LMFDB - Embedded newform 575.4.a.m.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,4,Mod(1,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	575.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:	$33.9260982533$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 37x^{5} + 123x^{4} + 304x^{3} - 1196x^{2} + 264x + 864 $ x^7 - 3x^6 - 37x^5 + 123x^4 + 304x^3 - 1196x^2 + 264x + 864
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$4.39200$ of defining polynomial
Character	$\chi$	$=$	575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.39200 q^{2} -3.27927 q^{3} +11.2896 q^{4} -14.4025 q^{6} +13.7978 q^{7} +14.4481 q^{8} -16.2464 q^{9} -16.7024 q^{11} -37.0218 q^{12} -68.7171 q^{13} +60.5997 q^{14} -26.8610 q^{16} -40.7204 q^{17} -71.3541 q^{18} +40.6147 q^{19} -45.2466 q^{21} -73.3568 q^{22} -23.0000 q^{23} -47.3793 q^{24} -301.805 q^{26} +141.817 q^{27} +155.772 q^{28} +41.4829 q^{29} -234.349 q^{31} -233.559 q^{32} +54.7716 q^{33} -178.844 q^{34} -183.416 q^{36} +101.472 q^{37} +178.380 q^{38} +225.342 q^{39} +9.09912 q^{41} -198.723 q^{42} -146.349 q^{43} -188.564 q^{44} -101.016 q^{46} -383.965 q^{47} +88.0845 q^{48} -152.622 q^{49} +133.533 q^{51} -775.792 q^{52} +430.635 q^{53} +622.858 q^{54} +199.352 q^{56} -133.187 q^{57} +182.193 q^{58} -441.067 q^{59} +73.6756 q^{61} -1029.26 q^{62} -224.164 q^{63} -810.901 q^{64} +240.557 q^{66} -245.433 q^{67} -459.719 q^{68} +75.4232 q^{69} -241.613 q^{71} -234.730 q^{72} +187.130 q^{73} +445.667 q^{74} +458.526 q^{76} -230.455 q^{77} +989.701 q^{78} +109.404 q^{79} -26.4023 q^{81} +39.9633 q^{82} -1198.92 q^{83} -510.818 q^{84} -642.765 q^{86} -136.034 q^{87} -241.318 q^{88} +148.942 q^{89} -948.142 q^{91} -259.662 q^{92} +768.492 q^{93} -1686.38 q^{94} +765.902 q^{96} +1829.68 q^{97} -670.314 q^{98} +271.353 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} + q^{3} + 27 q^{4} - 41 q^{6} + q^{7} - 57 q^{8} + 10 q^{9} - 52 q^{11} + 65 q^{12} + 45 q^{13} - 42 q^{14} - 85 q^{16} + 85 q^{17} + 18 q^{18} - 10 q^{19} - 202 q^{21} - 71 q^{22} - 161 q^{23}+ \cdots - 2143 q^{99}+O(q^{100}) $ 7 * q + 3 * q^2 + q^3 + 27 * q^4 - 41 * q^6 + q^7 - 57 * q^8 + 10 * q^9 - 52 * q^11 + 65 * q^12 + 45 * q^13 - 42 * q^14 - 85 * q^16 + 85 * q^17 + 18 * q^18 - 10 * q^19 - 202 * q^21 - 71 * q^22 - 161 * q^23 - 303 * q^24 - 26 * q^26 - 242 * q^27 + 186 * q^28 - 455 * q^29 - 690 * q^31 - 233 * q^32 - 203 * q^33 - 657 * q^34 + 204 * q^36 - 892 * q^37 + 421 * q^38 - 277 * q^39 - 230 * q^41 - 888 * q^42 + 217 * q^43 - 1003 * q^44 - 69 * q^46 - 166 * q^47 + 1225 * q^48 - 802 * q^49 - 695 * q^51 - 1640 * q^52 + 1148 * q^53 - 1247 * q^54 - 386 * q^56 - 845 * q^57 + 1454 * q^58 - 551 * q^59 - 1242 * q^61 - 296 * q^62 + 1111 * q^63 - 1501 * q^64 + 987 * q^66 + 201 * q^67 + 1575 * q^68 - 23 * q^69 + 186 * q^71 - 2964 * q^72 + 530 * q^73 - 1530 * q^74 - 1775 * q^76 - 1339 * q^77 + 1364 * q^78 - 2003 * q^79 - 365 * q^81 - 2063 * q^82 + 726 * q^83 - 1352 * q^84 + 1894 * q^86 + 263 * q^87 + 2145 * q^88 - 2629 * q^89 - 1295 * q^91 - 621 * q^92 - 549 * q^93 - 3468 * q^94 - 135 * q^96 + 472 * q^97 + 2389 * q^98 - 2143 * q^99

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 $

$2$	4.39200	1.55281	0.776403	−	0.630237i	$-0.217042\pi$
$2$	4.39200	1.55281	0.776403	+	0.630237i	$0.217042\pi$
$3$	−3.27927	−0.631096	−0.315548	−	0.948910i	$-0.602188\pi$
$3$	−3.27927	−0.631096	−0.315548	+	0.948910i	$0.602188\pi$
$4$	11.2896	1.41121
$5$	0	0
$5$	0	0
$6$	−14.4025	−0.979969
$7$	13.7978	0.745009	0.372505	−	0.928030i	$-0.378499\pi$
$7$	13.7978	0.745009	0.372505	+	0.928030i	$0.378499\pi$
$8$	14.4481	0.638523
$9$	−16.2464	−0.601718
$10$	0	0
$11$	−16.7024	−0.457814	−0.228907	−	0.973448i	$-0.573515\pi$
$11$	−16.7024	−0.457814	−0.228907	+	0.973448i	$0.573515\pi$
$12$	−37.0218	−0.890606
$13$	−68.7171	−1.46605	−0.733027	−	0.680200i	$-0.761893\pi$
$13$	−68.7171	−1.46605	−0.733027	+	0.680200i	$0.761893\pi$
$14$	60.5997	1.15685
$15$	0	0
$16$	−26.8610	−0.419703
$17$	−40.7204	−0.580950	−0.290475	−	0.956883i	$-0.593813\pi$
$17$	−40.7204	−0.580950	−0.290475	+	0.956883i	$0.593813\pi$
$18$	−71.3541	−0.934351
$19$	40.6147	0.490403	0.245201	−	0.969472i	$-0.421146\pi$
$19$	40.6147	0.490403	0.245201	+	0.969472i	$0.421146\pi$
$20$	0	0
$21$	−45.2466	−0.470172
$22$	−73.3568	−0.710897
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−47.3793	−0.402969
$25$	0	0
$26$	−301.805	−2.27650
$27$	141.817	1.01084
$28$	155.772	1.05136
$29$	41.4829	0.265627	0.132814	−	0.991141i	$-0.457599\pi$
$29$	41.4829	0.265627	0.132814	+	0.991141i	$0.457599\pi$
$30$	0	0
$31$	−234.349	−1.35775	−0.678875	−	0.734254i	$-0.737532\pi$
$31$	−234.349	−1.35775	−0.678875	+	0.734254i	$0.737532\pi$
$32$	−233.559	−1.29024
$33$	54.7716	0.288925
$34$	−178.844	−0.902103
$35$	0	0
$36$	−183.416	−0.849148
$37$	101.472	0.450864	0.225432	−	0.974259i	$-0.427621\pi$
$37$	101.472	0.450864	0.225432	+	0.974259i	$0.427621\pi$
$38$	178.380	0.761500
$39$	225.342	0.925220
$40$	0	0
$41$	9.09912	0.0346596	0.0173298	−	0.999850i	$-0.494483\pi$
$41$	9.09912	0.0346596	0.0173298	+	0.999850i	$0.494483\pi$
$42$	−198.723	−0.730086
$43$	−146.349	−0.519024	−0.259512	−	0.965740i	$-0.583562\pi$
$43$	−146.349	−0.519024	−0.259512	+	0.965740i	$0.583562\pi$
$44$	−188.564	−0.646070
$45$	0	0
$46$	−101.016	−0.323782
$47$	−383.965	−1.19164	−0.595820	−	0.803118i	$-0.703173\pi$
$47$	−383.965	−1.19164	−0.595820	+	0.803118i	$0.703173\pi$
$48$	88.0845	0.264873
