LMFDB - Embedded newform 575.4.b.j.24.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,4,Mod(24,575)]

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([1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("575.24");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	575.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$33.9260982533$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 83x^{12} + 2715x^{10} + 44273x^{8} + 372280x^{6} + 1482448x^{4} + 2136384x^{2} + 746496 $ x^14 + 83x^12 + 2715x^10 + 44273x^8 + 372280x^6 + 1482448x^4 + 2136384x^2 + 746496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.2
Root			$-4.39200i$ of defining polynomial
Character	$\chi$	$=$	575.24
Dual form			575.4.b.j.24.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.39200i q^{2} -3.27927i q^{3} -11.2896 q^{4} -14.4025 q^{6} -13.7978i q^{7} +14.4481i q^{8} +16.2464 q^{9} -16.7024 q^{11} +37.0218i q^{12} -68.7171i q^{13} -60.5997 q^{14} -26.8610 q^{16} +40.7204i q^{17} -71.3541i q^{18} -40.6147 q^{19} -45.2466 q^{21} +73.3568i q^{22} -23.0000i q^{23} +47.3793 q^{24} -301.805 q^{26} -141.817i q^{27} +155.772i q^{28} -41.4829 q^{29} -234.349 q^{31} +233.559i q^{32} +54.7716i q^{33} +178.844 q^{34} -183.416 q^{36} -101.472i q^{37} +178.380i q^{38} -225.342 q^{39} +9.09912 q^{41} +198.723i q^{42} -146.349i q^{43} +188.564 q^{44} -101.016 q^{46} +383.965i q^{47} +88.0845i q^{48} +152.622 q^{49} +133.533 q^{51} +775.792i q^{52} +430.635i q^{53} -622.858 q^{54} +199.352 q^{56} +133.187i q^{57} +182.193i q^{58} +441.067 q^{59} +73.6756 q^{61} +1029.26i q^{62} -224.164i q^{63} +810.901 q^{64} +240.557 q^{66} +245.433i q^{67} -459.719i q^{68} -75.4232 q^{69} -241.613 q^{71} +234.730i q^{72} +187.130i q^{73} -445.667 q^{74} +458.526 q^{76} +230.455i q^{77} +989.701i q^{78} -109.404 q^{79} -26.4023 q^{81} -39.9633i q^{82} -1198.92i q^{83} +510.818 q^{84} -642.765 q^{86} +136.034i q^{87} -241.318i q^{88} -148.942 q^{89} -948.142 q^{91} +259.662i q^{92} +768.492i q^{93} +1686.38 q^{94} +765.902 q^{96} -1829.68i q^{97} -670.314i q^{98} -271.353 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 54 q^{4} - 82 q^{6} - 20 q^{9} - 104 q^{11} + 84 q^{14} - 170 q^{16} + 20 q^{19} - 404 q^{21} + 606 q^{24} - 52 q^{26} + 910 q^{29} - 1380 q^{31} + 1314 q^{34} + 408 q^{36} + 554 q^{39} - 460 q^{41}+ \cdots + 4286 q^{99}+O(q^{100}) $ 14 * q - 54 * q^4 - 82 * q^6 - 20 * q^9 - 104 * q^11 + 84 * q^14 - 170 * q^16 + 20 * q^19 - 404 * q^21 + 606 * q^24 - 52 * q^26 + 910 * q^29 - 1380 * q^31 + 1314 * q^34 + 408 * q^36 + 554 * q^39 - 460 * q^41 + 2006 * q^44 - 138 * q^46 + 1604 * q^49 - 1390 * q^51 + 2494 * q^54 - 772 * q^56 + 1102 * q^59 - 2484 * q^61 + 3002 * q^64 + 1974 * q^66 + 46 * q^69 + 372 * q^71 + 3060 * q^74 - 3550 * q^76 + 4006 * q^79 - 730 * q^81 + 2704 * q^84 + 3788 * q^86 + 5258 * q^89 - 2590 * q^91 + 6936 * q^94 - 270 * q^96 + 4286 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 4.39200i	− 1.55281i	−0.630237	−	0.776403i	$-0.717042\pi$
$2$	− 4.39200i	− 1.55281i	0.630237	−	0.776403i	$-0.282958\pi$
$3$	− 3.27927i	− 0.631096i	−0.948910	−	0.315548i	$-0.897812\pi$
$3$	− 3.27927i	− 0.631096i	0.948910	−	0.315548i	$-0.102188\pi$
$4$	−11.2896	−1.41121
$5$	0	0
$5$	0	0
$6$	−14.4025	−0.979969
$7$	− 13.7978i	− 0.745009i	−0.928030	−	0.372505i	$-0.878499\pi$
$7$	− 13.7978i	− 0.745009i	0.928030	−	0.372505i	$-0.121501\pi$
$8$	14.4481i	0.638523i
$9$	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	$0.573515\pi$
$12$	37.0218i	0.890606i
$13$	− 68.7171i	− 1.46605i	−0.680200	−	0.733027i	$-0.738107\pi$
$13$	− 68.7171i	− 1.46605i	0.680200	−	0.733027i	$-0.261893\pi$
$14$	−60.5997	−1.15685
$15$	0	0
$16$	−26.8610	−0.419703
$17$	40.7204i	0.580950i	0.956883	+	0.290475i	$0.0938134\pi$
$17$	40.7204i	0.580950i	−0.956883	+	0.290475i	$0.906187\pi$
$18$	− 71.3541i	− 0.934351i
$19$	−40.6147	−0.490403	−0.245201	−	0.969472i	$-0.578854\pi$
$19$	−40.6147	−0.490403	−0.245201	+	0.969472i	$0.578854\pi$
$20$	0	0
$21$	−45.2466	−0.470172
$22$	73.3568i	0.710897i
$23$	− 23.0000i	− 0.208514i
$23$	− 23.0000i	− 0.208514i
$24$	47.3793	0.402969
$25$	0	0
$26$	−301.805	−2.27650
$27$	− 141.817i	− 1.01084i
$28$	155.772i	1.05136i
$29$	−41.4829	−0.265627	−0.132814	−	0.991141i	$-0.542401\pi$
$29$	−41.4829	−0.265627	−0.132814	+	0.991141i	$0.542401\pi$
$30$	0	0
$31$	−234.349	−1.35775	−0.678875	−	0.734254i	$-0.737532\pi$
$31$	−234.349	−1.35775	−0.678875	+	0.734254i	$0.737532\pi$
$32$	233.559i	1.29024i
$33$	54.7716i	0.288925i
$34$	178.844	0.902103
$35$	0	0
$36$	−183.416	−0.849148
$37$	− 101.472i	− 0.450864i	−0.974259	−	0.225432i	$-0.927621\pi$
$37$	− 101.472i	− 0.450864i	0.974259	−	0.225432i	$-0.0723794\pi$
$38$	178.380i	0.761500i
$39$	−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	$0.494483\pi$
$42$	198.723i	0.730086i
$43$	− 146.349i	− 0.519024i	−0.965740	−	0.259512i	$-0.916438\pi$
$43$	− 146.349i	− 0.519024i	0.965740	−	0.259512i	$-0.0835617\pi$
$44$	188.564	0.646070
$45$	0	0
$46$	−101.016	−0.323782
$47$	383.965i	1.19164i	0.803118	+	0.595820i	$0.203173\pi$
$47$	383.965i	1.19164i	−0.803118	+	0.595820i	$0.796827\pi$
$48$	88.0845i	0.264873i
$49$	152.622	0.444961
$50$	0	0
$51$	133.533	0.366635
$52$	775.792i	2.06890i
