062\pi$
$257$	−6233.09	−1.51288	−0.756438	+	0.654065i	$0.773062\pi$
$258$	0	0
$259$	1171.74	0.281114
$260$	0	0
$261$	0	0
$262$	5455.99	1.28653
$263$	−1184.50	−0.277716	−0.138858	−	0.990312i	$-0.544343\pi$
$263$	−1184.50	−0.277716	−0.138858	+	0.990312i	$0.544343\pi$
$264$	0	0
$265$	0	0
$266$	−200.875	−0.0463024
$267$	0	0
$268$	−1237.33	−0.282022
$269$	1916.03	0.434283	0.217141	−	0.976140i	$-0.430327\pi$
$269$	1916.03	0.434283	0.217141	+	0.976140i	$0.430327\pi$
$270$	0	0
$271$	−1168.95	−0.262025	−0.131013	−	0.991381i	$-0.541823\pi$
$271$	−1168.95	−0.262025	−0.131013	+	0.991381i	$0.541823\pi$
$272$	−3478.69	−0.775465
$273$	0	0
$274$	473.088	0.104308
$275$	0	0
$276$	0	0
$277$	−7269.54	−1.57684	−0.788419	−	0.615138i	$-0.789100\pi$
$277$	−7269.54	−1.57684	−0.788419	+	0.615138i	$0.789100\pi$
$278$	2080.38	0.448824
$279$	0	0
$280$	0	0
$281$	−298.126	−0.0632908	−0.0316454	−	0.999499i	$-0.510075\pi$
$281$	−298.126	−0.0632908	−0.0316454	+	0.999499i	$0.510075\pi$
$282$	0	0
$283$	4496.30	0.944444	0.472222	−	0.881480i	$-0.343452\pi$
$283$	4496.30	0.944444	0.472222	+	0.881480i	$0.343452\pi$
$284$	325.556	0.0680218
$285$	0	0
$286$	−4824.56	−0.997490
$287$	−3053.00	−0.627919
$288$	0	0
$289$	9324.77	1.89798
$290$	0	0
$291$	0	0
$292$	400.804	0.0803264
$293$	−1644.33	−0.327859	−0.163929	−	0.986472i	$-0.552417\pi$
$293$	−1644.33	−0.327859	−0.163929	+	0.986472i	$0.552417\pi$
$294$	0	0
$295$	0	0
$296$	−4111.40	−0.807331
$297$	0	0
$298$	−3736.02	−0.726248
$299$	5079.31	0.982421
$300$	0	0
$301$	−2751.21	−0.526834
$302$	5560.11	1.05943
$303$	0	0
$304$	379.129	0.0715281
$305$	0	0
$306$	0	0
$307$	4726.18	0.878623	0.439312	−	0.898335i	$-0.355222\pi$
$307$	4726.18	0.878623	0.439312	+	0.898335i	$0.355222\pi$
$308$	1231.62	0.227851
$309$	0	0
$310$	0	0
$311$	4853.99	0.885031	0.442515	−	0.896761i	$-0.354086\pi$
$311$	4853.99	0.885031	0.442515	+	0.896761i	$0.354086\pi$
$312$	0	0
$313$	−1690.87	−0.305348	−0.152674	−	0.988277i	$-0.548788\pi$
$313$	−1690.87	−0.305348	−0.152674	+	0.988277i	$0.548788\pi$
$314$	3578.06	0.643063
$315$	0	0
$316$	2009.88	0.357799
$317$	−3878.38	−0.687166	−0.343583	−	0.939122i	$-0.611641\pi$
$317$	−3878.38	−0.687166	−0.343583	+	0.939122i	$0.611641\pi$
$318$	0	0
$319$	4383.25	0.769326
$320$	0	0
$321$	0	0
$322$	2016.78	0.349040
$323$	−1551.72	−0.267307
$324$	0	0
$325$	0	0
$326$	2041.59	0.346850
$327$	0	0
$328$	10712.3	1.80332
$329$	−2556.90	−0.428469
$330$	0	0
$331$	9927.71	1.64857	0.824284	−	0.566176i	$-0.191578\pi$
$331$	9927.71	1.64857	0.824284	+	0.566176i	$0.191578\pi$
$332$	1602.93	0.264976
$333$	0	0
$334$	−5917.26	−0.969396
$335$	0	0
$336$	0	0
$337$	5283.88	0.854099	0.427050	−	0.904228i	$-0.359553\pi$
$337$	5283.88	0.854099	0.427050	+	0.904228i	$0.359553\pi$
$338$	−1508.45	−0.242748
$339$	0	0
$340$	0	0
$341$	−3431.97	−0.545019
$342$	0	0
$343$	343.000	0.0539949
$344$	9653.42	1.51302
$345$	0	0
$346$	−634.615	−0.0986044
$347$	10548.3	1.63188	0.815940	−	0.578137i	$-0.196220\pi$
$347$	10548.3	1.63188	0.815940	+	0.578137i	$0.196220\pi$
$348$	0	0
$349$	628.411	0.0963841	0.0481921	−	0.998838i	$-0.484654\pi$
$349$	628.411	0.0963841	0.0481921	+	0.998838i	$0.484654\pi$
$350$	0	0
$351$	0	0
$352$	−7427.52	−1.12468
$353$	−2548.17	−0.384209	−0.192104	−	0.981375i	$-0.561531\pi$
$353$	−2548.17	−0.384209	−0.192104	+	0.981375i	$0.561531\pi$
$354$	0	0
$355$	0	0
$356$	3835.37	0.570995
$357$	0	0
$358$	−8128.17	−1.19996