$49$	−152.622	−0.444961
$50$	0	0
$51$	133.533	0.366635
$52$	−775.792	−2.06890
$53$	430.635	1.11608	0.558040	−	0.829814i	$-0.311553\pi$
$53$	430.635	1.11608	0.558040	+	0.829814i	$0.311553\pi$
$54$	622.858	1.56963
$55$	0	0
$56$	199.352	0.475706
$57$	−133.187	−0.309491
$58$	182.193	0.412467
$59$	−441.067	−0.973255	−0.486627	−	0.873610i	$-0.661773\pi$
$59$	−441.067	−0.973255	−0.486627	+	0.873610i	$0.661773\pi$
$60$	0	0
$61$	73.6756	0.154643	0.0773213	−	0.997006i	$-0.475363\pi$
$61$	73.6756	0.154643	0.0773213	+	0.997006i	$0.475363\pi$
$62$	−1029.26	−2.10832
$63$	−224.164	−0.448286
$64$	−810.901	−1.58379
$65$	0	0
$66$	240.557	0.448644
$67$	−245.433	−0.447529	−0.223765	−	0.974643i	$-0.571835\pi$
$67$	−245.433	−0.447529	−0.223765	+	0.974643i	$0.571835\pi$
$68$	−459.719	−0.819841
$69$	75.4232	0.131593
$70$	0	0
$71$	−241.613	−0.403862	−0.201931	−	0.979400i	$-0.564722\pi$
$71$	−241.613	−0.403862	−0.201931	+	0.979400i	$0.564722\pi$
$72$	−234.730	−0.384211
$73$	187.130	0.300026	0.150013	−	0.988684i	$-0.452068\pi$
$73$	187.130	0.300026	0.150013	+	0.988684i	$0.452068\pi$
$74$	445.667	0.700104
$75$	0	0
$76$	458.526	0.692059
$77$	−230.455	−0.341076
$78$	989.701	1.43669
$79$	109.404	0.155808	0.0779041	−	0.996961i	$-0.475177\pi$
$79$	109.404	0.155808	0.0779041	+	0.996961i	$0.475177\pi$
$80$	0	0
$81$	−26.4023	−0.0362171
$82$	39.9633	0.0538196
$83$	−1198.92	−1.58552	−0.792761	−	0.609533i	$-0.791357\pi$
$83$	−1198.92	−1.58552	−0.792761	+	0.609533i	$0.791357\pi$
$84$	−510.818	−0.663510
$85$	0	0
$86$	−642.765	−0.805944
$87$	−136.034	−0.167636
$88$	−241.318	−0.292325
$89$	148.942	0.177391	0.0886953	−	0.996059i	$-0.471730\pi$
$89$	148.942	0.177391	0.0886953	+	0.996059i	$0.471730\pi$
$90$	0	0
$91$	−948.142	−1.09222
$92$	−259.662	−0.294257
$93$	768.492	0.856870
$94$	−1686.38	−1.85039
$95$	0	0
$96$	765.902	0.814266
$97$	1829.68	1.91521	0.957606	−	0.288082i	$-0.0930176\pi$
$97$	1829.68	1.91521	0.957606	+	0.288082i	$0.0930176\pi$
$98$	−670.314	−0.690939
$99$	271.353	0.275475
$100$	0	0
$101$	−628.795	−0.619479	−0.309740	−	0.950821i	$-0.600242\pi$
$101$	−628.795	−0.619479	−0.309740	+	0.950821i	$0.600242\pi$
$102$	586.478	0.569313
$103$	1577.66	1.50924	0.754620	−	0.656162i	$-0.227821\pi$
$103$	1577.66	1.50924	0.754620	+	0.656162i	$0.227821\pi$
$104$	−992.834	−0.936109
$105$	0	0
$106$	1891.35	1.73306
$107$	1253.21	1.13226	0.566130	−	0.824316i	$-0.308440\pi$
$107$	1253.21	1.13226	0.566130	+	0.824316i	$0.308440\pi$
$108$	1601.06	1.42650
$109$	−1271.40	−1.11723	−0.558615	−	0.829427i	$-0.688667\pi$
$109$	−1271.40	−1.11723	−0.558615	+	0.829427i	$0.688667\pi$
$110$	0	0
$111$	−332.756	−0.284538
$112$	−370.622	−0.312683
$113$	1469.50	1.22336	0.611678	−	0.791107i	$-0.290495\pi$
$113$	1469.50	1.22336	0.611678	+	0.791107i	$0.290495\pi$
$114$	−584.955	−0.480579
$115$	0	0
$116$	468.328	0.374855
$117$	1116.40	0.882151
$118$	−1937.17	−1.51128
$119$	−561.851	−0.432813
$120$	0	0
$121$	−1052.03	−0.790406
$122$	323.583	0.240130
$123$	−29.8385	−0.0218735
$124$	−2645.71	−1.91607
$125$	0	0
$126$	−984.527	−0.696100
$127$	−445.611	−0.311351	−0.155676	−	0.987808i	$-0.549755\pi$
$127$	−445.611	−0.311351	−0.155676	+	0.987808i	$0.549755\pi$
$128$	−1693.01	−1.16908
$129$	479.918	0.327554
$130$	0	0
$131$	630.681	0.420632	0.210316	−	0.977633i	$-0.432551\pi$
$131$	630.681	0.420632	0.210316	+	0.977633i	$0.432551\pi$
$132$	618.352	0.407732
$133$	560.392	0.365354
$134$	−1077.94	−0.694926
$135$	0	0
$136$	−588.334	−0.370950
$137$	479.846	0.299241	0.149621	−	0.988743i	$-0.452195\pi$
$137$	479.846	0.299241	0.149621	+	0.988743i	$0.452195\pi$
$138$	331.259	0.204338
$139$	−202.400	−0.123506	−0.0617531	−	0.998091i	$-0.519669\pi$
$139$	−202.400	−0.123506	−0.0617531	+	0.998091i	$0.519669\pi$
$140$	0	0
$141$	1259.13	0.752039
$142$	−1061.16	−0.627119
$143$	1147.74	0.671180
$144$	436.395	0.252543
$145$	0	0
$146$	821.874	0.465882
$147$	500.488	0.280813
$148$	1145.59	0.636262
$149$	−739.957	−0.406843	−0.203422	−	0.979091i	$-0.565206\pi$
$149$	−739.957	−0.406843	−0.203422	+	0.979091i	$0.565206\pi$
$150$	0	0
$151$	−1278.96	−0.689274	−0.344637	−	0.938736i	$-0.611998\pi$
$151$	−1278.96	−0.689274	−0.344637	+	0.938736i	$0.611998\pi$
$152$	586.806	0.313133
$153$	661.560	0.349568
$154$	−1012.16	−0.529624
$155$	0	0
$156$	2544.03	1.30568
$157$	713.690	0.362794	0.181397	−	0.983410i	$-0.441938\pi$
$157$	713.690	0.362794	0.181397	+	0.983410i	$0.441938\pi$
$158$	480.500	0.241940
$159$	−1412.17	−0.704354
$160$	0	0
$161$	−317.349	−0.155345
$162$	−115.959	−0.0562382
$163$	215.300	0.103458	0.0517288	−	0.998661i	$-0.483527\pi$
$163$	215.300	0.103458	0.0517288	+	0.998661i	$0.483527\pi$
$164$	102.726	0.0489118
$165$	0	0
$166$	−5265.64	−2.46201
$167$	4086.64	1.89362	0.946808	−	0.321800i	$-0.104288\pi$
$167$	4086.64	1.89362	0.946808	+	0.321800i	$0.104288\pi$
$168$	−653.729	−0.300216
$169$	2525.04	1.14931
$170$	0	0
$171$	−659.842	−0.295084
$172$	−1652.23	−0.732450
$173$	858.878	0.377453	0.188726	−	0.982030i	$-0.439564\pi$
$173$	858.878	0.377453	0.188726	+	0.982030i	$0.439564\pi$
$174$	−597.460	−0.260306
$175$	0	0
$176$	448.643	0.192146
$177$	1446.38	0.614217
$178$	654.151	0.275453
$179$	1668.46	0.696683	0.348342	−	0.937368i	$-0.386745\pi$
$179$	1668.46	0.696683	0.348342	+	0.937368i	$0.386745\pi$
$180$	0	0
$181$	−347.888	−0.142864	−0.0714319	−	0.997445i	$-0.522757\pi$
$181$	−347.888	−0.142864	−0.0714319	+	0.997445i	$0.522757\pi$
$182$	−4164.24	−1.69601
$183$	−241.602	−0.0975942
$184$	−332.307	−0.133141
$185$	0	0