$53$	430.635i	1.11608i	0.829814	+	0.558040i	$0.188447\pi$
$53$	430.635i	1.11608i	−0.829814	+	0.558040i	$0.811553\pi$
$54$	−622.858	−1.56963
$55$	0	0
$56$	199.352	0.475706
$57$	133.187i	0.309491i
$58$	182.193i	0.412467i
$59$	441.067	0.973255	0.486627	−	0.873610i	$-0.338227\pi$
$59$	441.067	0.973255	0.486627	+	0.873610i	$0.338227\pi$
$60$	0	0
$61$	73.6756	0.154643	0.0773213	−	0.997006i	$-0.475363\pi$
$61$	73.6756	0.154643	0.0773213	+	0.997006i	$0.475363\pi$
$62$	1029.26i	2.10832i
$63$	− 224.164i	− 0.448286i
$64$	810.901	1.58379
$65$	0	0
$66$	240.557	0.448644
$67$	245.433i	0.447529i	0.974643	+	0.223765i	$0.0718347\pi$
$67$	245.433i	0.447529i	−0.974643	+	0.223765i	$0.928165\pi$
$68$	− 459.719i	− 0.819841i
$69$	−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	$0.564722\pi$
$72$	234.730i	0.384211i
$73$	187.130i	0.300026i	0.988684	+	0.150013i	$0.0479315\pi$
$73$	187.130i	0.300026i	−0.988684	+	0.150013i	$0.952068\pi$
$74$	−445.667	−0.700104
$75$	0	0
$76$	458.526	0.692059
$77$	230.455i	0.341076i
$78$	989.701i	1.43669i
$79$	−109.404	−0.155808	−0.0779041	−	0.996961i	$-0.524823\pi$
$79$	−109.404	−0.155808	−0.0779041	+	0.996961i	$0.524823\pi$
$80$	0	0
$81$	−26.4023	−0.0362171
$82$	− 39.9633i	− 0.0538196i
$83$	− 1198.92i	− 1.58552i	−0.609533	−	0.792761i	$-0.708643\pi$
$83$	− 1198.92i	− 1.58552i	0.609533	−	0.792761i	$-0.291357\pi$
$84$	510.818	0.663510
$85$	0	0
$86$	−642.765	−0.805944
$87$	136.034i	0.167636i
$88$	− 241.318i	− 0.292325i
$89$	−148.942	−0.177391	−0.0886953	−	0.996059i	$-0.528270\pi$
$89$	−148.942	−0.177391	−0.0886953	+	0.996059i	$0.528270\pi$
$90$	0	0
$91$	−948.142	−1.09222
$92$	259.662i	0.294257i
$93$	768.492i	0.856870i
$94$	1686.38	1.85039
$95$	0	0
$96$	765.902	0.814266
$97$	− 1829.68i	− 1.91521i	−0.288082	−	0.957606i	$-0.593018\pi$
$97$	− 1829.68i	− 1.91521i	0.288082	−	0.957606i	$-0.406982\pi$
$98$	− 670.314i	− 0.690939i
$99$	−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	$0.600242\pi$
$102$	− 586.478i	− 0.569313i
$103$	1577.66i	1.50924i	0.656162	+	0.754620i	$0.272179\pi$
$103$	1577.66i	1.50924i	−0.656162	+	0.754620i	$0.727821\pi$
$104$	992.834	0.936109
$105$	0	0
$106$	1891.35	1.73306
$107$	− 1253.21i	− 1.13226i	−0.824316	−	0.566130i	$-0.808440\pi$
$107$	− 1253.21i	− 1.13226i	0.824316	−	0.566130i	$-0.191560\pi$
$108$	1601.06i	1.42650i
$109$	1271.40	1.11723	0.558615	−	0.829427i	$-0.311333\pi$
$109$	1271.40	1.11723	0.558615	+	0.829427i	$0.311333\pi$
$110$	0	0
$111$	−332.756	−0.284538
$112$	370.622i	0.312683i
$113$	1469.50i	1.22336i	0.791107	+	0.611678i	$0.209505\pi$
$113$	1469.50i	1.22336i	−0.791107	+	0.611678i	$0.790495\pi$
$114$	584.955	0.480579
$115$	0	0
$116$	468.328	0.374855
$117$	− 1116.40i	− 0.882151i
$118$	− 1937.17i	− 1.51128i
$119$	561.851	0.432813
$120$	0	0
$121$	−1052.03	−0.790406
$122$	− 323.583i	− 0.240130i
$123$	− 29.8385i	− 0.0218735i
$124$	2645.71	1.91607
$125$	0	0
$126$	−984.527	−0.696100
$127$	445.611i	0.311351i	0.987808	+	0.155676i	$0.0497555\pi$
$127$	445.611i	0.311351i	−0.987808	+	0.155676i	$0.950245\pi$
$128$	− 1693.01i	− 1.16908i
$129$	−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	$0.432551\pi$
$132$	− 618.352i	− 0.407732i
$133$	560.392i	0.365354i
$134$	1077.94	0.694926
$135$	0	0
$136$	−588.334	−0.370950
$137$	− 479.846i	− 0.299241i	−0.988743	−	0.149621i	$-0.952195\pi$
$137$	− 479.846i	− 0.299241i	0.988743	−	0.149621i	$-0.0478052\pi$
$138$	331.259i	0.204338i
$139$	202.400	0.123506	0.0617531	−	0.998091i	$-0.480331\pi$
$139$	202.400	0.123506	0.0617531	+	0.998091i	$0.480331\pi$
$140$	0	0
$141$	1259.13	0.752039
$142$	1061.16i	0.627119i
$143$	1147.74i	0.671180i
$144$	−436.395	−0.252543
$145$	0	0
$146$	821.874	0.465882
$147$	− 500.488i	− 0.280813i
$148$	1145.59i	0.636262i
$149$	739.957	0.406843	0.203422	−	0.979091i	$-0.434794\pi$
$149$	739.957	0.406843	0.203422	+	0.979091i	$0.434794\pi$
$150$	0	0
$151$	−1278.96	−0.689274	−0.344637	−	0.938736i	$-0.611998\pi$
$151$	−1278.96	−0.689274	−0.344637	+	0.938736i	$0.611998\pi$
$152$	− 586.806i	− 0.313133i
$153$	661.560i	0.349568i
$154$	1012.16	0.529624
$155$	0	0
$156$	2544.03	1.30568
$157$	− 713.690i	− 0.362794i	−0.983410	−	0.181397i	$-0.941938\pi$
$157$	− 713.690i	− 0.362794i	0.983410	−	0.181397i	$-0.0580619\pi$
$158$	480.500i	0.241940i
$159$	1412.17	0.704354
$160$	0	0
$161$	−317.349	−0.155345
$162$	115.959i	0.0562382i
$163$	215.300i	0.103458i	0.998661	+	0.0517288i	$0.0164731\pi$
$163$	215.300i	0.103458i	−0.998661	+	0.0517288i	$0.983527\pi$
$164$	−102.726	−0.0489118
$165$	0	0
$166$	−5265.64	−2.46201
$167$	− 4086.64i	− 1.89362i	−0.321800	−	0.946808i	$-0.604288\pi$
$167$	− 4086.64i	− 1.89362i	0.321800	−	0.946808i	$-0.395712\pi$
$168$	− 653.729i	− 0.300216i
$169$	−2525.04	−1.14931
$170$	0	0
$171$	−659.842	−0.295084
$172$	1652.23i	0.732450i
$173$	858.878i	0.377453i	0.982030	+	0.188726i	$0.0604359\pi$
$173$	858.878i	0.377453i	−0.982030	+	0.188726i	$0.939564\pi$
$174$	597.460	0.260306
$175$	0	0
$176$	448.643	0.192146
$177$	− 1446.38i	− 0.614217i
$178$	654.151i	0.275453i
$179$	−1668.46	−0.696683	−0.348342	−	0.937368i	$-0.613255\pi$
$179$	−1668.46	−0.696683	−0.348342	+	0.937368i	$0.613255\pi$
$180$	0	0
$181$	−347.888	−0.142864	−0.0714319	−	0.997445i	$-0.522757\pi$
$181$	−347.888	−0.142864	−0.0714319	+	0.997445i	$0.522757\pi$