$359$	−13046.3	−1.91799	−0.958996	−	0.283420i	$-0.908531\pi$
$359$	−13046.3	−1.91799	−0.958996	+	0.283420i	$0.908531\pi$
$360$	0	0
$361$	−6689.88	−0.975344
$362$	6912.69	1.00366
$363$	0	0
$364$	−852.534	−0.122761
$365$	0	0
$366$	0	0
$367$	8068.23	1.14757	0.573785	−	0.819006i	$-0.305475\pi$
$367$	8068.23	1.14757	0.573785	+	0.819006i	$0.305475\pi$
$368$	−3806.46	−0.539199
$369$	0	0
$370$	0	0
$371$	1974.34	0.276287
$372$	0	0
$373$	−3623.32	−0.502972	−0.251486	−	0.967861i	$-0.580919\pi$
$373$	−3623.32	−0.502972	−0.251486	+	0.967861i	$0.580919\pi$
$374$	−14797.9	−2.04594
$375$	0	0
$376$	8971.62	1.23052
$377$	−3034.11	−0.414495
$378$	0	0
$379$	7486.58	1.01467	0.507335	−	0.861749i	$-0.330631\pi$
$379$	7486.58	1.01467	0.507335	+	0.861749i	$0.330631\pi$
$380$	0	0
$381$	0	0
$382$	−3501.81	−0.469027
$383$	−8926.58	−1.19093	−0.595466	−	0.803381i	$-0.703033\pi$
$383$	−8926.58	−1.19093	−0.595466	+	0.803381i	$0.703033\pi$
$384$	0	0
$385$	0	0
$386$	−11428.3	−1.50695
$387$	0	0
$388$	−584.681	−0.0765018
$389$	12600.1	1.64228	0.821141	−	0.570725i	$-0.193338\pi$
$389$	12600.1	1.64228	0.821141	+	0.570725i	$0.193338\pi$
$390$	0	0
$391$	15579.3	2.01503
$392$	−1203.52	−0.155068
$393$	0	0
$394$	−1994.37	−0.255013
$395$	0	0
$396$	0	0
$397$	−12713.4	−1.60722	−0.803612	−	0.595154i	$-0.797091\pi$
$397$	−12713.4	−1.60722	−0.803612	+	0.595154i	$0.797091\pi$
$398$	−2585.12	−0.325579
$399$	0	0
$400$	0	0
$401$	6133.51	0.763822	0.381911	−	0.924199i	$-0.375266\pi$
$401$	6133.51	0.763822	0.381911	+	0.924199i	$0.375266\pi$
$402$	0	0
$403$	2375.62	0.293643
$404$	5167.72	0.636395
$405$	0	0
$406$	−1204.72	−0.147264
$407$	−9407.56	−1.14574
$408$	0	0
$409$	10600.3	1.28154	0.640769	−	0.767733i	$-0.278615\pi$
$409$	10600.3	1.28154	0.640769	+	0.767733i	$0.278615\pi$
$410$	0	0
$411$	0	0
$412$	225.994	0.0270241
$413$	−2903.90	−0.345984
$414$	0	0
$415$	0	0
$416$	5141.37	0.605953
$417$	0	0
$418$	1612.77	0.188715
$419$	−4296.43	−0.500941	−0.250470	−	0.968124i	$-0.580585\pi$
$419$	−4296.43	−0.500941	−0.250470	+	0.968124i	$0.580585\pi$
$420$	0	0
$421$	3916.78	0.453425	0.226713	−	0.973962i	$-0.427202\pi$
$421$	3916.78	0.453425	0.226713	+	0.973962i	$0.427202\pi$
$422$	−2434.30	−0.280806
$423$	0	0
$424$	−6927.55	−0.793471
$425$	0	0
$426$	0	0
$427$	−3946.62	−0.447284
$428$	−3764.76	−0.425179
$429$	0	0
$430$	0	0
$431$	−13408.9	−1.49857	−0.749287	−	0.662246i	$-0.769604\pi$
$431$	−13408.9	−1.49857	−0.749287	+	0.662246i	$0.769604\pi$
$432$	0	0
$433$	7792.31	0.864837	0.432419	−	0.901673i	$-0.357660\pi$
$433$	7792.31	0.864837	0.432419	+	0.901673i	$0.357660\pi$
$434$	943.263	0.104327
$435$	0	0
$436$	4856.78	0.533481
$437$	−1697.93	−0.185865
$438$	0	0
$439$	1039.29	0.112990	0.0564948	−	0.998403i	$-0.482008\pi$
$439$	1039.29	0.112990	0.0564948	+	0.998403i	$0.482008\pi$
$440$	0	0
$441$	0	0
$442$	10243.2	1.10230
$443$	2846.33	0.305267	0.152633	−	0.988283i	$-0.451225\pi$
$443$	2846.33	0.305267	0.152633	+	0.988283i	$0.451225\pi$
$444$	0	0
$445$	0	0
$446$	9003.05	0.955845
$447$	0	0
$448$	3674.04	0.387460
$449$	−7472.64	−0.785425	−0.392713	−	0.919661i	$-0.628463\pi$
$449$	−7472.64	−0.785425	−0.392713	+	0.919661i	$0.628463\pi$
$450$	0	0
$451$	24511.6	2.55922
$452$	6514.57	0.677920
$453$	0	0
$454$	−2056.53	−0.212594
$455$	0	0
$456$	0	0
$457$	−11014.3	−1.12742	−0.563708	−	0.825974i	$-0.690626\pi$
$457$	−11014.3	−1.12742	−0.563708	+	0.825974i	$0.690626\pi$
$458$	−3247.21	−0.331293
