LMFDB - Embedded newform 175.4.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, 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("175.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	175.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:	$10.3253342510$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 27x^{3} + 7x^{2} + 120x + 60 $ x^5 - x^4 - 27x^3 + 7x^2 + 120*x + 60
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.555276$ of defining polynomial
Character	$\chi$	$=$	175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.55528 q^{2} -4.96149 q^{3} -5.58112 q^{4} -7.71648 q^{6} +7.00000 q^{7} -21.1224 q^{8} -2.38365 q^{9} +29.8232 q^{11} +27.6906 q^{12} +90.7316 q^{13} +10.8869 q^{14} +11.7978 q^{16} +29.5740 q^{17} -3.70723 q^{18} -62.3278 q^{19} -34.7304 q^{21} +46.3833 q^{22} +90.6198 q^{23} +104.798 q^{24} +141.113 q^{26} +145.787 q^{27} -39.0678 q^{28} +193.070 q^{29} -152.123 q^{31} +187.328 q^{32} -147.967 q^{33} +45.9957 q^{34} +13.3034 q^{36} +102.453 q^{37} -96.9368 q^{38} -450.164 q^{39} -266.744 q^{41} -54.0154 q^{42} +387.125 q^{43} -166.447 q^{44} +140.939 q^{46} -152.298 q^{47} -58.5346 q^{48} +49.0000 q^{49} -146.731 q^{51} -506.384 q^{52} -81.5982 q^{53} +226.738 q^{54} -147.857 q^{56} +309.238 q^{57} +300.277 q^{58} -235.884 q^{59} +510.453 q^{61} -236.593 q^{62} -16.6855 q^{63} +196.964 q^{64} -230.130 q^{66} -347.374 q^{67} -165.056 q^{68} -449.609 q^{69} +317.014 q^{71} +50.3483 q^{72} -709.901 q^{73} +159.343 q^{74} +347.858 q^{76} +208.762 q^{77} -700.129 q^{78} +1062.95 q^{79} -658.960 q^{81} -414.861 q^{82} +503.810 q^{83} +193.834 q^{84} +602.086 q^{86} -957.914 q^{87} -629.937 q^{88} +482.342 q^{89} +635.121 q^{91} -505.760 q^{92} +754.756 q^{93} -236.866 q^{94} -929.425 q^{96} +481.167 q^{97} +76.2085 q^{98} -71.0879 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 10 q^{3} + 18 q^{4} + 6 q^{6} + 35 q^{7} + 42 q^{8} + 23 q^{9} + 42 q^{11} + 136 q^{12} + 34 q^{13} + 28 q^{14} + 74 q^{16} + 238 q^{17} - 2 q^{18} - 36 q^{19} + 70 q^{21} + 358 q^{22} + 152 q^{23}+ \cdots + 2652 q^{99}+O(q^{100}) $ 5 * q + 4 * q^2 + 10 * q^3 + 18 * q^4 + 6 * q^6 + 35 * q^7 + 42 * q^8 + 23 * q^9 + 42 * q^11 + 136 * q^12 + 34 * q^13 + 28 * q^14 + 74 * q^16 + 238 * q^17 - 2 * q^18 - 36 * q^19 + 70 * q^21 + 358 * q^22 + 152 * q^23 - 36 * q^24 - 310 * q^26 + 334 * q^27 + 126 * q^28 - 44 * q^29 + 60 * q^31 + 710 * q^32 + 426 * q^33 - 482 * q^34 - 210 * q^36 + 312 * q^37 + 280 * q^38 - 106 * q^39 - 426 * q^41 + 42 * q^42 + 304 * q^43 + 712 * q^44 + 88 * q^46 + 