$182$	4164.24i	1.69601i
$183$	− 241.602i	− 0.0975942i
$184$	332.307	0.133141
$185$	0	0
$186$	3375.22	1.33055
$187$	− 680.128i	− 0.265967i
$188$	− 4334.83i	− 1.68165i
$189$	−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	$0.355411\pi$
$192$	− 2659.16i	− 0.999524i
$193$	1530.07i	0.570658i	0.958430	+	0.285329i	$0.0921028\pi$
$193$	1530.07i	0.570658i	−0.958430	+	0.285329i	$0.907897\pi$
$194$	−8035.94	−2.97395
$195$	0	0
$196$	−1723.05	−0.627932
$197$	1798.69i	0.650513i	0.945626	+	0.325257i	$0.105451\pi$
$197$	1798.69i	0.650513i	−0.945626	+	0.325257i	$0.894549\pi$
$198$	1191.78i	0.427759i
$199$	−3231.92	−1.15128	−0.575640	−	0.817703i	$-0.695247\pi$
$199$	−3231.92	−1.15128	−0.575640	+	0.817703i	$0.695247\pi$
$200$	0	0
$201$	804.842	0.282434
$202$	2761.67i	0.961931i
$203$	572.372i	0.197895i
$204$	−1507.54	−0.517398
$205$	0	0
$206$	6929.09	2.34356
$207$	− 373.667i	− 0.125467i
$208$	1845.81i	0.615308i
$209$	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	$0.822353\pi$
$212$	− 4861.72i	− 1.57502i
$213$	792.314i	0.254875i
$214$	−5504.07	−1.75818
$215$	0	0
$216$	2048.98	0.645443
$217$	3233.49i	1.01154i
$218$	− 5583.99i	− 1.73484i
$219$	613.649	0.189345
$220$	0	0
$221$	2798.19	0.851704
$222$	1461.46i	0.441833i
$223$	3144.43i	0.944246i	0.881533	+	0.472123i	$0.156512\pi$
$223$	3144.43i	0.944246i	−0.881533	+	0.472123i	$0.843488\pi$
$224$	3222.59	0.961242
$225$	0	0
$226$	6454.06	1.89964
$227$	262.640i	0.0767931i	0.999263	+	0.0383966i	$0.0122250\pi$
$227$	262.640i	0.0767931i	−0.999263	+	0.0383966i	$0.987775\pi$
$228$	− 1503.63i	− 0.436756i
$229$	5559.69	1.60434	0.802172	−	0.597093i	$-0.203678\pi$
$229$	5559.69	1.60434	0.802172	+	0.597093i	$0.203678\pi$
$230$	0	0
$231$	755.726	0.215251
$232$	− 599.351i	− 0.169609i
$233$	− 4405.25i	− 1.23862i	−0.785148	−	0.619308i	$-0.787413\pi$
$233$	− 4405.25i	− 1.23862i	0.785148	−	0.619308i	$-0.212587\pi$
$234$	−4903.25	−1.36981
$235$	0	0
$236$	−4979.49	−1.37346
$237$	358.764i	0.0983300i
$238$	− 2467.65i	− 0.672075i
$239$	−6444.99	−1.74432	−0.872158	−	0.489224i	$-0.837280\pi$
$239$	−6444.99	−1.74432	−0.872158	+	0.489224i	$0.837280\pi$
$240$	0	0
$241$	−6708.71	−1.79314	−0.896569	−	0.442905i	$-0.853948\pi$
$241$	−6708.71	−1.79314	−0.896569	+	0.442905i	$0.853948\pi$
$242$	4620.52i	1.22735i
$243$	− 3742.47i	− 0.987981i
$244$	−831.772	−0.218232
$245$	0	0
$246$	−131.051	−0.0339653
$247$	2790.92i	0.718956i
$248$	− 3385.90i	− 0.866955i
$249$	−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	$0.511585\pi$
$252$	2530.73i	0.632623i
$253$	384.155i	0.0954609i
$254$	1957.12	0.483468
$255$	0	0
$256$	−948.473	−0.231561
$257$	4040.24i	0.980635i	0.871544	+	0.490318i	$0.163119\pi$
$257$	4040.24i	0.980635i	−0.871544	+	0.490318i	$0.836881\pi$
$258$	2107.80i	0.508628i
$259$	−1400.09	−0.335898
$260$	0	0
$261$	−673.948	−0.159833
$262$	− 2769.95i	− 0.653160i
$263$	− 5250.65i	− 1.23106i	−0.788114	−	0.615530i	$-0.788942\pi$
$263$	− 5250.65i	− 1.23106i	0.788114	−	0.615530i	$-0.211058\pi$
$264$	−791.347	−0.184485
$265$	0	0
$266$	2461.24	0.567324
$267$	488.419i	0.111951i
$268$	− 2770.86i	− 0.631556i
$269$	2417.55	0.547958	0.273979	−	0.961736i	$-0.411660\pi$
$269$	2417.55	0.547958	0.273979	+	0.961736i	$0.411660\pi$
$270$	0	0
$271$	−6593.83	−1.47803	−0.739016	−	0.673688i	$-0.764709\pi$
$271$	−6593.83	−1.47803	−0.739016	+	0.673688i	$0.764709\pi$
$272$	− 1093.79i	− 0.243827i
$273$	3109.21i	0.689297i
$274$	−2107.48	−0.464663
$275$	0	0
$276$	851.501	0.185704
$277$	− 2826.91i	− 0.613186i	−0.951841	−	0.306593i	$-0.900811\pi$
$277$	− 2826.91i	− 0.613186i	0.951841	−	0.306593i	$-0.0991891\pi$
$278$	− 888.941i	− 0.191781i
$279$	−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	$0.569345\pi$
$282$	− 5530.08i	− 1.16777i
$283$	4470.17i	0.938954i	0.882945	+	0.469477i	$0.155558\pi$
$283$	4470.17i	0.938954i	−0.882945	+	0.469477i	$0.844442\pi$
$284$	2727.73	0.569932
$285$	0	0
$286$	5040.87	1.04221
$287$	− 125.548i	− 0.0258217i
$288$	3794.48i	0.776362i
$289$	3254.85	0.662497
$290$	0	0
$291$	−6000.00	−1.20868
$292$	− 2112.63i	− 0.423398i
$293$	− 9457.29i	− 1.88567i	−0.333263	−	0.942834i	$-0.608150\pi$
$293$	− 9457.29i	− 1.88567i	0.333263	−	0.942834i	$-0.391850\pi$
$294$	−2198.14	−0.436048
$295$	0	0
$296$	1466.09	0.287887
$297$	2368.67i	0.462776i
$298$	− 3249.89i	− 0.631749i
$299$	−1580.49	−0.305693
$300$	0	0
$301$	−2019.29	−0.386678
$302$	5617.19i	1.07031i
$303$	2061.99i	0.390951i
$304$	1090.95	0.205824
$305$	0	0
$306$	2905.57	0.542812
$307$	4576.54i	0.850804i	0.905004	+	0.425402i	$0.139867\pi$
$307$	4576.54i	0.850804i	−0.905004	+	0.425402i	$0.860133\pi$
$308$	− 2601.76i	− 0.481328i
$309$	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	$0.609246\pi$
$312$	− 3255.77i	− 0.590774i
$313$	− 9771.57i	− 1.76461i	−0.470682	−	0.882303i	$-0.655992\pi$
$313$	− 9771.57i	− 1.76461i	0.470682	−	0.882303i	$-0.344008\pi$
$314$	−3134.53	−0.563349
$315$	0	0
$316$	1235.13	0.219878
$317$	− 9653.06i	− 1.71032i	−0.518368	−	0.855158i	$-0.673460\pi$
$317$	− 9653.06i	− 1.71032i	0.518368	−	0.855158i	$-0.326540\pi$
$318$	− 6202.24i	− 1.09372i
$319$	692.864	0.121608
$320$	0	0
$321$	−4109.60	−0.714565
$322$	1393.79i	0.241221i
$323$	− 1653.85i	− 0.284900i
$324$	298.073	0.0511098
$325$	0	0