$459$	0	0
$460$	0	0
$461$	−7944.67	−0.802647	−0.401323	−	0.915936i	$-0.631450\pi$
$461$	−7944.67	−0.802647	−0.401323	+	0.915936i	$0.631450\pi$
$462$	0	0
$463$	−7627.25	−0.765591	−0.382795	−	0.923833i	$-0.625039\pi$
$463$	−7627.25	−0.765591	−0.382795	+	0.923833i	$0.625039\pi$
$464$	2273.77	0.227494
$465$	0	0
$466$	−5470.45	−0.543806
$467$	3284.28	0.325436	0.162718	−	0.986673i	$-0.447974\pi$
$467$	3284.28	0.325436	0.162718	+	0.986673i	$0.447974\pi$
$468$	0	0
$469$	2766.61	0.272389
$470$	0	0
$471$	0	0
$472$	10189.2	0.993633
$473$	22088.6	2.14722
$474$	0	0
$475$	0	0
$476$	−2614.89	−0.251793
$477$	0	0
$478$	2105.30	0.201453
$479$	2909.45	0.277528	0.138764	−	0.990325i	$-0.455687\pi$
$479$	2909.45	0.277528	0.138764	+	0.990325i	$0.455687\pi$
$480$	0	0
$481$	6511.96	0.617297
$482$	−11688.7	−1.10458
$483$	0	0
$484$	−5721.42	−0.537323
$485$	0	0
$486$	0	0
$487$	−2201.84	−0.204876	−0.102438	−	0.994739i	$-0.532664\pi$
$487$	−2201.84	−0.204876	−0.102438	+	0.994739i	$0.532664\pi$
$488$	13847.9	1.28456
$489$	0	0
$490$	0	0
$491$	11827.6	1.08711	0.543556	−	0.839373i	$-0.317078\pi$
$491$	11827.6	1.08711	0.543556	+	0.839373i	$0.317078\pi$
$492$	0	0
$493$	−9306.23	−0.850165
$494$	−1116.37	−0.101675
$495$	0	0
$496$	−1780.30	−0.161165
$497$	−727.928	−0.0656983
$498$	0	0
$499$	1408.66	0.126374	0.0631868	−	0.998002i	$-0.479874\pi$
$499$	1408.66	0.126374	0.0631868	+	0.998002i	$0.479874\pi$
$500$	0	0
$501$	0	0
$502$	−4093.74	−0.363969
$503$	11018.9	0.976758	0.488379	−	0.872632i	$-0.337588\pi$
$503$	11018.9	0.976758	0.488379	+	0.872632i	$0.337588\pi$
$504$	0	0
$505$	0	0
$506$	−16192.2	−1.42259
$507$	0	0
$508$	−4428.95	−0.386816
$509$	−7032.93	−0.612434	−0.306217	−	0.951962i	$-0.599063\pi$
$509$	−7032.93	−0.612434	−0.306217	+	0.951962i	$0.599063\pi$
$510$	0	0
$511$	−896.180	−0.0775825
$512$	−9581.46	−0.827041
$513$	0	0
$514$	−13754.3	−1.18030
$515$	0	0
$516$	0	0
$517$	20528.6	1.74632
$518$	2585.63	0.219317
$519$	0	0
$520$	0	0
$521$	−3049.04	−0.256394	−0.128197	−	0.991749i	$-0.540919\pi$
$521$	−3049.04	−0.256394	−0.128197	+	0.991749i	$0.540919\pi$
$522$	0	0
$523$	8714.06	0.728564	0.364282	−	0.931289i	$-0.381314\pi$
$523$	8714.06	0.728564	0.364282	+	0.931289i	$0.381314\pi$
$524$	−7740.58	−0.645322
$525$	0	0
$526$	−2613.79	−0.216666
$527$	7286.52	0.602288
$528$	0	0
$529$	4880.17	0.401099
$530$	0	0
$531$	0	0
$532$	284.987	0.0232251
$533$	−16967.1	−1.37885
$534$	0	0
$535$	0	0
$536$	−9707.48	−0.782275
$537$	0	0
$538$	4228.02	0.338815
$539$	−2753.85	−0.220068
$540$	0	0
$541$	−5999.45	−0.476778	−0.238389	−	0.971170i	$-0.576619\pi$
$541$	−5999.45	−0.476778	−0.238389	+	0.971170i	$0.576619\pi$
$542$	−2579.48	−0.204425
$543$	0	0
$544$	15769.6	1.24286
$545$	0	0
$546$	0	0
$547$	7759.95	0.606566	0.303283	−	0.952901i	$-0.401917\pi$
$547$	7759.95	0.606566	0.303283	+	0.952901i	$0.401917\pi$
$548$	−671.184	−0.0523203
$549$	0	0
$550$	0	0
$551$	1014.25	0.0784184
$552$	0	0
$553$	−4493.99	−0.345577
$554$	−16041.4	−1.23021
$555$	0	0
$556$	−2951.50	−0.225128
$557$	6392.82	0.486306	0.243153	−	0.969988i	$-0.421818\pi$
$557$	6392.82	0.486306	0.243153	+	0.969988i	$0.421818\pi$
$558$	0	0
$559$	−15289.9	−1.15687
$560$	0	0
$561$	0	0
$562$	−657.863	−0.0493777
$563$	7682.70	0.575111	0.287555	−	0.957764i	$-0.407157\pi$
$563$	7682.70	0.575111	0.287555	+	0.957764i	$0.407157\pi$
$564$	0	0
$565$	0	0
$566$	9921.81	0.736828
$567$	0	0
$568$	2554.15	0.188679