$186$	3375.22	1.33055
$187$	680.128	0.265967
$188$	−4334.83	−1.68165
$189$	1956.75	0.753083
$190$	0	0
$191$	2316.47	0.877559	0.438780	−	0.898595i	$-0.355411\pi$
$191$	2316.47	0.877559	0.438780	+	0.898595i	$0.355411\pi$
$192$	2659.16	0.999524
$193$	1530.07	0.570658	0.285329	−	0.958430i	$-0.407897\pi$
$193$	1530.07	0.570658	0.285329	+	0.958430i	$0.407897\pi$
$194$	8035.94	2.97395
$195$	0	0
$196$	−1723.05	−0.627932
$197$	−1798.69	−0.650513	−0.325257	−	0.945626i	$-0.605451\pi$
$197$	−1798.69	−0.650513	−0.325257	+	0.945626i	$0.605451\pi$
$198$	1191.78	0.427759
$199$	3231.92	1.15128	0.575640	−	0.817703i	$-0.304753\pi$
$199$	3231.92	1.15128	0.575640	+	0.817703i	$0.304753\pi$
$200$	0	0
$201$	804.842	0.282434
$202$	−2761.67	−0.961931
$203$	572.372	0.197895
$204$	1507.54	0.517398
$205$	0	0
$206$	6929.09	2.34356
$207$	373.667	0.125467
$208$	1845.81	0.615308
$209$	−678.362	−0.224513
$210$	0	0
$211$	−5199.79	−1.69653	−0.848266	−	0.529570i	$-0.822353\pi$
$211$	−5199.79	−1.69653	−0.848266	+	0.529570i	$0.822353\pi$
$212$	4861.72	1.57502
$213$	792.314	0.254875
$214$	5504.07	1.75818
$215$	0	0
$216$	2048.98	0.645443
$217$	−3233.49	−1.01154
$218$	−5583.99	−1.73484
$219$	−613.649	−0.189345
$220$	0	0
$221$	2798.19	0.851704
$222$	−1461.46	−0.441833
$223$	3144.43	0.944246	0.472123	−	0.881533i	$-0.343488\pi$
$223$	3144.43	0.944246	0.472123	+	0.881533i	$0.343488\pi$
$224$	−3222.59	−0.961242
$225$	0	0
$226$	6454.06	1.89964
$227$	−262.640	−0.0767931	−0.0383966	−	0.999263i	$-0.512225\pi$
$227$	−262.640	−0.0767931	−0.0383966	+	0.999263i	$0.512225\pi$
$228$	−1503.63	−0.436756
$229$	−5559.69	−1.60434	−0.802172	−	0.597093i	$-0.796322\pi$
$229$	−5559.69	−1.60434	−0.802172	+	0.597093i	$0.796322\pi$
$230$	0	0
$231$	755.726	0.215251
$232$	599.351	0.169609
$233$	−4405.25	−1.23862	−0.619308	−	0.785148i	$-0.712587\pi$
$233$	−4405.25	−1.23862	−0.619308	+	0.785148i	$0.712587\pi$
$234$	4903.25	1.36981
$235$	0	0
$236$	−4979.49	−1.37346
$237$	−358.764	−0.0983300
$238$	−2467.65	−0.672075
$239$	6444.99	1.74432	0.872158	−	0.489224i	$-0.162720\pi$
$239$	6444.99	1.74432	0.872158	+	0.489224i	$0.162720\pi$
$240$	0	0
$241$	−6708.71	−1.79314	−0.896569	−	0.442905i	$-0.853948\pi$
$241$	−6708.71	−1.79314	−0.896569	+	0.442905i	$0.853948\pi$
$242$	−4620.52	−1.22735
$243$	−3742.47	−0.987981
$244$	831.772	0.218232
$245$	0	0
$246$	−131.051	−0.0339653
$247$	−2790.92	−0.718956
$248$	−3385.90	−0.866955
$249$	3931.57	1.00062
$250$	0	0
$251$	−289.397	−0.0727752	−0.0363876	−	0.999338i	$-0.511585\pi$
$251$	−289.397	−0.0727752	−0.0363876	+	0.999338i	$0.511585\pi$
$252$	−2530.73	−0.632623
$253$	384.155	0.0954609
$254$	−1957.12	−0.483468
$255$	0	0
$256$	−948.473	−0.231561
$257$	−4040.24	−0.980635	−0.490318	−	0.871544i	$-0.663119\pi$
$257$	−4040.24	−0.980635	−0.490318	+	0.871544i	$0.663119\pi$
$258$	2107.80	0.508628
$259$	1400.09	0.335898
$260$	0	0
$261$	−673.948	−0.159833
$262$	2769.95	0.653160
$263$	−5250.65	−1.23106	−0.615530	−	0.788114i	$-0.711058\pi$
$263$	−5250.65	−1.23106	−0.615530	+	0.788114i	$0.711058\pi$
$264$	791.347	0.184485
$265$	0	0
$266$	2461.24	0.567324
$267$	−488.419	−0.111951
$268$	−2770.86	−0.631556
$269$	−2417.55	−0.547958	−0.273979	−	0.961736i	$-0.588340\pi$
$269$	−2417.55	−0.547958	−0.273979	+	0.961736i	$0.588340\pi$
$270$	0	0
$271$	−6593.83	−1.47803	−0.739016	−	0.673688i	$-0.764709\pi$
$271$	−6593.83	−1.47803	−0.739016	+	0.673688i	$0.764709\pi$
$272$	1093.79	0.243827
$273$	3109.21	0.689297
$274$	2107.48	0.464663
$275$	0	0
$276$	851.501	0.185704
$277$	2826.91	0.613186	0.306593	−	0.951841i	$-0.400811\pi$
$277$	2826.91	0.613186	0.306593	+	0.951841i	$0.400811\pi$
$278$	−888.941	−0.191781
$279$	3807.32	0.816983
$280$	0	0
$281$	−2036.17	−0.432269	−0.216134	−	0.976364i	$-0.569345\pi$
$281$	−2036.17	−0.432269	−0.216134	+	0.976364i	$0.569345\pi$
$282$	5530.08	1.16777
$283$	4470.17	0.938954	0.469477	−	0.882945i	$-0.344442\pi$
$283$	4470.17	0.938954	0.469477	+	0.882945i	$0.344442\pi$
$284$	−2727.73	−0.569932
$285$	0	0
$286$	5040.87	1.04221
$287$	125.548	0.0258217
$288$	3794.48	0.776362
$289$	−3254.85	−0.662497
$290$	0	0
$291$	−6000.00	−1.20868
$292$	2112.63	0.423398
$293$	−9457.29	−1.88567	−0.942834	−	0.333263i	$-0.891850\pi$
$293$	−9457.29	−1.88567	−0.942834	+	0.333263i	$0.891850\pi$
$294$	2198.14	0.436048
$295$	0	0
$296$	1466.09	0.287887
$297$	−2368.67	−0.462776
$298$	−3249.89	−0.631749
$299$	1580.49	0.305693
$300$	0	0
$301$	−2019.29	−0.386678
$302$	−5617.19	−1.07031
$303$	2061.99	0.390951
$304$	−1090.95	−0.205824
$305$	0	0
$306$	2905.57	0.542812
$307$	−4576.54	−0.850804	−0.425402	−	0.905004i	$-0.639867\pi$
$307$	−4576.54	−0.850804	−0.425402	+	0.905004i	$0.639867\pi$
$308$	−2601.76	−0.481328
$309$	−5173.58	−0.952475
$310$	0	0
$311$	−3691.20	−0.673018	−0.336509	−	0.941680i	$-0.609246\pi$
$311$	−3691.20	−0.673018	−0.336509	+	0.941680i	$0.609246\pi$
$312$	3255.77	0.590774
$313$	−9771.57	−1.76461	−0.882303	−	0.470682i	$-0.844008\pi$
$313$	−9771.57	−1.76461	−0.882303	+	0.470682i	$0.844008\pi$
$314$	3134.53	0.563349
$315$	0	0
$316$	1235.13	0.219878
$317$	9653.06	1.71032	0.855158	−	0.518368i	$-0.173460\pi$
$317$	9653.06	1.71032	0.855158	+	0.518368i	$0.173460\pi$
$318$	−6202.24	−1.09372
$319$	−692.864	−0.121608
$320$	0	0
$321$	−4109.60	−0.714565
$322$	−1393.79	−0.241221
$323$	−1653.85	−0.284900
$324$	−298.073	−0.0511098
$325$	0	0
$326$	945.597	0.160650
$327$	4169.26	0.705079
$328$	131.465	0.0221310
$329$	−5297.86	−0.887783
$330$	0	0