370 * q^47 - 696 * q^48 + 245 * q^49 + 638 * q^51 - 1156 * q^52 + 976 * q^53 + 498 * q^54 + 294 * q^56 - 588 * q^57 - 2722 * q^58 - 432 * q^59 - 442 * q^61 - 956 * q^62 + 161 * q^63 + 1362 * q^64 + 574 * q^66 - 804 * q^67 - 420 * q^68 - 2404 * q^69 + 440 * q^71 - 3150 * q^72 + 564 * q^73 - 1512 * q^74 - 1336 * q^76 + 294 * q^77 - 2742 * q^78 + 1790 * q^79 - 151 * q^81 - 3480 * q^82 + 1656 * q^83 + 952 * q^84 + 1216 * q^86 - 1674 * q^87 - 1092 * q^88 + 746 * q^89 + 238 * q^91 + 572 * q^92 + 676 * q^93 - 826 * q^94 + 2040 * q^96 + 518 * q^97 + 196 * q^98 + 2652 * 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 $

$2$	1.55528	0.549873	0.274937	−	0.961462i	$-0.411343\pi$
$2$	1.55528	0.549873	0.274937	+	0.961462i	$0.411343\pi$
$3$	−4.96149	−0.954839	−0.477419	−	0.878676i	$-0.658428\pi$
$3$	−4.96149	−0.954839	−0.477419	+	0.878676i	$0.658428\pi$
$4$	−5.58112	−0.697640
$5$	0	0
$5$	0	0
$6$	−7.71648	−0.525040
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−21.1224	−0.933486
$9$	−2.38365	−0.0882832
$10$	0	0
$11$	29.8232	0.817457	0.408728	−	0.912656i	$-0.365972\pi$
$11$	29.8232	0.817457	0.408728	+	0.912656i	$0.365972\pi$
$12$	27.6906	0.666133
$13$	90.7316	1.93572	0.967862	−	0.251481i	$-0.0809174\pi$
$13$	90.7316	1.93572	0.967862	+	0.251481i	$0.0809174\pi$
$14$	10.8869	0.207832
$15$	0	0
$16$	11.7978	0.184341
$17$	29.5740	0.421927	0.210963	−	0.977494i	$-0.432340\pi$
$17$	29.5740	0.421927	0.210963	+	0.977494i	$0.432340\pi$
$18$	−3.70723	−0.0485446
$19$	−62.3278	−0.752577	−0.376289	−	0.926503i	$-0.622800\pi$
$19$	−62.3278	−0.752577	−0.376289	+	0.926503i	$0.622800\pi$
$20$	0	0
$21$	−34.7304	−0.360895
$22$	46.3833	0.449498
$23$	90.6198	0.821545	0.410772	−	0.911738i	$-0.365259\pi$
$23$	90.6198	0.821545	0.410772	+	0.911738i	$0.365259\pi$
$24$	104.798	0.891329
$25$	0	0
$26$	141.113	1.06440
$27$	145.787	1.03913
$28$	−39.0678	−0.263683
$29$	193.070	1.23628	0.618141	−	0.786067i	$-0.287886\pi$
$29$	193.070	1.23628	0.618141	+	0.786067i	$0.287886\pi$
$30$	0	0
$31$	−152.123	−0.881357	−0.440679	−	0.897665i	$-0.645262\pi$
$31$	−152.123	−0.881357	−0.440679	+	0.897665i	$0.645262\pi$
$32$	187.328	1.03485
$33$	−147.967	−0.780539
$34$	45.9957	0.232006
$35$	0	0
$36$	13.3034	0.0615899
$37$	102.453	0.455222	0.227611	−	0.973752i	$-0.426908\pi$
$37$	102.453	0.455222	0.227611	+	0.973752i	$0.426908\pi$
$38$	−96.9368	−0.413822
$39$	−450.164	−1.84830
$40$	0	0
$41$	−266.744	−1.01606	−0.508030	−	0.861340i	$-0.669626\pi$
$41$	−266.744	−1.01606	−0.508030	+	0.861340i	$0.669626\pi$
$42$	−54.0154	−0.198446