$569$	−143.175	−0.0105487	−0.00527434	−	0.999986i	$-0.501679\pi$
$569$	−143.175	−0.0105487	−0.00527434	+	0.999986i	$0.501679\pi$
$570$	0	0
$571$	1077.72	0.0789863	0.0394932	−	0.999220i	$-0.487426\pi$
$571$	1077.72	0.0789863	0.0394932	+	0.999220i	$0.487426\pi$
$572$	6844.74	0.500338
$573$	0	0
$574$	−6736.92	−0.489884
$575$	0	0
$576$	0	0
$577$	−12651.7	−0.912818	−0.456409	−	0.889770i	$-0.650865\pi$
$577$	−12651.7	−0.912818	−0.456409	+	0.889770i	$0.650865\pi$
$578$	20576.6	1.48075
$579$	0	0
$580$	0	0
$581$	−3584.07	−0.255925
$582$	0	0
$583$	−15851.4	−1.12607
$584$	3144.51	0.222810
$585$	0	0
$586$	−3628.47	−0.255786
$587$	2920.89	0.205380	0.102690	−	0.994713i	$-0.467255\pi$
$587$	2920.89	0.205380	0.102690	+	0.994713i	$0.467255\pi$
$588$	0	0
$589$	−794.131	−0.0555545
$590$	0	0
$591$	0	0
$592$	−4880.09	−0.338802
$593$	9801.70	0.678765	0.339382	−	0.940648i	$-0.389782\pi$
$593$	9801.70	0.678765	0.339382	+	0.940648i	$0.389782\pi$
$594$	0	0
$595$	0	0
$596$	5300.41	0.364284
$597$	0	0
$598$	11208.3	0.766458
$599$	6992.54	0.476974	0.238487	−	0.971146i	$-0.423349\pi$
$599$	6992.54	0.476974	0.238487	+	0.971146i	$0.423349\pi$
$600$	0	0
$601$	−26159.1	−1.77546	−0.887730	−	0.460364i	$-0.847719\pi$
$601$	−26159.1	−1.77546	−0.887730	+	0.460364i	$0.847719\pi$
$602$	−6070.98	−0.411021
$603$	0	0
$604$	−7888.29	−0.531407
$605$	0	0
$606$	0	0
$607$	264.526	0.0176883	0.00884415	−	0.999961i	$-0.497185\pi$
$607$	264.526	0.0176883	0.00884415	+	0.999961i	$0.497185\pi$
$608$	−1718.67	−0.114640
$609$	0	0
$610$	0	0
$611$	−14210.0	−0.940875
$612$	0	0
$613$	29371.1	1.93521	0.967607	−	0.252461i	$-0.0812400\pi$
$613$	29371.1	1.93521	0.967607	+	0.252461i	$0.0812400\pi$
$614$	10429.1	0.685477
$615$	0	0
$616$	9662.68	0.632014
$617$	26226.1	1.71122	0.855609	−	0.517622i	$-0.173183\pi$
$617$	26226.1	1.71122	0.855609	+	0.517622i	$0.173183\pi$
$618$	0	0
$619$	−8903.12	−0.578105	−0.289052	−	0.957313i	$-0.593340\pi$
$619$	−8903.12	−0.578105	−0.289052	+	0.957313i	$0.593340\pi$
$620$	0	0
$621$	0	0
$622$	10711.1	0.690476
$623$	−8575.72	−0.551491
$624$	0	0
$625$	0	0
$626$	−3731.18	−0.238224
$627$	0	0
$628$	−5076.31	−0.322558
$629$	19973.5	1.26613
$630$	0	0
$631$	−14136.0	−0.891832	−0.445916	−	0.895075i	$-0.647122\pi$
$631$	−14136.0	−0.891832	−0.445916	+	0.895075i	$0.647122\pi$
$632$	15768.5	0.992464
$633$	0	0
$634$	−8558.27	−0.536108
$635$	0	0
$636$	0	0
$637$	1906.23	0.118567
$638$	9672.34	0.600206
$639$	0	0
$640$	0	0
$641$	−17665.7	−1.08854	−0.544270	−	0.838910i	$-0.683193\pi$
$641$	−17665.7	−1.08854	−0.544270	+	0.838910i	$0.683193\pi$
$642$	0	0
$643$	10890.0	0.667901	0.333951	−	0.942591i	$-0.391618\pi$
$643$	10890.0	0.667901	0.333951	+	0.942591i	$0.391618\pi$
$644$	−2861.27	−0.175078
$645$	0	0
$646$	−3424.12	−0.208545
$647$	−24281.0	−1.47540	−0.737701	−	0.675128i	$-0.764089\pi$
$647$	−24281.0	−1.47540	−0.737701	+	0.675128i	$0.764089\pi$
$648$	0	0
$649$	23314.5	1.41013
$650$	0	0
$651$	0	0
$652$	−2896.46	−0.173979
$653$	−1865.16	−0.111776	−0.0558878	−	0.998437i	$-0.517799\pi$
$653$	−1865.16	−0.111776	−0.0558878	+	0.998437i	$0.517799\pi$
$654$	0	0
$655$	0	0
$656$	12715.2	0.756775
$657$	0	0
$658$	−5642.20	−0.334279
$659$	8327.14	0.492230	0.246115	−	0.969241i	$-0.420846\pi$
$659$	8327.14	0.492230	0.246115	+	0.969241i	$0.420846\pi$
$660$	0	0
$661$	−20665.7	−1.21604	−0.608021	−	0.793921i	$-0.708036\pi$
$661$	−20665.7	−1.21604	−0.608021	+	0.793921i	$0.708036\pi$