$326$	945.597	0.160650
$327$	− 4169.26i	− 0.705079i
$328$	131.465i	0.0221310i
$329$	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	$0.422035\pi$
$332$	13535.4i	2.23750i
$333$	− 1648.56i	− 0.271293i
$334$	−17948.5	−2.94042
$335$	0	0
$336$	1215.37	0.197333
$337$	− 7794.72i	− 1.25996i	−0.776613	−	0.629978i	$-0.783064\pi$
$337$	− 7794.72i	− 1.25996i	0.776613	−	0.629978i	$-0.216936\pi$
$338$	11090.0i	1.78466i
$339$	4818.90	0.772055
$340$	0	0
$341$	3914.18	0.621597
$342$	2898.03i	0.458208i
$343$	− 6838.47i	− 1.07651i
$344$	2114.47	0.331409
$345$	0	0
$346$	3772.19	0.586111
$347$	− 7025.93i	− 1.08695i	−0.839425	−	0.543475i	$-0.817108\pi$
$347$	− 7025.93i	− 1.08695i	0.839425	−	0.543475i	$-0.182892\pi$
$348$	− 1535.77i	− 0.236569i
$349$	−6338.40	−0.972168	−0.486084	−	0.873912i	$-0.661575\pi$
$349$	−6338.40	−0.972168	−0.486084	+	0.873912i	$0.661575\pi$
$350$	0	0
$351$	−9745.22	−1.48194
$352$	− 3900.98i	− 0.590691i
$353$	− 8834.31i	− 1.33202i	−0.745943	−	0.666010i	$-0.768001\pi$
$353$	− 8834.31i	− 1.33202i	0.745943	−	0.666010i	$-0.231999\pi$
$354$	−6352.49	−0.953760
$355$	0	0
$356$	1681.50	0.250335
$357$	− 1842.46i	− 0.273147i
$358$	7327.86i	1.08181i
$359$	917.755	0.134923	0.0674613	−	0.997722i	$-0.478510\pi$
$359$	917.755	0.134923	0.0674613	+	0.997722i	$0.478510\pi$
$360$	0	0
$361$	−5209.45	−0.759505
$362$	1527.93i	0.221840i
$363$	3449.89i	0.498822i
$364$	10704.2	1.54135
$365$	0	0
$366$	−1061.12	−0.151545
$367$	− 5451.98i	− 0.775453i	−0.921774	−	0.387727i	$-0.873260\pi$
$367$	− 5451.98i	− 0.775453i	0.921774	−	0.387727i	$-0.126740\pi$
$368$	617.804i	0.0875142i
$369$	147.828	0.0208553
$370$	0	0
$371$	5941.80	0.831491
$372$	− 8676.01i	− 1.20922i
$373$	9583.17i	1.33029i	0.746715	+	0.665144i	$0.231630\pi$
$373$	9583.17i	1.33029i	−0.746715	+	0.665144i	$0.768370\pi$
$374$	−2987.12	−0.412996
$375$	0	0
$376$	−5547.58	−0.760890
$377$	2850.59i	0.389424i
$378$	8594.05i	1.16939i
$379$	−5560.92	−0.753681	−0.376841	−	0.926278i	$-0.622990\pi$
$379$	−5560.92	−0.753681	−0.376841	+	0.926278i	$0.622990\pi$
$380$	0	0
$381$	1461.28	0.196493
$382$	− 10173.9i	− 1.36268i
$383$	− 4847.50i	− 0.646725i	−0.946275	−	0.323363i	$-0.895187\pi$
$383$	− 4847.50i	− 0.646725i	0.946275	−	0.323363i	$-0.104813\pi$
$384$	−5551.82	−0.737800
$385$	0	0
$386$	6720.07	0.886120
$387$	− 2377.65i	− 0.312306i
$388$	20656.4i	2.70276i
$389$	11572.6	1.50837	0.754184	−	0.656663i	$-0.228033\pi$
$389$	11572.6	1.50837	0.754184	+	0.656663i	$0.228033\pi$
$390$	0	0
$391$	936.570	0.121137
$392$	2205.10i	0.284118i
$393$	− 2068.17i	− 0.265459i
$394$	7899.83	1.01012
$395$	0	0
$396$	3063.48	0.388752
$397$	5767.82i	0.729165i	0.931171	+	0.364582i	$0.118788\pi$
$397$	5767.82i	0.729165i	−0.931171	+	0.364582i	$0.881212\pi$
$398$	14194.6i	1.78771i
$399$	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	$0.116847\pi$
$402$	− 3534.87i	− 0.438565i
$403$	16103.8i	1.99053i
$404$	7098.87	0.874213
$405$	0	0
$406$	2513.86	0.307292
$407$	1694.83i	0.206412i
$408$	1929.31i	0.234105i
$409$	1487.95	0.179888	0.0899442	−	0.995947i	$-0.471331\pi$
$409$	1487.95	0.179888	0.0899442	+	0.995947i	$0.471331\pi$
$410$	0	0
$411$	−1573.55	−0.188850
$412$	− 17811.2i	− 2.12985i
$413$	− 6085.74i	− 0.725084i
$414$	−1641.14	−0.194826
$415$	0	0
$416$	16049.5	1.89156
$417$	− 663.725i	− 0.0779442i
$418$	− 2979.36i	− 0.348625i
$419$	2739.23	0.319380	0.159690	−	0.987167i	$-0.448951\pi$
$419$	2739.23	0.319380	0.159690	+	0.987167i	$0.448951\pi$
$420$	0	0
$421$	2205.79	0.255353	0.127676	−	0.991816i	$-0.459248\pi$
$421$	2205.79	0.255353	0.127676	+	0.991816i	$0.459248\pi$
$422$	22837.5i	2.63439i
$423$	6238.05i	0.717032i
$424$	−6221.87	−0.712644
$425$	0	0
$426$	3479.84	0.395772
$427$	− 1016.56i	− 0.115210i
$428$	14148.2i	1.59785i
$429$	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	$0.256887\pi$
$432$	3809.34i	0.424252i
$433$	− 11610.5i	− 1.28860i	−0.764774	−	0.644299i	$-0.777149\pi$
$433$	− 11610.5i	− 1.28860i	0.764774	−	0.644299i	$-0.222851\pi$
$434$	14201.5	1.57072
$435$	0	0
$436$	−14353.7	−1.57664
$437$	934.138i	0.102256i
$438$	− 2695.15i	− 0.294016i
$439$	10585.8	1.15087	0.575433	−	0.817849i	$-0.304833\pi$
$439$	10585.8	1.15087	0.575433	+	0.817849i	$0.304833\pi$
$440$	0	0
$441$	2479.55	0.267741
$442$	− 12289.6i	− 1.32253i
$443$	13572.0i	1.45559i	0.685794	+	0.727796i	$0.259455\pi$
$443$	13572.0i	1.45559i	−0.685794	+	0.727796i	$0.740545\pi$
$444$	3756.69	0.401542
$445$	0	0
$446$	13810.3	1.46623
$447$	− 2426.52i	− 0.256757i
$448$	− 11188.6i	− 1.17994i
$449$	−8401.73	−0.883079	−0.441539	−	0.897242i	$-0.645567\pi$
$449$	−8401.73	−0.883079	−0.441539	+	0.897242i	$0.645567\pi$
$450$	0	0
$451$	−151.977	−0.0158677
$452$	− 16590.2i	− 1.72641i
$453$	4194.06i	0.434998i
$454$	1153.52	0.119245
$455$	0	0
$456$	−1924.30	−0.197617
$457$	10406.0i	1.06515i	0.846384	+	0.532573i	$0.178775\pi$
$457$	10406.0i	1.06515i	−0.846384	+	0.532573i	$0.821225\pi$
$458$	− 24418.2i	− 2.49124i
$459$	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	$0.296935\pi$
$462$	− 3319.15i	− 0.334244i
$463$	− 15713.0i	− 1.57721i	−0.614902	−	0.788604i	$-0.710804\pi$
$463$	− 15713.0i	− 1.57721i	0.614902	−	0.788604i	$-0.289196\pi$
$464$	1114.27	0.111485
$465$	0	0
$466$	−19347.9	−1.92333
$467$	5213.88i	0.516637i	0.966060	+	0.258319i	$0.0831685\pi$