$43$	387.125	1.37293	0.686465	−	0.727163i	$-0.259162\pi$
$43$	387.125	1.37293	0.686465	+	0.727163i	$0.259162\pi$
$44$	−166.447	−0.570290
$45$	0	0
$46$	140.939	0.451745
$47$	−152.298	−0.472660	−0.236330	−	0.971673i	$-0.575945\pi$
$47$	−152.298	−0.472660	−0.236330	+	0.971673i	$0.575945\pi$
$48$	−58.5346	−0.176016
$49$	49.0000	0.142857
$50$	0	0
$51$	−146.731	−0.402872
$52$	−506.384	−1.35044
$53$	−81.5982	−0.211479	−0.105739	−	0.994394i	$-0.533721\pi$
$53$	−81.5982	−0.211479	−0.105739	+	0.994394i	$0.533721\pi$
$54$	226.738	0.571392
$55$	0	0
$56$	−147.857	−0.352825
$57$	309.238	0.718590
$58$	300.277	0.679798
$59$	−235.884	−0.520499	−0.260250	−	0.965541i	$-0.583805\pi$
$59$	−235.884	−0.520499	−0.260250	+	0.965541i	$0.583805\pi$
$60$	0	0
$61$	510.453	1.07142	0.535712	−	0.844401i	$-0.320043\pi$
$61$	510.453	1.07142	0.535712	+	0.844401i	$0.320043\pi$
$62$	−236.593	−0.484635
$63$	−16.6855	−0.0333679
$64$	196.964	0.384696
$65$	0	0
$66$	−230.130	−0.429198
$67$	−347.374	−0.633410	−0.316705	−	0.948524i	$-0.602577\pi$
$67$	−347.374	−0.633410	−0.316705	+	0.948524i	$0.602577\pi$
$68$	−165.056	−0.294353
$69$	−449.609	−0.784443
$70$	0	0
$71$	317.014	0.529896	0.264948	−	0.964263i	$-0.414645\pi$
$71$	317.014	0.529896	0.264948	+	0.964263i	$0.414645\pi$
$72$	50.3483	0.0824112
$73$	−709.901	−1.13819	−0.569093	−	0.822273i	$-0.692706\pi$
$73$	−709.901	−1.13819	−0.569093	+	0.822273i	$0.692706\pi$
$74$	159.343	0.250315
$75$	0	0
$76$	347.858	0.525028
$77$	208.762	0.308970
$78$	−700.129	−1.01633
$79$	1062.95	1.51382	0.756908	−	0.653522i	$-0.226709\pi$
$79$	1062.95	1.51382	0.756908	+	0.653522i	$0.226709\pi$
$80$	0	0
$81$	−658.960	−0.903923
$82$	−414.861	−0.558704
$83$	503.810	0.666270	0.333135	−	0.942879i	$-0.391894\pi$
$83$	503.810	0.666270	0.333135	+	0.942879i	$0.391894\pi$
$84$	193.834	0.251775
$85$	0	0
$86$	602.086	0.754937
$87$	−957.914	−1.18045
$88$	−629.937	−0.763085
$89$	482.342	0.574474	0.287237	−	0.957860i	$-0.407263\pi$
$89$	482.342	0.574474	0.287237	+	0.957860i	$0.407263\pi$
$90$	0	0
$91$	635.121	0.731635
$92$	−505.760	−0.573142
$93$	754.756	0.841554
$94$	−236.866	−0.259903
$95$	0	0
$96$	−929.425	−0.988115
$97$	481.167	0.503661	0.251831	−	0.967771i	$-0.418967\pi$
$97$	481.167	0.503661	0.251831	+	0.967771i	$0.418967\pi$
$98$	76.2085	0.0785533
$99$	−71.0879	−0.0721677
$100$	0	0
$101$	1065.65	1.04986	0.524931	−	0.851145i	$-0.324091\pi$
$101$	1065.65	1.04986	0.524931	+	0.851145i	$0.324091\pi$
$102$	−228.207	−0.221528
$103$	−94.8594	−0.0907454	−0.0453727	−	0.998970i	$-0.514448\pi$
$103$	−94.8594	−0.0907454	−0.0453727	+	0.998970i	$0.514448\pi$