$662$	21907.1	1.28617
$663$	0	0
$664$	12575.8	0.734991
$665$	0	0
$666$	0	0
$667$	−10183.1	−0.591140
$668$	8395.00	0.486246
$669$	0	0
$670$	0	0
$671$	31686.2	1.82300
$672$	0	0
$673$	−1283.48	−0.0735136	−0.0367568	−	0.999324i	$-0.511703\pi$
$673$	−1283.48	−0.0735136	−0.0367568	+	0.999324i	$0.511703\pi$
$674$	11659.7	0.666344
$675$	0	0
$676$	2140.08	0.121762
$677$	13783.2	0.782467	0.391234	−	0.920291i	$-0.372048\pi$
$677$	13783.2	0.782467	0.391234	+	0.920291i	$0.372048\pi$
$678$	0	0
$679$	1307.32	0.0738886
$680$	0	0
$681$	0	0
$682$	−7573.18	−0.425208
$683$	−10796.5	−0.604856	−0.302428	−	0.953172i	$-0.597797\pi$
$683$	−10796.5	−0.604856	−0.302428	+	0.953172i	$0.597797\pi$
$684$	0	0
$685$	0	0
$686$	756.884	0.0421253
$687$	0	0
$688$	11458.3	0.634947
$689$	10972.4	0.606699
$690$	0	0
$691$	−12082.4	−0.665173	−0.332587	−	0.943073i	$-0.607921\pi$
$691$	−12082.4	−0.665173	−0.332587	+	0.943073i	$0.607921\pi$
$692$	900.348	0.0494597
$693$	0	0
$694$	23276.5	1.27315
$695$	0	0
$696$	0	0
$697$	−52041.4	−2.82813
$698$	1386.69	0.0751962
$699$	0	0
$700$	0	0
$701$	28753.5	1.54922	0.774610	−	0.632439i	$-0.217946\pi$
$701$	28753.5	1.54922	0.774610	+	0.632439i	$0.217946\pi$
$702$	0	0
$703$	−2176.84	−0.116787
$704$	−29497.8	−1.57917
$705$	0	0
$706$	−5622.95	−0.299749
$707$	−11554.8	−0.614657
$708$	0	0
$709$	4577.21	0.242455	0.121228	−	0.992625i	$-0.461317\pi$
$709$	4577.21	0.242455	0.121228	+	0.992625i	$0.461317\pi$
$710$	0	0
$711$	0	0
$712$	30090.4	1.58383
$713$	7973.07	0.418785
$714$	0	0
$715$	0	0
$716$	11531.7	0.601898
$717$	0	0
$718$	−28788.8	−1.49636
$719$	−30875.9	−1.60150	−0.800749	−	0.599000i	$-0.795565\pi$
$719$	−30875.9	−1.60150	−0.800749	+	0.599000i	$0.795565\pi$
$720$	0	0
$721$	−505.313	−0.0261010
$722$	−14762.3	−0.760936
$723$	0	0
$724$	−9807.25	−0.503430
$725$	0	0
$726$	0	0
$727$	−520.090	−0.0265324	−0.0132662	−	0.999912i	$-0.504223\pi$
$727$	−520.090	−0.0265324	−0.0132662	+	0.999912i	$0.504223\pi$
$728$	−6688.56	−0.340514
$729$	0	0
$730$	0	0
$731$	−46897.1	−2.37285
$732$	0	0
$733$	393.396	0.0198232	0.00991160	−	0.999951i	$-0.496845\pi$
$733$	393.396	0.0198232	0.00991160	+	0.999951i	$0.496845\pi$
$734$	17803.8	0.895301
$735$	0	0
$736$	17255.5	0.864191
$737$	−22212.3	−1.11018
$738$	0	0
$739$	−9348.92	−0.465366	−0.232683	−	0.972553i	$-0.574750\pi$
$739$	−9348.92	−0.465366	−0.232683	+	0.972553i	$0.574750\pi$
$740$	0	0
$741$	0	0
$742$	4356.69	0.215552
$743$	33710.8	1.66451	0.832254	−	0.554394i	$-0.187050\pi$
$743$	33710.8	1.66451	0.832254	+	0.554394i	$0.187050\pi$
$744$	0	0
$745$	0	0
$746$	−7995.44	−0.392405
$747$	0	0
$748$	20994.2	1.02624
$749$	8417.83	0.410655
$750$	0	0
$751$	−21116.6	−1.02604	−0.513019	−	0.858377i	$-0.671473\pi$
$751$	−21116.6	−1.02604	−0.513019	+	0.858377i	$0.671473\pi$
$752$	10649.0	0.516396
$753$	0	0
$754$	−6695.24	−0.323377
$755$	0	0
$756$	0	0
$757$	7385.25	0.354586	0.177293	−	0.984158i	$-0.443266\pi$
$757$	7385.25	0.354586	0.177293	+	0.984158i	$0.443266\pi$
$758$	16520.3	0.791616
$759$	0	0
$760$	0	0
$761$	27682.0	1.31862	0.659311	−	0.751871i	$-0.270848\pi$
$761$	27682.0	1.31862	0.659311	+	0.751871i	$0.270848\pi$
$762$	0	0
$763$	−10859.5	−0.515258
$764$	4968.12	0.235262
$765$	0	0
$766$	−19697.9	−0.929132
$767$	−16138.4	−0.759746
$768$	0	0
$769$	−22248.6	−1.04331	−0.521654	−	0.853157i	$-0.674685\pi$
$769$	−22248.6	−1.04331	−0.521654	+	0.853157i	$0.674685\pi$
$770$	0	0
$771$	0	0