$331$	2920.59	0.484985	0.242492	−	0.970153i	$-0.422035\pi$
$331$	2920.59	0.484985	0.242492	+	0.970153i	$0.422035\pi$
$332$	−13535.4	−2.23750
$333$	−1648.56	−0.271293
$334$	17948.5	2.94042
$335$	0	0
$336$	1215.37	0.197333
$337$	7794.72	1.25996	0.629978	−	0.776613i	$-0.283064\pi$
$337$	7794.72	1.25996	0.629978	+	0.776613i	$0.283064\pi$
$338$	11090.0	1.78466
$339$	−4818.90	−0.772055
$340$	0	0
$341$	3914.18	0.621597
$342$	−2898.03	−0.458208
$343$	−6838.47	−1.07651
$344$	−2114.47	−0.331409
$345$	0	0
$346$	3772.19	0.586111
$347$	7025.93	1.08695	0.543475	−	0.839425i	$-0.317108\pi$
$347$	7025.93	1.08695	0.543475	+	0.839425i	$0.317108\pi$
$348$	−1535.77	−0.236569
$349$	6338.40	0.972168	0.486084	−	0.873912i	$-0.338425\pi$
$349$	6338.40	0.972168	0.486084	+	0.873912i	$0.338425\pi$
$350$	0	0
$351$	−9745.22	−1.48194
$352$	3900.98	0.590691
$353$	−8834.31	−1.33202	−0.666010	−	0.745943i	$-0.731999\pi$
$353$	−8834.31	−1.33202	−0.666010	+	0.745943i	$0.731999\pi$
$354$	6352.49	0.953760
$355$	0	0
$356$	1681.50	0.250335
$357$	1842.46	0.273147
$358$	7327.86	1.08181
$359$	−917.755	−0.134923	−0.0674613	−	0.997722i	$-0.521490\pi$
$359$	−917.755	−0.134923	−0.0674613	+	0.997722i	$0.521490\pi$
$360$	0	0
$361$	−5209.45	−0.759505
$362$	−1527.93	−0.221840
$363$	3449.89	0.498822
$364$	−10704.2	−1.54135
$365$	0	0
$366$	−1061.12	−0.151545
$367$	5451.98	0.775453	0.387727	−	0.921774i	$-0.373260\pi$
$367$	5451.98	0.775453	0.387727	+	0.921774i	$0.373260\pi$
$368$	617.804	0.0875142
$369$	−147.828	−0.0208553
$370$	0	0
$371$	5941.80	0.831491
$372$	8676.01	1.20922
$373$	9583.17	1.33029	0.665144	−	0.746715i	$-0.268370\pi$
$373$	9583.17	1.33029	0.665144	+	0.746715i	$0.268370\pi$
$374$	2987.12	0.412996
$375$	0	0
$376$	−5547.58	−0.760890
$377$	−2850.59	−0.389424
$378$	8594.05	1.16939
$379$	5560.92	0.753681	0.376841	−	0.926278i	$-0.377010\pi$
$379$	5560.92	0.753681	0.376841	+	0.926278i	$0.377010\pi$
$380$	0	0
$381$	1461.28	0.196493
$382$	10173.9	1.36268
$383$	−4847.50	−0.646725	−0.323363	−	0.946275i	$-0.604813\pi$
$383$	−4847.50	−0.646725	−0.323363	+	0.946275i	$0.604813\pi$
$384$	5551.82	0.737800
$385$	0	0
$386$	6720.07	0.886120
$387$	2377.65	0.312306
$388$	20656.4	2.70276
$389$	−11572.6	−1.50837	−0.754184	−	0.656663i	$-0.771967\pi$
$389$	−11572.6	−1.50837	−0.754184	+	0.656663i	$0.771967\pi$
$390$	0	0
$391$	936.570	0.121137
$392$	−2205.10	−0.284118
$393$	−2068.17	−0.265459
$394$	−7899.83	−1.01012
$395$	0	0
$396$	3063.48	0.388752
$397$	−5767.82	−0.729165	−0.364582	−	0.931171i	$-0.618788\pi$
$397$	−5767.82	−0.729165	−0.364582	+	0.931171i	$0.618788\pi$
$398$	14194.6	1.78771
$399$	−1837.68	−0.230574
$400$	0	0
$401$	14990.1	1.86675	0.933377	−	0.358897i	$-0.116847\pi$
$401$	14990.1	1.86675	0.933377	+	0.358897i	$0.116847\pi$
$402$	3534.87	0.438565
$403$	16103.8	1.99053
$404$	−7098.87	−0.874213
$405$	0	0
$406$	2513.86	0.307292
$407$	−1694.83	−0.206412
$408$	1929.31	0.234105
$409$	−1487.95	−0.179888	−0.0899442	−	0.995947i	$-0.528669\pi$
$409$	−1487.95	−0.179888	−0.0899442	+	0.995947i	$0.528669\pi$
$410$	0	0
$411$	−1573.55	−0.188850
$412$	17811.2	2.12985
$413$	−6085.74	−0.725084
$414$	1641.14	0.194826
$415$	0	0
$416$	16049.5	1.89156
$417$	663.725	0.0779442
$418$	−2979.36	−0.348625
$419$	−2739.23	−0.319380	−0.159690	−	0.987167i	$-0.551049\pi$
$419$	−2739.23	−0.319380	−0.159690	+	0.987167i	$0.551049\pi$
$420$	0	0
$421$	2205.79	0.255353	0.127676	−	0.991816i	$-0.459248\pi$
$421$	2205.79	0.255353	0.127676	+	0.991816i	$0.459248\pi$
$422$	−22837.5	−2.63439
$423$	6238.05	0.717032
$424$	6221.87	0.712644
$425$	0	0
$426$	3479.84	0.395772
$427$	1016.56	0.115210
$428$	14148.2	1.59785
$429$	−3763.75	−0.423579
$430$	0	0
$431$	12377.3	1.38329	0.691643	−	0.722240i	$-0.256887\pi$
$431$	12377.3	1.38329	0.691643	+	0.722240i	$0.256887\pi$
$432$	−3809.34	−0.424252
$433$	−11610.5	−1.28860	−0.644299	−	0.764774i	$-0.722851\pi$
$433$	−11610.5	−1.28860	−0.644299	+	0.764774i	$0.722851\pi$
$434$	−14201.5	−1.57072
$435$	0	0
$436$	−14353.7	−1.57664
$437$	−934.138	−0.102256
$438$	−2695.15	−0.294016
$439$	−10585.8	−1.15087	−0.575433	−	0.817849i	$-0.695167\pi$
$439$	−10585.8	−1.15087	−0.575433	+	0.817849i	$0.695167\pi$
$440$	0	0
$441$	2479.55	0.267741
$442$	12289.6	1.32253
$443$	13572.0	1.45559	0.727796	−	0.685794i	$-0.240545\pi$
$443$	13572.0	1.45559	0.727796	+	0.685794i	$0.240545\pi$
$444$	−3756.69	−0.401542
$445$	0	0
$446$	13810.3	1.46623
$447$	2426.52	0.256757
$448$	−11188.6	−1.17994
$449$	8401.73	0.883079	0.441539	−	0.897242i	$-0.354433\pi$
$449$	8401.73	0.883079	0.441539	+	0.897242i	$0.354433\pi$
$450$	0	0
$451$	−151.977	−0.0158677
$452$	16590.2	1.72641
$453$	4194.06	0.434998
$454$	−1153.52	−0.119245
$455$	0	0
$456$	−1924.30	−0.197617
$457$	−10406.0	−1.06515	−0.532573	−	0.846384i	$-0.678775\pi$
$457$	−10406.0	−1.06515	−0.532573	+	0.846384i	$0.678775\pi$
$458$	−24418.2	−2.49124
$459$	−5774.83	−0.587246
$460$	0	0
$461$	11789.6	1.19110	0.595548	−	0.803320i	$-0.296935\pi$
$461$	11789.6	1.19110	0.595548	+	0.803320i	$0.296935\pi$
$462$	3319.15	0.334244
$463$	−15713.0	−1.57721	−0.788604	−	0.614902i	$-0.789196\pi$
$463$	−15713.0	−1.57721	−0.788604	+	0.614902i	$0.789196\pi$
$464$	−1114.27	−0.111485
$465$	0	0
$466$	−19347.9	−1.92333
$467$	−5213.88	−0.516637	−0.258319	−	0.966060i	$-0.583168\pi$
$467$	−5213.88	−0.516637	−0.258319	+	0.966060i	$0.583168\pi$
$468$	12603.8	1.24490
$469$	−3386.43	−0.333413
$470$	0	0
$471$	−2340.38	−0.228958
$472$	−6372.59	−0.621446
$473$	2444.38	0.237617
$474$	−1575.69	−0.152687