$467$	5213.88i	0.516637i	−0.966060	+	0.258319i	$0.916832\pi$
$468$	12603.8i	1.24490i
$469$	3386.43	0.333413
$470$	0	0
$471$	−2340.38	−0.228958
$472$	6372.59i	0.621446i
$473$	2444.38i	0.237617i
$474$	1575.69	0.152687
$475$	0	0
$476$	−6343.10	−0.610789
$477$	6996.27i	0.671566i
$478$	28306.4i	2.70858i
$479$	7912.84	0.754796	0.377398	−	0.926051i	$-0.376819\pi$
$479$	7912.84	0.754796	0.377398	+	0.926051i	$0.376819\pi$
$480$	0	0
$481$	−6972.89	−0.660991
$482$	29464.6i	2.78439i
$483$	1040.67i	0.0980377i
$484$	11877.1	1.11543
$485$	0	0
$486$	−16436.9	−1.53414
$487$	2710.59i	0.252215i	0.992017	+	0.126108i	$0.0402484\pi$
$487$	2710.59i	0.252215i	−0.992017	+	0.126108i	$0.959752\pi$
$488$	1064.47i	0.0987428i
$489$	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	$0.792156\pi$
$492$	336.866i	0.0308681i
$493$	− 1689.20i	− 0.154316i
$494$	12257.7	1.11640
$495$	0	0
$496$	6294.84	0.569853
$497$	3333.72i	0.300881i
$498$	17267.5i	1.55376i
$499$	−16521.7	−1.48219	−0.741096	−	0.671399i	$-0.765694\pi$
$499$	−16521.7	−1.48219	−0.741096	+	0.671399i	$0.765694\pi$
$500$	0	0
$501$	−13401.2	−1.19505
$502$	1271.03i	0.113006i
$503$	− 16269.1i	− 1.44215i	−0.692857	−	0.721075i	$-0.743648\pi$
$503$	− 16269.1i	− 1.44215i	0.692857	−	0.721075i	$-0.256352\pi$
$504$	3238.75	0.286241
$505$	0	0
$506$	1687.21	0.148232
$507$	8280.29i	0.725326i
$508$	− 5030.80i	− 0.439381i
$509$	9299.84	0.809839	0.404920	−	0.914352i	$-0.367299\pi$
$509$	9299.84	0.809839	0.404920	+	0.914352i	$0.367299\pi$
$510$	0	0
$511$	2581.97	0.223522
$512$	− 9378.36i	− 0.809509i
$513$	5759.84i	0.495717i
$514$	17744.7	1.52274
$515$	0	0
$516$	5418.11	0.462246
$517$	− 6413.13i	− 0.545550i
$518$	6149.21i	0.521584i
$519$	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	$0.922165\pi$
$522$	2959.98i	0.248189i
$523$	− 10348.3i	− 0.865200i	−0.901586	−	0.432600i	$-0.857596\pi$
$523$	− 10348.3i	− 0.865200i	0.901586	−	0.432600i	$-0.142404\pi$
$524$	−7120.17	−0.593599
$525$	0	0
$526$	−23060.8	−1.91160
$527$	− 9542.78i	− 0.788785i
$528$	− 1471.22i	− 0.121263i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	7165.75	0.585625
$532$	− 6326.63i	− 0.515590i
$533$	− 625.265i	− 0.0508128i
$534$	2145.14	0.173837
$535$	0	0
$536$	−3546.05	−0.285758
$537$	5471.32i	0.439674i
$538$	− 10617.9i	− 0.850872i
$539$	−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	$0.473682\pi$
$542$	28960.1i	2.29510i
$543$	1140.82i	0.0901607i
$544$	−9510.61	−0.749566
$545$	0	0
$546$	13655.7	1.07035
$547$	− 16340.5i	− 1.27727i	−0.769509	−	0.638637i	$-0.779499\pi$
$547$	− 16340.5i	− 1.27727i	0.769509	−	0.638637i	$-0.220501\pi$
$548$	5417.29i	0.422291i
$549$	1196.96	0.0930512
$550$	0	0
$551$	1684.82	0.130264
$552$	− 1089.72i	− 0.0840249i
$553$	1509.52i	0.116079i
$554$	−12415.8	−0.952159
$555$	0	0
$556$	−2285.03	−0.174293
$557$	4738.43i	0.360455i	0.983625	+	0.180228i	$0.0576835\pi$
$557$	4738.43i	0.360455i	−0.983625	+	0.180228i	$0.942317\pi$
$558$	16721.7i	1.26862i
$559$	−10056.7	−0.760917
$560$	0	0
$561$	−2230.32	−0.167851
$562$	8942.84i	0.671230i
$563$	− 19003.9i	− 1.42259i	−0.702894	−	0.711294i	$-0.748109\pi$
$563$	− 19003.9i	− 1.42259i	0.702894	−	0.711294i	$-0.251891\pi$
$564$	−14215.1	−1.06128
$565$	0	0
$566$	19633.0	1.45801
$567$	364.293i	0.0269821i
$568$	− 3490.86i	− 0.257875i
$569$	2789.77	0.205542	0.102771	−	0.994705i	$-0.467229\pi$
$569$	2789.77	0.205542	0.102771	+	0.994705i	$0.467229\pi$
$570$	0	0
$571$	−1154.31	−0.0845998	−0.0422999	−	0.999105i	$-0.513468\pi$
$571$	−1154.31	−0.0845998	−0.0422999	+	0.999105i	$0.513468\pi$
$572$	− 12957.6i	− 0.947173i
$573$	− 7596.33i	− 0.553824i
$574$	−551.404	−0.0400961
$575$	0	0
$576$	13174.2	0.952996
$577$	− 2360.98i	− 0.170345i	−0.996366	−	0.0851723i	$-0.972856\pi$
$577$	− 2360.98i	− 0.170345i	0.996366	−	0.0851723i	$-0.0271441\pi$
$578$	− 14295.3i	− 1.02873i
$579$	5017.52	0.360140
$580$	0	0
$581$	−16542.4	−1.18123
$582$	26352.0i	1.87685i
$583$	− 7192.63i	− 0.510958i
$584$	−2703.68	−0.191573
$585$	0	0
$586$	−41536.4	−2.92808
$587$	− 6641.99i	− 0.467026i	−0.972354	−	0.233513i	$-0.924978\pi$
$587$	− 6641.99i	− 0.467026i	0.972354	−	0.233513i	$-0.0750221\pi$
$588$	5650.33i	0.396285i
$589$	9518.00	0.665844
$590$	0	0
$591$	5898.38	0.410536
$592$	2725.65i	0.189229i
$593$	− 3879.68i	− 0.268667i	−0.990936	−	0.134334i	$-0.957111\pi$
$593$	− 3879.68i	− 0.268667i	0.990936	−	0.134334i	$-0.0428893\pi$
$594$	10403.2	0.718601
$595$	0	0
$596$	−8353.86	−0.574140
$597$	10598.3i	0.726568i
$598$	6941.52i	0.474682i
$599$	8316.59	0.567290	0.283645	−	0.958929i	$-0.408456\pi$
$599$	8316.59	0.567290	0.283645	+	0.958929i	$0.408456\pi$
$600$	0	0
$601$	14526.1	0.985908	0.492954	−	0.870055i	$-0.335917\pi$
$601$	14526.1	0.985908	0.492954	+	0.870055i	$0.335917\pi$
$602$	8868.72i	0.600435i
$603$	3987.41i	0.269286i
$604$	14439.0	0.972707
$605$	0	0
$606$	9056.25	0.607071
$607$	12982.3i	0.868097i	0.900890	+	0.434049i	$0.142915\pi$
$607$	12982.3i	0.868097i	−0.900890	+	0.434049i	$0.857085\pi$
$608$	− 9485.91i	− 0.632738i
$609$	1876.96	0.124890
$610$	0	0
$611$	26385.0	1.74701
$612$	− 7468.78i	− 0.493313i
$613$	− 23179.5i	− 1.52726i	−0.645653	−	0.763631i	$-0.723415\pi$
$613$	− 23179.5i	− 1.52726i	0.645653	−	0.763631i	$-0.276585\pi$
$614$	20100.2	1.32113