$772$	16213.6	0.755883
$773$	−11372.3	−0.529152	−0.264576	−	0.964365i	$-0.585232\pi$
$773$	−11372.3	−0.529152	−0.264576	+	0.964365i	$0.585232\pi$
$774$	0	0
$775$	0	0
$776$	−4587.12	−0.212201
$777$	0	0
$778$	27804.0	1.28126
$779$	5671.80	0.260864
$780$	0	0
$781$	5844.32	0.267767
$782$	34378.1	1.57207
$783$	0	0
$784$	−1428.53	−0.0650754
$785$	0	0
$786$	0	0
$787$	36561.9	1.65602	0.828012	−	0.560710i	$-0.189472\pi$
$787$	36561.9	1.65602	0.828012	+	0.560710i	$0.189472\pi$
$788$	2829.48	0.127914
$789$	0	0
$790$	0	0
$791$	−14566.3	−0.654763
$792$	0	0
$793$	−21933.4	−0.982190
$794$	−28054.2	−1.25391
$795$	0	0
$796$	3667.59	0.163309
$797$	−7436.80	−0.330521	−0.165260	−	0.986250i	$-0.552846\pi$
$797$	−7436.80	−0.330521	−0.165260	+	0.986250i	$0.552846\pi$
$798$	0	0
$799$	−43584.9	−1.92982
$800$	0	0
$801$	0	0
$802$	13534.6	0.595913
$803$	7195.17	0.316204
$804$	0	0
$805$	0	0
$806$	5242.20	0.229092
$807$	0	0
$808$	40543.4	1.76524
$809$	30585.4	1.32920	0.664601	−	0.747198i	$-0.268601\pi$
$809$	30585.4	1.32920	0.664601	+	0.747198i	$0.268601\pi$
$810$	0	0
$811$	23756.0	1.02859	0.514294	−	0.857614i	$-0.328054\pi$
$811$	23756.0	1.02859	0.514294	+	0.857614i	$0.328054\pi$
$812$	1709.17	0.0738672
$813$	0	0
$814$	−20759.3	−0.893873
$815$	0	0
$816$	0	0
$817$	5111.14	0.218869
$818$	23391.2	0.999821
$819$	0	0
$820$	0	0
$821$	33842.2	1.43861	0.719306	−	0.694694i	$-0.244460\pi$
$821$	33842.2	1.43861	0.719306	+	0.694694i	$0.244460\pi$
$822$	0	0
$823$	18730.9	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$823$	18730.9	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$824$	1773.04	0.0749597
$825$	0	0
$826$	−6407.91	−0.269927
$827$	−19107.1	−0.803409	−0.401705	−	0.915769i	$-0.631582\pi$
$827$	−19107.1	−0.803409	−0.401705	+	0.915769i	$0.631582\pi$
$828$	0	0
$829$	9942.59	0.416551	0.208275	−	0.978070i	$-0.433215\pi$
$829$	9942.59	0.416551	0.208275	+	0.978070i	$0.433215\pi$
$830$	0	0
$831$	0	0
$832$	20418.5	0.850822
$833$	5846.78	0.243192
$834$	0	0
$835$	0	0
$836$	−2288.08	−0.0946590
$837$	0	0
$838$	−9480.75	−0.390820
$839$	−37144.9	−1.52847	−0.764233	−	0.644940i	$-0.776882\pi$
$839$	−37144.9	−1.52847	−0.764233	+	0.644940i	$0.776882\pi$
$840$	0	0
$841$	−18306.2	−0.750591
$842$	8642.99	0.353750
$843$	0	0
$844$	3453.62	0.140851
$845$	0	0
$846$	0	0
$847$	12792.8	0.518969
$848$	−8222.78	−0.332985
$849$	0	0
$850$	0	0
$851$	21855.4	0.880370
$852$	0	0
$853$	31377.2	1.25948	0.629739	−	0.776807i	$-0.283162\pi$
$853$	31377.2	1.25948	0.629739	+	0.776807i	$0.283162\pi$
$854$	−8708.84	−0.348958
$855$	0	0
$856$	−29536.4	−1.17936
$857$	−24561.3	−0.978993	−0.489496	−	0.872005i	$-0.662819\pi$
$857$	−24561.3	−0.978993	−0.489496	+	0.872005i	$0.662819\pi$
$858$	0	0
$859$	−39819.1	−1.58162	−0.790809	−	0.612063i	$-0.790340\pi$
$859$	−39819.1	−1.58162	−0.790809	+	0.612063i	$0.790340\pi$
$860$	0	0
$861$	0	0
$862$	−29588.9	−1.16914
$863$	14702.2	0.579919	0.289959	−	0.957039i	$-0.406358\pi$
$863$	14702.2	0.579919	0.289959	+	0.957039i	$0.406358\pi$
$864$	0	0
$865$	0	0
$866$	17195.0	0.674722
$867$	0	0
$868$	−1338.24	−0.0523303
$869$	36081.0	1.40847
$870$	0	0
$871$	15375.5	0.598138
$872$	38103.9	1.47977
$873$	0	0
$874$	−3746.74	−0.145006
$875$	0	0
$876$	0	0
$877$	−7208.31	−0.277546	−0.138773	−	0.990324i	$-0.544316\pi$
$877$	−7208.31	−0.277546	−0.138773	+	0.990324i	$0.544316\pi$
$878$	2293.35	0.0881512
$879$	0	0
$880$	0	0