$104$	−1916.47	−1.80697
$105$	0	0
$106$	−126.908	−0.116286
$107$	−1288.94	−1.16455	−0.582274	−	0.812993i	$-0.697837\pi$
$107$	−1288.94	−1.16455	−0.582274	+	0.812993i	$0.697837\pi$
$108$	−813.652	−0.724942
$109$	343.006	0.301413	0.150707	−	0.988579i	$-0.451845\pi$
$109$	343.006	0.301413	0.150707	+	0.988579i	$0.451845\pi$
$110$	0	0
$111$	−508.321	−0.434664
$112$	82.5846	0.0696742
$113$	1444.58	1.20261	0.601306	−	0.799019i	$-0.294647\pi$
$113$	1444.58	1.20261	0.601306	+	0.799019i	$0.294647\pi$
$114$	480.951	0.395133
$115$	0	0
$116$	−1077.55	−0.862479
$117$	−216.272	−0.170892
$118$	−366.864	−0.286208
$119$	207.018	0.159473
$120$	0	0
$121$	−441.578	−0.331764
$122$	793.895	0.589147
$123$	1323.45	0.970173
$124$	849.016	0.614870
$125$	0	0
$126$	−25.9506	−0.0183481
$127$	1786.52	1.24825	0.624125	−	0.781324i	$-0.285455\pi$
$127$	1786.52	1.24825	0.624125	+	0.781324i	$0.285455\pi$
$128$	−1192.29	−0.823317
$129$	−1920.71	−1.31093
$130$	0	0
$131$	−885.502	−0.590585	−0.295293	−	0.955407i	$-0.595417\pi$
$131$	−885.502	−0.590585	−0.295293	+	0.955407i	$0.595417\pi$
$132$	825.823	0.544535
$133$	−436.294	−0.284447
$134$	−540.263	−0.348295
$135$	0	0
$136$	−624.674	−0.393863
$137$	3019.24	1.88285	0.941426	−	0.337219i	$-0.109486\pi$
$137$	3019.24	1.88285	0.941426	+	0.337219i	$0.109486\pi$
$138$	−699.266	−0.431344
$139$	−2725.65	−1.66322	−0.831608	−	0.555363i	$-0.812579\pi$
$139$	−2725.65	−1.66322	−0.831608	+	0.555363i	$0.812579\pi$
$140$	0	0
$141$	755.626	0.451314
$142$	493.043	0.291375
$143$	2705.90	1.58237
$144$	−28.1218	−0.0162742
$145$	0	0
$146$	−1104.09	−0.625858
$147$	−243.113	−0.136406
$148$	−571.804	−0.317581
$149$	2963.88	1.62960	0.814801	−	0.579741i	$-0.196846\pi$
$149$	2963.88	1.62960	0.814801	+	0.579741i	$0.196846\pi$
$150$	0	0
$151$	434.482	0.234157	0.117078	−	0.993123i	$-0.462647\pi$
$151$	434.482	0.234157	0.117078	+	0.993123i	$0.462647\pi$
$152$	1316.51	0.702520
$153$	−70.4940	−0.0372490
$154$	324.683	0.169894
$155$	0	0
$156$	2512.42	1.28945
$157$	496.409	0.252342	0.126171	−	0.992008i	$-0.459731\pi$
$157$	496.409	0.252342	0.126171	+	0.992008i	$0.459731\pi$
$158$	1653.18	0.832406
$159$	404.848	0.201928
$160$	0	0
$161$	634.339	0.310515
$162$	−1024.86	−0.497043
$163$	−2966.73	−1.42560	−0.712799	−	0.701368i	$-0.752573\pi$
$163$	−2966.73	−1.42560	−0.712799	+	0.701368i	$0.752573\pi$
$164$	1488.73	0.708843
$165$	0	0
$166$	783.564	0.366364
$167$	−1295.44	−0.600265	−0.300132	−	0.953898i	$-0.597031\pi$
$167$	−1295.44	−0.600265	−0.300132	+	0.953898i	$0.597031\pi$
$168$	733.589	0.336891
$169$	6035.23	2.74703