$615$	0	0
$616$	−3329.65	−0.217785
$617$	6322.17i	0.412514i	0.978498	+	0.206257i	$0.0661283\pi$
$617$	6322.17i	0.412514i	−0.978498	+	0.206257i	$0.933872\pi$
$618$	− 22722.4i	− 1.47901i
$619$	−10500.8	−0.681849	−0.340924	−	0.940091i	$-0.610740\pi$
$619$	−10500.8	−0.681849	−0.340924	+	0.940091i	$0.610740\pi$
$620$	0	0
$621$	−3261.78	−0.210774
$622$	16211.7i	1.04507i
$623$	2055.06i	0.132158i
$624$	6052.91	0.388318
$625$	0	0
$626$	−42916.7	−2.74009
$627$	− 2224.53i	− 0.141689i
$628$	8057.31i	0.511977i
$629$	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	$0.338537\pi$
$632$	− 1580.68i	− 0.0994872i
$633$	17051.5i	1.07067i
$634$	−42396.2	−2.65579
$635$	0	0
$636$	−15942.9	−0.993989
$637$	− 10487.7i	− 0.652337i
$638$	− 3043.06i	− 0.188833i
$639$	−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	$0.471788\pi$
$642$	18049.3i	1.10958i
$643$	− 15141.3i	− 0.928638i	−0.885668	−	0.464319i	$-0.846299\pi$
$643$	− 15141.3i	− 0.928638i	0.885668	−	0.464319i	$-0.153701\pi$
$644$	3582.75	0.219224
$645$	0	0
$646$	−7263.70	−0.442394
$647$	5086.61i	0.309081i	0.987986	+	0.154540i	$0.0493897\pi$
$647$	5086.61i	0.309081i	−0.987986	+	0.154540i	$0.950610\pi$
$648$	− 381.464i	− 0.0231255i
$649$	−7366.87	−0.445570
$650$	0	0
$651$	10603.5	0.638376
$652$	− 2430.66i	− 0.146000i
$653$	15214.0i	0.911744i	0.890045	+	0.455872i	$0.150673\pi$
$653$	15214.0i	0.911744i	−0.890045	+	0.455872i	$0.849327\pi$
$654$	−18311.4	−1.09485
$655$	0	0
$656$	−244.412	−0.0145468
$657$	3040.18i	0.180531i
$658$	− 23268.2i	− 1.37855i
$659$	−20029.9	−1.18400	−0.591999	−	0.805938i	$-0.701661\pi$
$659$	−20029.9	−1.18400	−0.591999	+	0.805938i	$0.701661\pi$
$660$	0	0
$661$	−92.3018	−0.00543135	−0.00271568	−	0.999996i	$-0.500864\pi$
$661$	−92.3018	−0.00543135	−0.00271568	+	0.999996i	$0.500864\pi$
$662$	− 12827.2i	− 0.753087i
$663$	− 9176.02i	− 0.537507i
$664$	17322.1	1.01239
$665$	0	0
$666$	−7240.48	−0.421265
$667$	954.107i	0.0553871i
$668$	46136.7i	2.67228i
$669$	10311.4	0.595910
$670$	0	0
$671$	−1230.56	−0.0707975
$672$	− 10567.7i	− 0.606635i
$673$	− 15452.2i	− 0.885048i	−0.896757	−	0.442524i	$-0.854083\pi$
$673$	− 15452.2i	− 0.885048i	0.896757	−	0.442524i	$-0.145917\pi$
$674$	−34234.4	−1.95647
$675$	0	0
$676$	28506.8	1.62192
$677$	19199.7i	1.08996i	0.838449	+	0.544980i	$0.183463\pi$
$677$	19199.7i	1.08996i	−0.838449	+	0.544980i	$0.816537\pi$
$678$	− 21164.6i	− 1.19885i
$679$	−25245.4	−1.42685
$680$	0	0
$681$	861.268	0.0484638
$682$	− 17191.1i	− 0.965220i
$683$	20090.5i	1.12554i	0.826614	+	0.562769i	$0.190264\pi$
$683$	20090.5i	1.12554i	−0.826614	+	0.562769i	$0.809736\pi$
$684$	7449.38	0.416424
$685$	0	0
$686$	−30034.6	−1.67161
$687$	− 18231.7i	− 1.01250i
$688$	3931.09i	0.217836i
$689$	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	$0.341248\pi$
$692$	− 9696.43i	− 0.532663i
$693$	3744.07i	0.205231i
$694$	−30857.9	−1.68782
$695$	0	0
$696$	−1965.43	−0.107040
$697$	370.520i	0.0201355i
$698$	27838.2i	1.50959i
$699$	−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	$0.402070\pi$
$702$	42801.0i	2.30117i
$703$	4121.27i	0.221105i
$704$	−13544.0	−0.725082
$705$	0	0
$706$	−38800.3	−2.06837
$707$	8675.96i	0.461518i
$708$	16329.1i	0.866787i
$709$	35667.3	1.88930	0.944649	−	0.328082i	$-0.106402\pi$
$709$	35667.3	1.88930	0.944649	+	0.328082i	$0.106402\pi$
$710$	0	0
$711$	−1777.41	−0.0937527
$712$	− 2151.93i	− 0.113268i
$713$	5390.02i	0.283111i
$714$	−8092.08	−0.424144
$715$	0	0
$716$	18836.3	0.983164
$717$	21134.9i	1.10083i
$718$	− 4030.78i	− 0.209509i
$719$	25533.1	1.32437	0.662187	−	0.749339i	$-0.269629\pi$
$719$	25533.1	1.32437	0.662187	+	0.749339i	$0.269629\pi$
$720$	0	0
$721$	21768.2	1.12440
$722$	22879.9i	1.17936i
$723$	21999.7i	1.13164i
$724$	3927.54	0.201610
$725$	0	0
$726$	15151.9	0.774574
$727$	27933.1i	1.42501i	0.701668	+	0.712504i	$0.252439\pi$
$727$	27933.1i	1.42501i	−0.701668	+	0.712504i	$0.747561\pi$
$728$	− 13698.9i	− 0.697410i
$729$	−12985.4	−0.659728
$730$	0	0
$731$	5959.40	0.301527
$732$	2727.60i	0.137726i
$733$	29359.9i	1.47945i	0.672912	+	0.739723i	$0.265043\pi$
$733$	29359.9i	1.47945i	−0.672912	+	0.739723i	$0.734957\pi$
$734$	−23945.1	−1.20413
$735$	0	0
$736$	5371.85	0.269034
$737$	− 4099.32i	− 0.204885i
$738$	− 649.260i	− 0.0323843i
$739$	29310.8	1.45902	0.729510	−	0.683970i	$-0.239748\pi$
$739$	29310.8	1.45902	0.729510	+	0.683970i	$0.239748\pi$
$740$	0	0
$741$	9152.19	0.453730
$742$	− 26096.4i	− 1.29114i
$743$	− 17147.7i	− 0.846688i	−0.905969	−	0.423344i	$-0.860856\pi$
$743$	− 17147.7i	− 0.846688i	0.905969	−	0.423344i	$-0.139144\pi$
$744$	−11103.3	−0.547132
$745$	0	0
$746$	42089.3	2.06568
$747$	− 19478.1i	− 0.954037i
$748$	7678.41i	0.375335i
$749$	−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	$0.857511\pi$
$752$	− 10313.7i	− 0.500136i
$753$	949.011i	0.0459281i
$754$	12519.8	0.604699
$755$	0	0
$756$	22091.0	1.06276
$757$	− 26843.4i	− 1.28883i	−0.764678	−	0.644413i	$-0.777102\pi$
$757$	− 26843.4i	− 1.28883i	0.764678	−	0.644413i	$-0.222898\pi$
$758$	24423.5i	1.17032i
$759$	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	$0.861168\pi$
$762$	− 6417.94i	− 0.305115i
$763$	− 17542.5i	− 0.832347i
$764$	−26152.1	−1.23842
$765$	0	0
$766$	−21290.2	−1.00424
$767$	− 30308.8i	− 1.42684i
$768$	3110.30i	0.146137i