$475$	0	0
$476$	−6343.10	−0.610789
$477$	−6996.27	−0.671566
$478$	28306.4	2.70858
$479$	−7912.84	−0.754796	−0.377398	−	0.926051i	$-0.623181\pi$
$479$	−7912.84	−0.754796	−0.377398	+	0.926051i	$0.623181\pi$
$480$	0	0
$481$	−6972.89	−0.660991
$482$	−29464.6	−2.78439
$483$	1040.67	0.0980377
$484$	−11877.1	−1.11543
$485$	0	0
$486$	−16436.9	−1.53414
$487$	−2710.59	−0.252215	−0.126108	−	0.992017i	$-0.540248\pi$
$487$	−2710.59	−0.252215	−0.126108	+	0.992017i	$0.540248\pi$
$488$	1064.47	0.0987428
$489$	−706.026	−0.0652917
$490$	0	0
$491$	−17283.4	−1.58858	−0.794288	−	0.607541i	$-0.792156\pi$
$491$	−17283.4	−1.58858	−0.794288	+	0.607541i	$0.792156\pi$
$492$	−336.866	−0.0308681
$493$	−1689.20	−0.154316
$494$	−12257.7	−1.11640
$495$	0	0
$496$	6294.84	0.569853
$497$	−3333.72	−0.300881
$498$	17267.5	1.55376
$499$	16521.7	1.48219	0.741096	−	0.671399i	$-0.234306\pi$
$499$	16521.7	1.48219	0.741096	+	0.671399i	$0.234306\pi$
$500$	0	0
$501$	−13401.2	−1.19505
$502$	−1271.03	−0.113006
$503$	−16269.1	−1.44215	−0.721075	−	0.692857i	$-0.756352\pi$
$503$	−16269.1	−1.44215	−0.721075	+	0.692857i	$0.756352\pi$
$504$	−3238.75	−0.286241
$505$	0	0
$506$	1687.21	0.148232
$507$	−8280.29	−0.725326
$508$	−5030.80	−0.439381
$509$	−9299.84	−0.809839	−0.404920	−	0.914352i	$-0.632701\pi$
$509$	−9299.84	−0.809839	−0.404920	+	0.914352i	$0.632701\pi$
$510$	0	0
$511$	2581.97	0.223522
$512$	9378.36	0.809509
$513$	5759.84	0.495717
$514$	−17744.7	−1.52274
$515$	0	0
$516$	5418.11	0.462246
$517$	6413.13	0.545550
$518$	6149.21	0.521584
$519$	−2816.49	−0.238209
$520$	0	0
$521$	−23076.6	−1.94050	−0.970252	−	0.242096i	$-0.922165\pi$
$521$	−23076.6	−1.94050	−0.970252	+	0.242096i	$0.922165\pi$
$522$	−2959.98	−0.248189
$523$	−10348.3	−0.865200	−0.432600	−	0.901586i	$-0.642404\pi$
$523$	−10348.3	−0.865200	−0.432600	+	0.901586i	$0.642404\pi$
$524$	7120.17	0.593599
$525$	0	0
$526$	−23060.8	−1.91160
$527$	9542.78	0.788785
$528$	−1471.22	−0.121263
$529$	529.000	0.0434783
$530$	0	0
$531$	7165.75	0.585625
$532$	6326.63	0.515590
$533$	−625.265	−0.0508128
$534$	−2145.14	−0.173837
$535$	0	0
$536$	−3546.05	−0.285758
$537$	−5471.32	−0.439674
$538$	−10617.9	−0.850872
$539$	2549.15	0.203710
$540$	0	0
$541$	2078.43	0.165173	0.0825867	−	0.996584i	$-0.473682\pi$
$541$	2078.43	0.165173	0.0825867	+	0.996584i	$0.473682\pi$
$542$	−28960.1	−2.29510
$543$	1140.82	0.0901607
$544$	9510.61	0.749566
$545$	0	0
$546$	13655.7	1.07035
$547$	16340.5	1.27727	0.638637	−	0.769509i	$-0.279499\pi$
$547$	16340.5	1.27727	0.638637	+	0.769509i	$0.279499\pi$
$548$	5417.29	0.422291
$549$	−1196.96	−0.0930512
$550$	0	0
$551$	1684.82	0.130264
$552$	1089.72	0.0840249
$553$	1509.52	0.116079
$554$	12415.8	0.952159
$555$	0	0
$556$	−2285.03	−0.174293
$557$	−4738.43	−0.360455	−0.180228	−	0.983625i	$-0.557683\pi$
$557$	−4738.43	−0.360455	−0.180228	+	0.983625i	$0.557683\pi$
$558$	16721.7	1.26862
$559$	10056.7	0.760917
$560$	0	0
$561$	−2230.32	−0.167851
$562$	−8942.84	−0.671230
$563$	−19003.9	−1.42259	−0.711294	−	0.702894i	$-0.751891\pi$
$563$	−19003.9	−1.42259	−0.711294	+	0.702894i	$0.751891\pi$
$564$	14215.1	1.06128
$565$	0	0
$566$	19633.0	1.45801
$567$	−364.293	−0.0269821
$568$	−3490.86	−0.257875
$569$	−2789.77	−0.205542	−0.102771	−	0.994705i	$-0.532771\pi$
$569$	−2789.77	−0.205542	−0.102771	+	0.994705i	$0.532771\pi$
$570$	0	0
$571$	−1154.31	−0.0845998	−0.0422999	−	0.999105i	$-0.513468\pi$
$571$	−1154.31	−0.0845998	−0.0422999	+	0.999105i	$0.513468\pi$
$572$	12957.6	0.947173
$573$	−7596.33	−0.553824
$574$	551.404	0.0400961
$575$	0	0
$576$	13174.2	0.952996
$577$	2360.98	0.170345	0.0851723	−	0.996366i	$-0.472856\pi$
$577$	2360.98	0.170345	0.0851723	+	0.996366i	$0.472856\pi$
$578$	−14295.3	−1.02873
$579$	−5017.52	−0.360140
$580$	0	0
$581$	−16542.4	−1.18123
$582$	−26352.0	−1.87685
$583$	−7192.63	−0.510958
$584$	2703.68	0.191573
$585$	0	0
$586$	−41536.4	−2.92808
$587$	6641.99	0.467026	0.233513	−	0.972354i	$-0.424978\pi$
$587$	6641.99	0.467026	0.233513	+	0.972354i	$0.424978\pi$
$588$	5650.33	0.396285
$589$	−9518.00	−0.665844
$590$	0	0
$591$	5898.38	0.410536
$592$	−2725.65	−0.189229
$593$	−3879.68	−0.268667	−0.134334	−	0.990936i	$-0.542889\pi$
$593$	−3879.68	−0.268667	−0.134334	+	0.990936i	$0.542889\pi$
$594$	−10403.2	−0.718601
$595$	0	0
$596$	−8353.86	−0.574140
$597$	−10598.3	−0.726568
$598$	6941.52	0.474682
$599$	−8316.59	−0.567290	−0.283645	−	0.958929i	$-0.591544\pi$
$599$	−8316.59	−0.567290	−0.283645	+	0.958929i	$0.591544\pi$
$600$	0	0
$601$	14526.1	0.985908	0.492954	−	0.870055i	$-0.335917\pi$
$601$	14526.1	0.985908	0.492954	+	0.870055i	$0.335917\pi$
$602$	−8868.72	−0.600435
$603$	3987.41	0.269286
$604$	−14439.0	−0.972707
$605$	0	0
$606$	9056.25	0.607071
$607$	−12982.3	−0.868097	−0.434049	−	0.900890i	$-0.642915\pi$
$607$	−12982.3	−0.868097	−0.434049	+	0.900890i	$0.642915\pi$
$608$	−9485.91	−0.632738
$609$	−1876.96	−0.124890
$610$	0	0
$611$	26385.0	1.74701
$612$	7468.78	0.493313
$613$	−23179.5	−1.52726	−0.763631	−	0.645653i	$-0.776585\pi$
$613$	−23179.5	−1.52726	−0.763631	+	0.645653i	$0.776585\pi$
$614$	−20100.2	−1.32113
$615$	0	0
$616$	−3329.65	−0.217785
$617$	−6322.17	−0.412514	−0.206257	−	0.978498i	$-0.566128\pi$
$617$	−6322.17	−0.412514	−0.206257	+	0.978498i	$0.566128\pi$
$618$	−22722.4	−1.47901
$619$	10500.8	0.681849	0.340924	−	0.940091i	$-0.389260\pi$
$619$	10500.8	0.681849	0.340924	+	0.940091i	$0.389260\pi$
$620$	0	0
$621$	−3261.78	−0.210774
$622$	−16211.7	−1.04507
$623$	2055.06	0.132158