$170$	0	0
$171$	148.567	0.0664399
$172$	−2160.59	−0.957810
$173$	3195.32	1.40425	0.702126	−	0.712052i	$-0.252234\pi$
$173$	3195.32	1.40425	0.702126	+	0.712052i	$0.252234\pi$
$174$	−1489.82	−0.649098
$175$	0	0
$176$	351.848	0.150691
$177$	1170.33	0.496993
$178$	750.176	0.315888
$179$	−4294.90	−1.79338	−0.896692	−	0.442655i	$-0.854037\pi$
$179$	−4294.90	−1.79338	−0.896692	+	0.442655i	$0.854037\pi$
$180$	0	0
$181$	−1017.02	−0.417648	−0.208824	−	0.977953i	$-0.566964\pi$
$181$	−1017.02	−0.417648	−0.208824	+	0.977953i	$0.566964\pi$
$182$	987.789	0.402306
$183$	−2532.61	−1.02304
$184$	−1914.11	−0.766901
$185$	0	0
$186$	1173.85	0.462748
$187$	881.991	0.344907
$188$	849.995	0.329746
$189$	1020.51	0.392756
$190$	0	0
$191$	−1072.62	−0.406347	−0.203174	−	0.979143i	$-0.565126\pi$
$191$	−1072.62	−0.406347	−0.203174	+	0.979143i	$0.565126\pi$
$192$	−977.235	−0.367322
$193$	−1382.74	−0.515710	−0.257855	−	0.966184i	$-0.583016\pi$
$193$	−1382.74	−0.515710	−0.257855	+	0.966184i	$0.583016\pi$
$194$	748.348	0.276950
$195$	0	0
$196$	−273.475	−0.0996628
$197$	933.061	0.337451	0.168725	−	0.985663i	$-0.446035\pi$
$197$	933.061	0.337451	0.168725	+	0.985663i	$0.446035\pi$
$198$	−110.561	−0.0396831
$199$	126.253	0.0449740	0.0224870	−	0.999747i	$-0.492842\pi$
$199$	126.253	0.0449740	0.0224870	+	0.999747i	$0.492842\pi$
$200$	0	0
$201$	1723.49	0.604805
$202$	1657.38	0.577291
$203$	1351.49	0.467271
$204$	818.923	0.281059
$205$	0	0
$206$	−147.533	−0.0498985
$207$	−216.006	−0.0725286
$208$	1070.43	0.356833
$209$	−1858.81	−0.615199
$210$	0	0
$211$	−862.984	−0.281565	−0.140783	−	0.990041i	$-0.544962\pi$
$211$	−862.984	−0.281565	−0.140783	+	0.990041i	$0.544962\pi$
$212$	455.409	0.147536
$213$	−1572.86	−0.505965
$214$	−2004.66	−0.640353
$215$	0	0
$216$	−3079.36	−0.970018
$217$	−1064.86	−0.333122
$218$	533.469	0.165739
$219$	3522.16	1.08678
$220$	0	0
$221$	2683.30	0.816734
$222$	−790.579	−0.239010
$223$	−4533.19	−1.36128	−0.680639	−	0.732619i	$-0.738298\pi$
$223$	−4533.19	−1.36128	−0.680639	+	0.732619i	$0.738298\pi$
$224$	1311.30	0.391137
$225$	0	0
$226$	2246.73	0.661284
$227$	3806.46	1.11297	0.556484	−	0.830858i	$-0.312150\pi$
$227$	3806.46	1.11297	0.556484	+	0.830858i	$0.312150\pi$
$228$	−1725.90	−0.501317
$229$	−3073.77	−0.886989	−0.443495	−	0.896277i	$-0.646261\pi$
$229$	−3073.77	−0.886989	−0.443495	+	0.896277i	$0.646261\pi$
$230$	0	0
$231$	−1035.77	−0.295016
$232$	−4078.10	−1.15405
$233$	1527.66	0.429530	0.214765	−	0.976666i	$-0.431101\pi$
$233$	1527.66	0.429530	0.214765	+	0.976666i	$0.431101\pi$
$234$	−336.363	−0.0939689
$235$	0	0