$769$	−11248.9	−0.527496	−0.263748	−	0.964592i	$-0.584959\pi$
$769$	−11248.9	−0.527496	−0.263748	+	0.964592i	$0.584959\pi$
$770$	0	0
$771$	13249.0	0.618875
$772$	− 17274.0i	− 0.805315i
$773$	22704.0i	1.05641i	0.849117	+	0.528205i	$0.177135\pi$
$773$	22704.0i	1.05641i	−0.849117	+	0.528205i	$0.822865\pi$
$774$	−10442.6	−0.484951
$775$	0	0
$776$	26435.4	1.22291
$777$	4591.28i	0.211984i
$778$	− 50826.9i	− 2.34220i
$779$	−369.558	−0.0169972
$780$	0	0
$781$	4035.51	0.184894
$782$	− 4113.41i	− 0.188101i
$783$	5882.97i	0.268506i
$784$	−4099.58	−0.186752
$785$	0	0
$786$	−9083.41	−0.412207
$787$	− 24801.6i	− 1.12336i	−0.827356	−	0.561678i	$-0.810156\pi$
$787$	− 24801.6i	− 1.12336i	0.827356	−	0.561678i	$-0.189844\pi$
$788$	− 20306.5i	− 0.918008i
$789$	−17218.3	−0.776917
$790$	0	0
$791$	20275.9	0.911412
$792$	− 3920.55i	− 0.175897i
$793$	− 5062.77i	− 0.226714i
$794$	25332.2	1.13225
$795$	0	0
$796$	36487.2	1.62469
$797$	− 10289.9i	− 0.457323i	−0.973506	−	0.228662i	$-0.926565\pi$
$797$	− 10289.9i	− 0.457323i	0.973506	−	0.228662i	$-0.0734349\pi$
$798$	− 8071.07i	− 0.358036i
$799$	−15635.2	−0.692284
$800$	0	0
$801$	−2419.76	−0.106739
$802$	− 65836.4i	− 2.89871i
$803$	− 3125.51i	− 0.137356i
$804$	−9086.38	−0.398572
$805$	0	0
$806$	70727.7	3.09091
$807$	− 7927.80i	− 0.345814i
$808$	− 9084.91i	− 0.395552i
$809$	17890.5	0.777500	0.388750	−	0.921343i	$-0.372907\pi$
$809$	17890.5	0.777500	0.388750	+	0.921343i	$0.372907\pi$
$810$	0	0
$811$	−19915.9	−0.862322	−0.431161	−	0.902275i	$-0.641896\pi$
$811$	−19915.9	−0.862322	−0.431161	+	0.902275i	$0.641896\pi$
$812$	− 6461.87i	− 0.279270i
$813$	21622.9i	0.932779i
$814$	7443.70	0.320518
$815$	0	0
$816$	−3586.84	−0.153878
$817$	5943.93i	0.254531i
$818$	− 6535.07i	− 0.279332i
$819$	−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	$0.594117\pi$
$822$	6911.01i	0.293247i
$823$	− 26399.7i	− 1.11815i	−0.829118	−	0.559074i	$-0.811157\pi$
$823$	− 26399.7i	− 1.11815i	0.829118	−	0.559074i	$-0.188843\pi$
$824$	−22794.3	−0.963684
$825$	0	0
$826$	−26728.6	−1.12591
$827$	− 18410.9i	− 0.774135i	−0.922051	−	0.387067i	$-0.873488\pi$
$827$	− 18410.9i	− 0.774135i	0.922051	−	0.387067i	$-0.126512\pi$
$828$	4218.57i	0.177060i
$829$	−8828.44	−0.369873	−0.184936	−	0.982750i	$-0.559208\pi$
$829$	−8828.44	−0.369873	−0.184936	+	0.982750i	$0.559208\pi$
$830$	0	0
$831$	−9270.20	−0.386979
$832$	− 55722.8i	− 2.32192i
$833$	6214.82i	0.258500i
$834$	−2915.08	−0.121032
$835$	0	0
$836$	−7658.47	−0.316834
$837$	33234.5i	1.37246i
$838$	− 12030.7i	− 0.495935i
$839$	18062.0	0.743231	0.371616	−	0.928387i	$-0.378804\pi$
$839$	18062.0	0.743231	0.371616	+	0.928387i	$0.378804\pi$
$840$	0	0
$841$	−22668.2	−0.929442
$842$	− 9687.81i	− 0.396513i
$843$	6677.14i	0.272803i
$844$	58703.8	2.39416
$845$	0	0
$846$	27397.5	1.11341
$847$	14515.7i	0.588860i
$848$	− 11567.3i	− 0.468423i
$849$	14658.9	0.592570
$850$	0	0
$851$	−2333.87	−0.0940116
$852$	− 8944.95i	− 0.359682i
$853$	− 5289.59i	− 0.212324i	−0.994349	−	0.106162i	$-0.966144\pi$
$853$	− 5289.59i	− 0.212324i	0.994349	−	0.106162i	$-0.0338562\pi$
$854$	−4464.72	−0.178899
$855$	0	0
$856$	18106.5	0.722975
$857$	− 11311.0i	− 0.450846i	−0.974261	−	0.225423i	$-0.927624\pi$
$857$	− 11311.0i	− 0.450846i	0.974261	−	0.225423i	$-0.0723764\pi$
$858$	− 16530.4i	− 0.657736i
$859$	−2466.99	−0.0979892	−0.0489946	−	0.998799i	$-0.515602\pi$
$859$	−2466.99	−0.0979892	−0.0489946	+	0.998799i	$0.515602\pi$
$860$	0	0
$861$	−411.704	−0.0162960
$862$	− 54361.3i	− 2.14797i
$863$	8715.03i	0.343758i	0.985118	+	0.171879i	$0.0549838\pi$
$863$	8715.03i	0.343758i	−0.985118	+	0.171879i	$0.945016\pi$
$864$	33122.5	1.30422
$865$	0	0
$866$	−50993.1	−2.00094
$867$	− 10673.5i	− 0.418099i
$868$	− 36504.9i	− 1.42749i
$869$	1827.30	0.0713313
$870$	0	0
$871$	16865.5	0.656102
$872$	18369.4i	0.713377i
$873$	− 29725.6i	− 1.15242i
$874$	4102.73	0.158784
$875$	0	0
$876$	−6927.88	−0.267205
$877$	− 2707.70i	− 0.104256i	−0.998640	−	0.0521280i	$-0.983400\pi$
$877$	− 2707.70i	− 0.104256i	0.998640	−	0.0521280i	$-0.0166004\pi$
$878$	− 46492.6i	− 1.78707i
$879$	−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	$0.0868675\pi$
$882$	− 10890.2i	− 0.415750i
$883$	29515.0i	1.12487i	0.826842	+	0.562434i	$0.190135\pi$
$883$	29515.0i	1.12487i	−0.826842	+	0.562434i	$0.809865\pi$
$884$	−31590.6	−1.20193
$885$	0	0
$886$	59608.4	2.26025
$887$	− 16506.7i	− 0.624849i	−0.949943	−	0.312425i	$-0.898859\pi$
$887$	− 16506.7i	− 0.624849i	0.949943	−	0.312425i	$-0.101141\pi$
$888$	− 4807.70i	− 0.181684i
$889$	6148.44	0.231960
$890$	0	0
$891$	440.981	0.0165807
$892$	− 35499.5i	− 1.33253i
$893$	− 15594.6i	− 0.584383i
$894$	−10657.3	−0.398694
$895$	0	0
$896$	−23359.7	−0.870974
$897$	5182.86i	0.192922i
$898$	36900.4i	1.37125i
$899$	9721.47	0.360655
$900$	0	0
$901$	−17535.6	−0.648388
$902$	667.482i	0.0246394i
$903$	6621.80i	0.244031i
$904$	−21231.6	−0.781142
$905$	0	0
$906$	18420.3	0.675467
$907$	− 9107.76i	− 0.333427i	−0.986005	−	0.166713i	$-0.946685\pi$
$907$	− 9107.76i	− 0.333427i	0.986005	−	0.166713i	$-0.0533155\pi$
$908$	− 2965.12i	− 0.108371i
$909$	−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	$0.358399\pi$
$912$	− 3577.53i	− 0.129894i
$913$	20024.8i	0.725874i
$914$	45703.1	1.65396
$915$	0	0