$624$	−6052.91	−0.388318
$625$	0	0
$626$	−42916.7	−2.74009
$627$	2224.53	0.141689
$628$	8057.31	0.511977
$629$	−4132.00	−0.261930
$630$	0	0
$631$	15399.7	0.971554	0.485777	−	0.874083i	$-0.338537\pi$
$631$	15399.7	0.971554	0.485777	+	0.874083i	$0.338537\pi$
$632$	1580.68	0.0994872
$633$	17051.5	1.07067
$634$	42396.2	2.65579
$635$	0	0
$636$	−15942.9	−0.993989
$637$	10487.7	0.652337
$638$	−3043.06	−0.188833
$639$	3925.34	0.243011
$640$	0	0
$641$	2873.00	0.177030	0.0885152	−	0.996075i	$-0.471788\pi$
$641$	2873.00	0.177030	0.0885152	+	0.996075i	$0.471788\pi$
$642$	−18049.3	−1.10958
$643$	−15141.3	−0.928638	−0.464319	−	0.885668i	$-0.653701\pi$
$643$	−15141.3	−0.928638	−0.464319	+	0.885668i	$0.653701\pi$
$644$	−3582.75	−0.219224
$645$	0	0
$646$	−7263.70	−0.442394
$647$	−5086.61	−0.309081	−0.154540	−	0.987986i	$-0.549390\pi$
$647$	−5086.61	−0.309081	−0.154540	+	0.987986i	$0.549390\pi$
$648$	−381.464	−0.0231255
$649$	7366.87	0.445570
$650$	0	0
$651$	10603.5	0.638376
$652$	2430.66	0.146000
$653$	15214.0	0.911744	0.455872	−	0.890045i	$-0.349327\pi$
$653$	15214.0	0.911744	0.455872	+	0.890045i	$0.349327\pi$
$654$	18311.4	1.09485
$655$	0	0
$656$	−244.412	−0.0145468
$657$	−3040.18	−0.180531
$658$	−23268.2	−1.37855
$659$	20029.9	1.18400	0.591999	−	0.805938i	$-0.298339\pi$
$659$	20029.9	1.18400	0.591999	+	0.805938i	$0.298339\pi$
$660$	0	0
$661$	−92.3018	−0.00543135	−0.00271568	−	0.999996i	$-0.500864\pi$
$661$	−92.3018	−0.00543135	−0.00271568	+	0.999996i	$0.500864\pi$
$662$	12827.2	0.753087
$663$	−9176.02	−0.537507
$664$	−17322.1	−1.01239
$665$	0	0
$666$	−7240.48	−0.421265
$667$	−954.107	−0.0553871
$668$	46136.7	2.67228
$669$	−10311.4	−0.595910
$670$	0	0
$671$	−1230.56	−0.0707975
$672$	10567.7	0.606635
$673$	−15452.2	−0.885048	−0.442524	−	0.896757i	$-0.645917\pi$
$673$	−15452.2	−0.885048	−0.442524	+	0.896757i	$0.645917\pi$
$674$	34234.4	1.95647
$675$	0	0
$676$	28506.8	1.62192
$677$	−19199.7	−1.08996	−0.544980	−	0.838449i	$-0.683463\pi$
$677$	−19199.7	−1.08996	−0.544980	+	0.838449i	$0.683463\pi$
$678$	−21164.6	−1.19885
$679$	25245.4	1.42685
$680$	0	0
$681$	861.268	0.0484638
$682$	17191.1	0.965220
$683$	20090.5	1.12554	0.562769	−	0.826614i	$-0.309736\pi$
$683$	20090.5	1.12554	0.562769	+	0.826614i	$0.309736\pi$
$684$	−7449.38	−0.416424
$685$	0	0
$686$	−30034.6	−1.67161
$687$	18231.7	1.01250
$688$	3931.09	0.217836
$689$	−29592.0	−1.63623
$690$	0	0
$691$	17376.4	0.956629	0.478315	−	0.878189i	$-0.341248\pi$
$691$	17376.4	0.956629	0.478315	+	0.878189i	$0.341248\pi$
$692$	9696.43	0.532663
$693$	3744.07	0.205231
$694$	30857.9	1.68782
$695$	0	0
$696$	−1965.43	−0.107040
$697$	−370.520	−0.0201355
$698$	27838.2	1.50959
$699$	14446.0	0.781686
$700$	0	0
$701$	11240.9	0.605651	0.302825	−	0.953046i	$-0.402070\pi$
$701$	11240.9	0.605651	0.302825	+	0.953046i	$0.402070\pi$
$702$	−42801.0	−2.30117
$703$	4121.27	0.221105
$704$	13544.0	0.725082
$705$	0	0
$706$	−38800.3	−2.06837
$707$	−8675.96	−0.461518
$708$	16329.1	0.866787
$709$	−35667.3	−1.88930	−0.944649	−	0.328082i	$-0.893598\pi$
$709$	−35667.3	−1.88930	−0.944649	+	0.328082i	$0.893598\pi$
$710$	0	0
$711$	−1777.41	−0.0937527
$712$	2151.93	0.113268
$713$	5390.02	0.283111
$714$	8092.08	0.424144
$715$	0	0
$716$	18836.3	0.983164
$717$	−21134.9	−1.10083
$718$	−4030.78	−0.209509
$719$	−25533.1	−1.32437	−0.662187	−	0.749339i	$-0.730371\pi$
$719$	−25533.1	−1.32437	−0.662187	+	0.749339i	$0.730371\pi$
$720$	0	0
$721$	21768.2	1.12440
$722$	−22879.9	−1.17936
$723$	21999.7	1.13164
$724$	−3927.54	−0.201610
$725$	0	0
$726$	15151.9	0.774574
$727$	−27933.1	−1.42501	−0.712504	−	0.701668i	$-0.752439\pi$
$727$	−27933.1	−1.42501	−0.712504	+	0.701668i	$0.752439\pi$
$728$	−13698.9	−0.697410
$729$	12985.4	0.659728
$730$	0	0
$731$	5959.40	0.301527
$732$	−2727.60	−0.137726
$733$	29359.9	1.47945	0.739723	−	0.672912i	$-0.234957\pi$
$733$	29359.9	1.47945	0.739723	+	0.672912i	$0.234957\pi$
$734$	23945.1	1.20413
$735$	0	0
$736$	5371.85	0.269034
$737$	4099.32	0.204885
$738$	−649.260	−0.0323843
$739$	−29310.8	−1.45902	−0.729510	−	0.683970i	$-0.760252\pi$
$739$	−29310.8	−1.45902	−0.729510	+	0.683970i	$0.760252\pi$
$740$	0	0
$741$	9152.19	0.453730
$742$	26096.4	1.29114
$743$	−17147.7	−0.846688	−0.423344	−	0.905969i	$-0.639144\pi$
$743$	−17147.7	−0.846688	−0.423344	+	0.905969i	$0.639144\pi$
$744$	11103.3	0.547132
$745$	0	0
$746$	42089.3	2.06568
$747$	19478.1	0.954037
$748$	7678.41	0.375335
$749$	17291.4	0.843545
$750$	0	0
$751$	−37105.7	−1.80294	−0.901469	−	0.432843i	$-0.857511\pi$
$751$	−37105.7	−1.80294	−0.901469	+	0.432843i	$0.857511\pi$
$752$	10313.7	0.500136
$753$	949.011	0.0459281
$754$	−12519.8	−0.604699
$755$	0	0
$756$	22091.0	1.06276
$757$	26843.4	1.28883	0.644413	−	0.764678i	$-0.277102\pi$
$757$	26843.4	1.28883	0.644413	+	0.764678i	$0.277102\pi$
$758$	24423.5	1.17032
$759$	−1259.75	−0.0602449
$760$	0	0
$761$	−38055.7	−1.81277	−0.906384	−	0.422455i	$-0.861168\pi$
$761$	−38055.7	−1.81277	−0.906384	+	0.422455i	$0.861168\pi$
$762$	6417.94	0.305115
$763$	−17542.5	−0.832347
$764$	26152.1	1.23842
$765$	0	0
$766$	−21290.2	−1.00424
$767$	30308.8	1.42684
$768$	3110.30	0.146137
$769$	11248.9	0.527496	0.263748	−	0.964592i	$-0.415041\pi$
$769$	11248.9	0.527496	0.263748	+	0.964592i	$0.415041\pi$
$770$	0	0
$771$	13249.0	0.618875
$772$	17274.0	0.805315
$773$	22704.0	1.05641	0.528205	−	0.849117i	$-0.322865\pi$
$773$	22704.0	1.05641	0.528205	+	0.849117i	$0.322865\pi$
$774$	10442.6	0.484951
$775$	0	0
$776$	26435.4	1.22291