$881$	−3789.00	−0.144898	−0.0724488	−	0.997372i	$-0.523081\pi$
$881$	−3789.00	−0.144898	−0.0724488	+	0.997372i	$0.523081\pi$
$882$	0	0
$883$	26953.7	1.02725	0.513626	−	0.858014i	$-0.328302\pi$
$883$	26953.7	1.02725	0.513626	+	0.858014i	$0.328302\pi$
$884$	−14532.3	−0.552912
$885$	0	0
$886$	6280.88	0.238160
$887$	−37869.8	−1.43353	−0.716766	−	0.697314i	$-0.754379\pi$
$887$	−37869.8	−1.43353	−0.716766	+	0.697314i	$0.754379\pi$
$888$	0	0
$889$	9902.92	0.373603
$890$	0	0
$891$	0	0
$892$	−12772.9	−0.479449
$893$	4750.15	0.178004
$894$	0	0
$895$	0	0
$896$	706.385	0.0263378
$897$	0	0
$898$	−16489.6	−0.612767
$899$	−4762.69	−0.176690
$900$	0	0
$901$	33654.6	1.24439
$902$	54088.8	1.99663
$903$	0	0
$904$	51110.1	1.88042
$905$	0	0
$906$	0	0
$907$	−11717.5	−0.428968	−0.214484	−	0.976728i	$-0.568807\pi$
$907$	−11717.5	−0.428968	−0.214484	+	0.976728i	$0.568807\pi$
$908$	2917.66	0.106636
$909$	0	0
$910$	0	0
$911$	15909.2	0.578589	0.289294	−	0.957240i	$-0.406579\pi$
$911$	15909.2	0.578589	0.289294	+	0.957240i	$0.406579\pi$
$912$	0	0
$913$	28775.4	1.04308
$914$	−24304.9	−0.879578
$915$	0	0
$916$	4606.92	0.166176
$917$	17307.6	0.623279
$918$	0	0
$919$	−32933.3	−1.18212	−0.591060	−	0.806628i	$-0.701290\pi$
$919$	−32933.3	−1.18212	−0.591060	+	0.806628i	$0.701290\pi$
$920$	0	0
$921$	0	0
$922$	−17531.2	−0.626203
$923$	−4045.47	−0.144267
$924$	0	0
$925$	0	0
$926$	−16830.7	−0.597292
$927$	0	0
$928$	−10307.5	−0.364612
$929$	30912.6	1.09172	0.545861	−	0.837876i	$-0.316203\pi$
$929$	30912.6	1.09172	0.545861	+	0.837876i	$0.316203\pi$
$930$	0	0
$931$	−637.219	−0.0224318
$932$	7761.09	0.272771
$933$	0	0
$934$	7247.29	0.253896
$935$	0	0
$936$	0	0
$937$	45737.6	1.59465	0.797323	−	0.603553i	$-0.206249\pi$
$937$	45737.6	1.59465	0.797323	+	0.603553i	$0.206249\pi$
$938$	6104.97	0.212510
$939$	0	0
$940$	0	0
$941$	2137.79	0.0740595	0.0370297	−	0.999314i	$-0.488210\pi$
$941$	2137.79	0.0740595	0.0370297	+	0.999314i	$0.488210\pi$
$942$	0	0
$943$	−56944.8	−1.96647
$944$	12094.2	0.416984
$945$	0	0
$946$	48742.1	1.67520
$947$	42513.2	1.45881	0.729405	−	0.684082i	$-0.239797\pi$
$947$	42513.2	1.45881	0.729405	+	0.684082i	$0.239797\pi$
$948$	0	0
$949$	−4980.53	−0.170363
$950$	0	0
$951$	0	0
$952$	−20515.2	−0.698425
$953$	−9012.85	−0.306353	−0.153177	−	0.988199i	$-0.548950\pi$
$953$	−9012.85	−0.306353	−0.153177	+	0.988199i	$0.548950\pi$
$954$	0	0
$955$	0	0
$956$	−2986.86	−0.101048
$957$	0	0
$958$	6420.15	0.216520
$959$	1500.74	0.0505332
$960$	0	0
$961$	−26061.9	−0.874826
$962$	14369.7	0.481598
$963$	0	0
$964$	16583.1	0.554053
$965$	0	0
$966$	0	0
$967$	−29024.2	−0.965207	−0.482604	−	0.875839i	$-0.660309\pi$
$967$	−29024.2	−0.965207	−0.482604	+	0.875839i	$0.660309\pi$
$968$	−44887.4	−1.49043
$969$	0	0
$970$	0	0
$971$	−12874.4	−0.425498	−0.212749	−	0.977107i	$-0.568242\pi$
$971$	−12874.4	−0.425498	−0.212749	+	0.977107i	$0.568242\pi$
$972$	0	0
$973$	6599.42	0.217438
$974$	−4858.70	−0.159839
$975$	0	0
$976$	16437.0	0.539072
$977$	−15195.8	−0.497600	−0.248800	−	0.968555i	$-0.580036\pi$
$977$	−15195.8	−0.497600	−0.248800	+	0.968555i	$0.580036\pi$
$978$	0	0
$979$	68851.9	2.24772
$980$	0	0
$981$	0	0
$982$	26099.5	0.848134
$983$	12042.2	0.390728	0.195364	−	0.980731i	$-0.437411\pi$
$983$	12042.2	0.390728	0.195364	+	0.980731i	$0.437411\pi$
$984$	0	0
$985$	0	0
$986$	−20535.7	−0.663275
$987$	0	0
$988$	1583.82	0.0510001
$989$	−51315.8	−1.64990
$990$	0	0