$916$	−62767.0	−2.26406
$917$	− 8701.99i	− 0.313375i
$918$	− 25363.1i	− 0.911880i
$919$	−30834.5	−1.10679	−0.553393	−	0.832920i	$-0.686667\pi$
$919$	−30834.5	−1.10679	−0.553393	+	0.832920i	$0.686667\pi$
$920$	0	0
$921$	15007.7	0.536939
$922$	− 51779.8i	− 1.84954i
$923$	16602.9i	0.592083i
$924$	−8531.88	−0.303764
$925$	0	0
$926$	−69011.7	−2.44910
$927$	25631.3i	0.908137i
$928$	− 9688.70i	− 0.342723i
$929$	−17648.9	−0.623297	−0.311648	−	0.950197i	$-0.600881\pi$
$929$	−17648.9	−0.623297	−0.311648	+	0.950197i	$0.600881\pi$
$930$	0	0
$931$	−6198.68	−0.218210
$932$	49733.8i	1.74794i
$933$	12104.4i	0.424739i
$934$	22899.4	0.802238
$935$	0	0
$936$	16130.0	0.563274
$937$	23621.9i	0.823580i	0.911279	+	0.411790i	$0.135096\pi$
$937$	23621.9i	0.823580i	−0.911279	+	0.411790i	$0.864904\pi$
$938$	− 14873.2i	− 0.517726i
$939$	−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	$0.544975\pi$
$942$	10279.0i	0.355527i
$943$	− 209.280i	− 0.00722703i
$944$	−11847.5	−0.408478
$945$	0	0
$946$	10735.7	0.368972
$947$	− 52681.5i	− 1.80773i	−0.427818	−	0.903865i	$-0.640718\pi$
$947$	− 52681.5i	− 1.80773i	0.427818	−	0.903865i	$-0.359282\pi$
$948$	− 4050.31i	− 0.138764i
$949$	12859.0	0.439854
$950$	0	0
$951$	−31655.0	−1.07937
$952$	8117.69i	0.276361i
$953$	48109.3i	1.63527i	0.575735	+	0.817636i	$0.304716\pi$
$953$	48109.3i	1.63527i	−0.575735	+	0.817636i	$0.695284\pi$
$954$	30727.6	1.04281
$955$	0	0
$956$	72761.6	2.46159
$957$	− 2272.09i	− 0.0767462i
$958$	− 34753.2i	− 1.17205i
$959$	−6620.80	−0.222937
$960$	0	0
$961$	25128.3	0.843486
$962$	30624.9i	1.02639i
$963$	− 20360.1i	− 0.681302i
$964$	75739.0	2.53049
$965$	0	0
$966$	4570.63	0.152233
$967$	41498.5i	1.38004i	0.723789	+	0.690022i	$0.242399\pi$
$967$	41498.5i	1.38004i	−0.723789	+	0.690022i	$0.757601\pi$
$968$	− 15199.9i	− 0.504693i
$969$	−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	$0.489318\pi$
$972$	42251.1i	1.39424i
$973$	− 2792.67i	− 0.0920132i
$974$	11904.9	0.391641
$975$	0	0
$976$	−1979.00	−0.0649040
$977$	− 42161.7i	− 1.38063i	−0.723511	−	0.690313i	$-0.757473\pi$
$977$	− 42161.7i	− 1.38063i	0.723511	−	0.690313i	$-0.242527\pi$
$978$	− 3100.87i	− 0.101385i
$979$	2487.68	0.0812120
$980$	0	0
$981$	20655.7	0.672258
$982$	75908.9i	2.46675i
$983$	44619.0i	1.44774i	0.689938	+	0.723869i	$0.257638\pi$
$983$	44619.0i	1.44774i	−0.689938	+	0.723869i	$0.742362\pi$
$984$	431.110	0.0139668
$985$	0	0
$986$	−7418.98	−0.239623
$987$	− 17373.1i	− 0.560276i
$988$	− 31508.5i	− 1.01460i
$989$	−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	$0.597092\pi$
$992$	− 54734.1i	− 1.75183i
$993$	− 9577.39i	− 0.306072i
$994$	14641.7	0.467209
$995$	0	0
$996$	44386.1	1.41208
$997$	48629.2i	1.54474i	0.635175	+	0.772368i	$0.280928\pi$
$997$	48629.2i	1.54474i	−0.635175	+	0.772368i	$0.719072\pi$
$998$	72563.3i	2.30156i
$999$	−14390.5	−0.455750

Twists

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

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

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.39200i q^{2} -3.27927i q^{3} -11.2896 q^{4} -14.4025 q^{6} -13.7978i q^{7} +14.4481i q^{8} +16.2464 q^{9} -16.7024 q^{11} +37.0218i q^{12} -68.7171i q^{13} -60.5997 q^{14} -26.8610 q^{16} +40.7204i q^{17} -71.3541i q^{18} -40.6147 q^{19} -45.2466 q^{21} +73.3568i q^{22} -23.0000i q^{23} +47.3793 q^{24} -301.805 q^{26} -141.817i q^{27} +155.772i q^{28} -41.4829 q^{29} -234.349 q^{31} +233.559i q^{32} +54.7716i q^{33} +178.844 q^{34} -183.416 q^{36} -101.472i q^{37} +178.380i q^{38} -225.342 q^{39} +9.09912 q^{41} +198.723i q^{42} -146.349i q^{43} +188.564 q^{44} -101.016 q^{46} +383.965i q^{47} +88.0845i q^{48} +152.622 q^{49} +133.533 q^{51} +775.792i q^{52} +430.635i q^{53} -622.858 q^{54} +199.352 q^{56} +133.187i q^{57} +182.193i q^{58} +441.067 q^{59} +73.6756 q^{61} +1029.26i q^{62} -224.164i q^{63} +810.901 q^{64} +240.557 q^{66} +245.433i q^{67} -459.719i q^{68} -75.4232 q^{69} -241.613 q^{71} +234.730i q^{72} +187.130i q^{73} -445.667 q^{74} +458.526 q^{76} +230.455i q^{77} +989.701i q^{78} -109.404 q^{79} -26.4023 q^{81} -39.9633i q^{82} -1198.92i q^{83} +510.818 q^{84} -642.765 q^{86} +136.034i q^{87} -241.318i q^{88} -148.942 q^{89} -948.142 q^{91} +259.662i q^{92} +768.492i q^{93} +1686.38 q^{94} +765.902 q^{96} -1829.68i q^{97} -670.314i q^{98} -271.353 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 54 q^{4} - 82 q^{6} - 20 q^{9} - 104 q^{11} + 84 q^{14} - 170 q^{16} + 20 q^{19} - 404 q^{21} + 606 q^{24} - 52 q^{26} + 910 q^{29} - 1380 q^{31} + 1314 q^{34} + 408 q^{36} + 554 q^{39} - 460 q^{41}+ \cdots + 4286 q^{99}+O(q^{100}) \) 14 * q - 54 * q^4 - 82 * q^6 - 20 * q^9 - 104 * q^11 + 84 * q^14 - 170 * q^16 + 20 * q^19 - 404 * q^21 + 606 * q^24 - 52 * q^26 + 910 * q^29 - 1380 * q^31 + 1314 * q^34 + 408 * q^36 + 554 * q^39 - 460 * q^41 + 2006 * q^44 - 138 * q^46 + 1604 * q^49 - 1390 * q^51 + 2494 * q^54 - 772 * q^56 + 1102 * q^59 - 2484 * q^61 + 3002 * q^64 + 1974 * q^66 + 46 * q^69 + 372 * q^71 + 3060 * q^74 - 3550 * q^76 + 4006 * q^79 - 730 * q^81 + 2704 * q^84 + 3788 * q^86 + 5258 * q^89 - 2590 * q^91 + 6936 * q^94 - 270 * q^96 + 4286 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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