$777$	−4591.28	−0.211984
$778$	−50826.9	−2.34220
$779$	369.558	0.0169972
$780$	0	0
$781$	4035.51	0.184894
$782$	4113.41	0.188101
$783$	5882.97	0.268506
$784$	4099.58	0.186752
$785$	0	0
$786$	−9083.41	−0.412207
$787$	24801.6	1.12336	0.561678	−	0.827356i	$-0.310156\pi$
$787$	24801.6	1.12336	0.561678	+	0.827356i	$0.310156\pi$
$788$	−20306.5	−0.918008
$789$	17218.3	0.776917
$790$	0	0
$791$	20275.9	0.911412
$792$	3920.55	0.175897
$793$	−5062.77	−0.226714
$794$	−25332.2	−1.13225
$795$	0	0
$796$	36487.2	1.62469
$797$	10289.9	0.457323	0.228662	−	0.973506i	$-0.426565\pi$
$797$	10289.9	0.457323	0.228662	+	0.973506i	$0.426565\pi$
$798$	−8071.07	−0.358036
$799$	15635.2	0.692284
$800$	0	0
$801$	−2419.76	−0.106739
$802$	65836.4	2.89871
$803$	−3125.51	−0.137356
$804$	9086.38	0.398572
$805$	0	0
$806$	70727.7	3.09091
$807$	7927.80	0.345814
$808$	−9084.91	−0.395552
$809$	−17890.5	−0.777500	−0.388750	−	0.921343i	$-0.627093\pi$
$809$	−17890.5	−0.777500	−0.388750	+	0.921343i	$0.627093\pi$
$810$	0	0
$811$	−19915.9	−0.862322	−0.431161	−	0.902275i	$-0.641896\pi$
$811$	−19915.9	−0.862322	−0.431161	+	0.902275i	$0.641896\pi$
$812$	6461.87	0.279270
$813$	21622.9	0.932779
$814$	−7443.70	−0.320518
$815$	0	0
$816$	−3586.84	−0.153878
$817$	−5943.93	−0.254531
$818$	−6535.07	−0.279332
$819$	15403.9	0.657211
$820$	0	0
$821$	−13709.3	−0.582776	−0.291388	−	0.956605i	$-0.594117\pi$
$821$	−13709.3	−0.582776	−0.291388	+	0.956605i	$0.594117\pi$
$822$	−6911.01	−0.293247
$823$	−26399.7	−1.11815	−0.559074	−	0.829118i	$-0.688843\pi$
$823$	−26399.7	−1.11815	−0.559074	+	0.829118i	$0.688843\pi$
$824$	22794.3	0.963684
$825$	0	0
$826$	−26728.6	−1.12591
$827$	18410.9	0.774135	0.387067	−	0.922051i	$-0.373488\pi$
$827$	18410.9	0.774135	0.387067	+	0.922051i	$0.373488\pi$
$828$	4218.57	0.177060
$829$	8828.44	0.369873	0.184936	−	0.982750i	$-0.440792\pi$
$829$	8828.44	0.369873	0.184936	+	0.982750i	$0.440792\pi$
$830$	0	0
$831$	−9270.20	−0.386979
$832$	55722.8	2.32192
$833$	6214.82	0.258500
$834$	2915.08	0.121032
$835$	0	0
$836$	−7658.47	−0.316834
$837$	−33234.5	−1.37246
$838$	−12030.7	−0.495935
$839$	−18062.0	−0.743231	−0.371616	−	0.928387i	$-0.621196\pi$
$839$	−18062.0	−0.743231	−0.371616	+	0.928387i	$0.621196\pi$
$840$	0	0
$841$	−22668.2	−0.929442
$842$	9687.81	0.396513
$843$	6677.14	0.272803
$844$	−58703.8	−2.39416
$845$	0	0
$846$	27397.5	1.11341
$847$	−14515.7	−0.588860
$848$	−11567.3	−0.468423
$849$	−14658.9	−0.592570
$850$	0	0
$851$	−2333.87	−0.0940116
$852$	8944.95	0.359682
$853$	−5289.59	−0.212324	−0.106162	−	0.994349i	$-0.533856\pi$
$853$	−5289.59	−0.212324	−0.106162	+	0.994349i	$0.533856\pi$
$854$	4464.72	0.178899
$855$	0	0
$856$	18106.5	0.722975
$857$	11311.0	0.450846	0.225423	−	0.974261i	$-0.427624\pi$
$857$	11311.0	0.450846	0.225423	+	0.974261i	$0.427624\pi$
$858$	−16530.4	−0.657736
$859$	2466.99	0.0979892	0.0489946	−	0.998799i	$-0.484398\pi$
$859$	2466.99	0.0979892	0.0489946	+	0.998799i	$0.484398\pi$
$860$	0	0
$861$	−411.704	−0.0162960
$862$	54361.3	2.14797
$863$	8715.03	0.343758	0.171879	−	0.985118i	$-0.445016\pi$
$863$	8715.03	0.343758	0.171879	+	0.985118i	$0.445016\pi$
$864$	−33122.5	−1.30422
$865$	0	0
$866$	−50993.1	−2.00094
$867$	10673.5	0.418099
$868$	−36504.9	−1.42749
$869$	−1827.30	−0.0713313
$870$	0	0
$871$	16865.5	0.656102
$872$	−18369.4	−0.713377
$873$	−29725.6	−1.15242
$874$	−4102.73	−0.158784
$875$	0	0
$876$	−6927.88	−0.267205
$877$	2707.70	0.104256	0.0521280	−	0.998640i	$-0.483400\pi$
$877$	2707.70	0.104256	0.0521280	+	0.998640i	$0.483400\pi$
$878$	−46492.6	−1.78707
$879$	31013.0	1.19004
$880$	0	0
$881$	50363.6	1.92599	0.962993	−	0.269527i	$-0.0868675\pi$
$881$	50363.6	1.92599	0.962993	+	0.269527i	$0.0868675\pi$
$882$	10890.2	0.415750
$883$	29515.0	1.12487	0.562434	−	0.826842i	$-0.309865\pi$
$883$	29515.0	1.12487	0.562434	+	0.826842i	$0.309865\pi$
$884$	31590.6	1.20193
$885$	0	0
$886$	59608.4	2.26025
$887$	16506.7	0.624849	0.312425	−	0.949943i	$-0.398859\pi$
$887$	16506.7	0.624849	0.312425	+	0.949943i	$0.398859\pi$
$888$	−4807.70	−0.181684
$889$	−6148.44	−0.231960
$890$	0	0
$891$	440.981	0.0165807
$892$	35499.5	1.33253
$893$	−15594.6	−0.584383
$894$	10657.3	0.398694
$895$	0	0
$896$	−23359.7	−0.870974
$897$	−5182.86	−0.192922
$898$	36900.4	1.37125
$899$	−9721.47	−0.360655
$900$	0	0
$901$	−17535.6	−0.648388
$902$	−667.482	−0.0246394
$903$	6621.80	0.244031
$904$	21231.6	0.781142
$905$	0	0
$906$	18420.3	0.675467
$907$	9107.76	0.333427	0.166713	−	0.986005i	$-0.446685\pi$
$907$	9107.76	0.333427	0.166713	+	0.986005i	$0.446685\pi$
$908$	−2965.12	−0.108371
$909$	10215.6	0.372752
$910$	0	0
$911$	23664.9	0.860652	0.430326	−	0.902674i	$-0.358399\pi$
$911$	23664.9	0.860652	0.430326	+	0.902674i	$0.358399\pi$
$912$	3577.53	0.129894
$913$	20024.8	0.725874
$914$	−45703.1	−1.65396
$915$	0	0
$916$	−62767.0	−2.26406
$917$	8701.99	0.313375
$918$	−25363.1	−0.911880
$919$	30834.5	1.10679	0.553393	−	0.832920i	$-0.313333\pi$
$919$	30834.5	1.10679	0.553393	+	0.832920i	$0.313333\pi$
$920$	0	0
$921$	15007.7	0.536939
$922$	51779.8	1.84954
$923$	16602.9	0.592083
$924$	8531.88	0.303764
$925$	0	0
$926$	−69011.7	−2.44910
$927$	−25631.3	−0.908137
$928$	−9688.70	−0.342723
$929$	17648.9	0.623297	0.311648	−	0.950197i	$-0.399119\pi$
$929$	17648.9	0.623297	0.311648	+	0.950197i	$0.399119\pi$
$930$	0	0
$931$	−6198.68	−0.218210
$932$	−49733.8	−1.74794
$933$	12104.4	0.424739
$934$	−22899.4	−0.802238
$935$	0	0
$936$	16130.0	0.563274
$937$	−23621.9	−0.823580	−0.411790	−	0.911279i	$-0.635096\pi$