$991$	11490.7	0.368330	0.184165	−	0.982895i	$-0.441042\pi$
$991$	11490.7	0.368330	0.184165	+	0.982895i	$0.441042\pi$
$992$	8070.49	0.258305
$993$	0	0
$994$	−1606.29	−0.0512560
$995$	0	0
$996$	0	0
$997$	−4076.85	−0.129504	−0.0647518	−	0.997901i	$-0.520626\pi$
$997$	−4076.85	−0.129504	−0.0647518	+	0.997901i	$0.520626\pi$
$998$	3108.44	0.0985931
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bo.1.4		5
3.2	odd	2		525.4.a.x.1.2		5
5.2	odd	4		315.4.d.b.64.7		10
5.3	odd	4		315.4.d.b.64.4		10
5.4	even	2		1575.4.a.bp.1.2		5
15.2	even	4		105.4.d.b.64.4	✓	10
15.8	even	4		105.4.d.b.64.7	yes	10
15.14	odd	2		525.4.a.w.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.d.b.64.4	✓	10	15.2	even	4
105.4.d.b.64.7	yes	10	15.8	even	4
315.4.d.b.64.4		10	5.3	odd	4
315.4.d.b.64.7		10	5.2	odd	4
525.4.a.w.1.4		5	15.14	odd	2
525.4.a.x.1.2		5	3.2	odd	2
1575.4.a.bo.1.4		5	1.1	even	1	trivial
1575.4.a.bp.1.2		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+2.20666 q^{2} -3.13065 q^{4} +7.00000 q^{7} -24.5616 q^{8} +O(q^{10})\)	\(q+2.20666 q^{2} -3.13065 q^{4} +7.00000 q^{7} -24.5616 q^{8} -56.2010 q^{11} +38.9026 q^{13} +15.4466 q^{14} -29.1538 q^{16} +119.322 q^{17} -13.0045 q^{19} -124.016 q^{22} +130.565 q^{23} +85.8448 q^{26} -21.9146 q^{28} -77.9925 q^{29} +61.0660 q^{31} +132.160 q^{32} +263.303 q^{34} +167.391 q^{37} -28.6964 q^{38} -436.142 q^{41} -393.030 q^{43} +175.946 q^{44} +288.112 q^{46} -365.271 q^{47} +49.0000 q^{49} -121.791 q^{52} +282.048 q^{53} -171.931 q^{56} -172.103 q^{58} -414.842 q^{59} -563.802 q^{61} +134.752 q^{62} +524.862 q^{64} +395.230 q^{67} -373.556 q^{68} -103.990 q^{71} -128.026 q^{73} +369.376 q^{74} +40.7125 q^{76} -393.407 q^{77} -641.999 q^{79} -962.417 q^{82} -512.010 q^{83} -867.283 q^{86} +1380.38 q^{88} -1225.10 q^{89} +272.318 q^{91} -408.753 q^{92} -806.028 q^{94} +186.760 q^{97} +108.126 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8}+O(q^{10}) \) 5 * q - q^2 + 27 * q^4 + 35 * q^7 - 33 * q^8	\( 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8} - 66 q^{11} - 2 q^{13} - 7 q^{14} + 155 q^{16} - 108 q^{17} + 174 q^{19} - 506 q^{22} + 116 q^{23} - 446 q^{26} + 189 q^{28} - 370 q^{29} + 342 q^{31} + 55 q^{32} + 112 q^{34} - 408 q^{37} + 34 q^{38} - 802 q^{41} - 584 q^{43} - 290 q^{44} + 640 q^{46} - 716 q^{47} + 245 q^{49} + 338 q^{52} - 98 q^{53} - 231 q^{56} + 482 q^{58} - 704 q^{59} + 650 q^{61} - 2070 q^{62} + 75 q^{64} + 180 q^{67} - 4520 q^{68} - 1470 q^{71} - 534 q^{73} + 1312 q^{74} + 4370 q^{76} - 462 q^{77} - 820 q^{79} - 1338 q^{82} - 1520 q^{83} - 832 q^{86} - 3258 q^{88} - 286 q^{89} - 14 q^{91} - 1288 q^{92} + 2540 q^{94} + 278 q^{97} - 49 q^{98}+O(q^{100}) \) 5 * q - q^2 + 27 * q^4 + 35 * q^7 - 33 * q^8 - 66 * q^11 - 2 * q^13 - 7 * q^14 + 155 * q^16 - 108 * q^17 + 174 * q^19 - 506 * q^22 + 116 * q^23 - 446 * q^26 + 189 * q^28 - 370 * q^29 + 342 * q^31 + 55 * q^32 + 112 * q^34 - 408 * q^37 + 34 * q^38 - 802 * q^41 - 584 * q^43 - 290 * q^44 + 640 * q^46 - 716 * q^47 + 245 * q^49 + 338 * q^52 - 98 * q^53 - 231 * q^56 + 482 * q^58 - 704 * q^59 + 650 * q^61 - 2070 * q^62 + 75 * q^64 + 180 * q^67 - 4520 * q^68 - 1470 * q^71 - 534 * q^73 + 1312 * q^74 + 4370 * q^76 - 462 * q^77 - 820 * q^79 - 1338 * q^82 - 1520 * q^83 - 832 * q^86 - 3258 * q^88 - 286 * q^89 - 14 * q^91 - 1288 * q^92 + 2540 * q^94 + 278 * q^97 - 49 * q^98

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