$937$	−23621.9	−0.823580	−0.411790	+	0.911279i	$0.635096\pi$
$938$	−14873.2	−0.517726
$939$	32043.6	1.11364
$940$	0	0
$941$	−8129.95	−0.281646	−0.140823	−	0.990035i	$-0.544975\pi$
$941$	−8129.95	−0.281646	−0.140823	+	0.990035i	$0.544975\pi$
$942$	−10279.0	−0.355527
$943$	−209.280	−0.00722703
$944$	11847.5	0.408478
$945$	0	0
$946$	10735.7	0.368972
$947$	52681.5	1.80773	0.903865	−	0.427818i	$-0.140718\pi$
$947$	52681.5	1.80773	0.903865	+	0.427818i	$0.140718\pi$
$948$	−4050.31	−0.138764
$949$	−12859.0	−0.439854
$950$	0	0
$951$	−31655.0	−1.07937
$952$	−8117.69	−0.276361
$953$	48109.3	1.63527	0.817636	−	0.575735i	$-0.195284\pi$
$953$	48109.3	1.63527	0.817636	+	0.575735i	$0.195284\pi$
$954$	−30727.6	−1.04281
$955$	0	0
$956$	72761.6	2.46159
$957$	2272.09	0.0767462
$958$	−34753.2	−1.17205
$959$	6620.80	0.222937
$960$	0	0
$961$	25128.3	0.843486
$962$	−30624.9	−1.02639
$963$	−20360.1	−0.681302
$964$	−75739.0	−2.53049
$965$	0	0
$966$	4570.63	0.152233
$967$	−41498.5	−1.38004	−0.690022	−	0.723789i	$-0.742399\pi$
$967$	−41498.5	−1.38004	−0.690022	+	0.723789i	$0.742399\pi$
$968$	−15199.9	−0.504693
$969$	5423.41	0.179799
$970$	0	0
$971$	2030.35	0.0671031	0.0335515	−	0.999437i	$-0.489318\pi$
$971$	2030.35	0.0671031	0.0335515	+	0.999437i	$0.489318\pi$
$972$	−42251.1	−1.39424
$973$	−2792.67	−0.0920132
$974$	−11904.9	−0.391641
$975$	0	0
$976$	−1979.00	−0.0649040
$977$	42161.7	1.38063	0.690313	−	0.723511i	$-0.257473\pi$
$977$	42161.7	1.38063	0.690313	+	0.723511i	$0.257473\pi$
$978$	−3100.87	−0.101385
$979$	−2487.68	−0.0812120
$980$	0	0
$981$	20655.7	0.672258
$982$	−75908.9	−2.46675
$983$	44619.0	1.44774	0.723869	−	0.689938i	$-0.242362\pi$
$983$	44619.0	1.44774	0.723869	+	0.689938i	$0.242362\pi$
$984$	−431.110	−0.0139668
$985$	0	0
$986$	−7418.98	−0.239623
$987$	17373.1	0.560276
$988$	−31508.5	−1.01460
$989$	3366.03	0.108224
$990$	0	0
$991$	−18737.8	−0.600633	−0.300316	−	0.953840i	$-0.597092\pi$
$991$	−18737.8	−0.600633	−0.300316	+	0.953840i	$0.597092\pi$
$992$	54734.1	1.75183
$993$	−9577.39	−0.306072
$994$	−14641.7	−0.467209
$995$	0	0
$996$	44386.1	1.41208
$997$	−48629.2	−1.54474	−0.772368	−	0.635175i	$-0.780928\pi$
$997$	−48629.2	−1.54474	−0.772368	+	0.635175i	$0.780928\pi$
$998$	72563.3	2.30156
$999$	14390.5	0.455750

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.4.a.m.1.7	yes	7
5.2	odd	4		575.4.b.j.24.13		14
5.3	odd	4		575.4.b.j.24.2		14
5.4	even	2		575.4.a.l.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
575.4.a.l.1.1	✓	7	5.4	even	2
575.4.a.m.1.7	yes	7	1.1	even	1	trivial
575.4.b.j.24.2		14	5.3	odd	4
575.4.b.j.24.13		14	5.2	odd	4

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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	575.a (trivial)

\(f(q)\)	\(=\)	\(q+4.39200 q^{2} -3.27927 q^{3} +11.2896 q^{4} -14.4025 q^{6} +13.7978 q^{7} +14.4481 q^{8} -16.2464 q^{9} -16.7024 q^{11} -37.0218 q^{12} -68.7171 q^{13} +60.5997 q^{14} -26.8610 q^{16} -40.7204 q^{17} -71.3541 q^{18} +40.6147 q^{19} -45.2466 q^{21} -73.3568 q^{22} -23.0000 q^{23} -47.3793 q^{24} -301.805 q^{26} +141.817 q^{27} +155.772 q^{28} +41.4829 q^{29} -234.349 q^{31} -233.559 q^{32} +54.7716 q^{33} -178.844 q^{34} -183.416 q^{36} +101.472 q^{37} +178.380 q^{38} +225.342 q^{39} +9.09912 q^{41} -198.723 q^{42} -146.349 q^{43} -188.564 q^{44} -101.016 q^{46} -383.965 q^{47} +88.0845 q^{48} -152.622 q^{49} +133.533 q^{51} -775.792 q^{52} +430.635 q^{53} +622.858 q^{54} +199.352 q^{56} -133.187 q^{57} +182.193 q^{58} -441.067 q^{59} +73.6756 q^{61} -1029.26 q^{62} -224.164 q^{63} -810.901 q^{64} +240.557 q^{66} -245.433 q^{67} -459.719 q^{68} +75.4232 q^{69} -241.613 q^{71} -234.730 q^{72} +187.130 q^{73} +445.667 q^{74} +458.526 q^{76} -230.455 q^{77} +989.701 q^{78} +109.404 q^{79} -26.4023 q^{81} +39.9633 q^{82} -1198.92 q^{83} -510.818 q^{84} -642.765 q^{86} -136.034 q^{87} -241.318 q^{88} +148.942 q^{89} -948.142 q^{91} -259.662 q^{92} +768.492 q^{93} -1686.38 q^{94} +765.902 q^{96} +1829.68 q^{97} -670.314 q^{98} +271.353 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} + q^{3} + 27 q^{4} - 41 q^{6} + q^{7} - 57 q^{8} + 10 q^{9} - 52 q^{11} + 65 q^{12} + 45 q^{13} - 42 q^{14} - 85 q^{16} + 85 q^{17} + 18 q^{18} - 10 q^{19} - 202 q^{21} - 71 q^{22} - 161 q^{23}+ \cdots - 2143 q^{99}+O(q^{100}) \) 7 * q + 3 * q^2 + q^3 + 27 * q^4 - 41 * q^6 + q^7 - 57 * q^8 + 10 * q^9 - 52 * q^11 + 65 * q^12 + 45 * q^13 - 42 * q^14 - 85 * q^16 + 85 * q^17 + 18 * q^18 - 10 * q^19 - 202 * q^21 - 71 * q^22 - 161 * q^23 - 303 * q^24 - 26 * q^26 - 242 * q^27 + 186 * q^28 - 455 * q^29 - 690 * q^31 - 233 * q^32 - 203 * q^33 - 657 * q^34 + 204 * q^36 - 892 * q^37 + 421 * q^38 - 277 * q^39 - 230 * q^41 - 888 * q^42 + 217 * q^43 - 1003 * q^44 - 69 * q^46 - 166 * q^47 + 1225 * q^48 - 802 * q^49 - 695 * q^51 - 1640 * q^52 + 1148 * q^53 - 1247 * q^54 - 386 * q^56 - 845 * q^57 + 1454 * q^58 - 551 * q^59 - 1242 * q^61 - 296 * q^62 + 1111 * q^63 - 1501 * q^64 + 987 * q^66 + 201 * q^67 + 1575 * q^68 - 23 * q^69 + 186 * q^71 - 2964 * q^72 + 530 * q^73 - 1530 * q^74 - 1775 * q^76 - 1339 * q^77 + 1364 * q^78 - 2003 * q^79 - 365 * q^81 - 2063 * q^82 + 726 * q^83 - 1352 * q^84 + 1894 * q^86 + 263 * q^87 + 2145 * q^88 - 2629 * q^89 - 1295 * q^91 - 621 * q^92 - 549 * q^93 - 3468 * q^94 - 135 * q^96 + 472 * q^97 + 2389 * q^98 - 2143 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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