$236$	1316.49	0.363121
$237$	−5273.82	−1.44545
$238$	321.970	0.0876900
$239$	−2356.10	−0.637672	−0.318836	−	0.947810i	$-0.603292\pi$
$239$	−2356.10	−0.637672	−0.318836	+	0.947810i	$0.603292\pi$
$240$	0	0
$241$	−6297.25	−1.68316	−0.841581	−	0.540132i	$-0.818374\pi$
$241$	−6297.25	−1.68316	−0.841581	+	0.540132i	$0.818374\pi$
$242$	−686.776	−0.182428
$243$	−666.817	−0.176034
$244$	−2848.90	−0.747467
$245$	0	0
$246$	2058.33	0.533472
$247$	−5655.10	−1.45678
$248$	3213.20	0.822735
$249$	−2499.65	−0.636180
$250$	0	0
$251$	−3531.12	−0.887978	−0.443989	−	0.896032i	$-0.646437\pi$
$251$	−3531.12	−0.887978	−0.443989	+	0.896032i	$0.646437\pi$
$252$	93.1239	0.0232788
$253$	2702.57	0.671578
$254$	2778.53	0.686379
$255$	0	0
$256$	−3430.05	−0.837415
$257$	−2186.27	−0.530645	−0.265323	−	0.964160i	$-0.585478\pi$
$257$	−2186.27	−0.530645	−0.265323	+	0.964160i	$0.585478\pi$
$258$	−2987.24	−0.720843
$259$	717.173	0.172058
$260$	0	0
$261$	−460.210	−0.109143
$262$	−1377.20	−0.324747
$263$	−2432.29	−0.570271	−0.285135	−	0.958487i	$-0.592039\pi$
$263$	−2432.29	−0.570271	−0.285135	+	0.958487i	$0.592039\pi$
$264$	3125.42	0.728623
$265$	0	0
$266$	−678.558	−0.156410
$267$	−2393.14	−0.548530
$268$	1938.74	0.441892
$269$	−1025.45	−0.232428	−0.116214	−	0.993224i	$-0.537076\pi$
$269$	−1025.45	−0.232428	−0.116214	+	0.993224i	$0.537076\pi$
$270$	0	0
$271$	3859.89	0.865209	0.432604	−	0.901584i	$-0.357595\pi$
$271$	3859.89	0.865209	0.432604	+	0.901584i	$0.357595\pi$
$272$	348.908	0.0777782
$273$	−3151.15	−0.698594
$274$	4695.75	1.03533
$275$	0	0
$276$	2509.32	0.547258
$277$	−5352.31	−1.16097	−0.580485	−	0.814271i	$-0.697137\pi$
$277$	−5352.31	−1.16097	−0.580485	+	0.814271i	$0.697137\pi$
$278$	−4239.15	−0.914558
$279$	362.607	0.0778090
$280$	0	0
$281$	1405.89	0.298464	0.149232	−	0.988802i	$-0.452320\pi$
$281$	1405.89	0.298464	0.149232	+	0.988802i	$0.452320\pi$
$282$	1175.21	0.248165
$283$	−1886.24	−0.396203	−0.198102	−	0.980181i	$-0.563478\pi$
$283$	−1886.24	−0.396203	−0.198102	+	0.980181i	$0.563478\pi$
$284$	−1769.29	−0.369676
$285$	0	0
$286$	4208.43	0.870104
$287$	−1867.21	−0.384034
$288$	−446.523	−0.0913599
$289$	−4038.38	−0.821978
$290$	0	0
$291$	−2387.31	−0.480915
$292$	3962.04	0.794044
$293$	6265.03	1.24917	0.624585	−	0.780956i	$-0.285268\pi$
$293$	6265.03	1.24917	0.624585	+	0.780956i	$0.285268\pi$
$294$	−378.108	−0.0750057
$295$	0	0
$296$	−2164.06	−0.424944
$297$	4347.82	0.849448
$298$	4609.65	0.896074
$299$	8222.08	1.59028
$300$	0	0
$301$	2709.87	0.518919
$302$	675.739	0.128756
$303$	−5287.21	−1.00245
$304$	−735.331	−0.138731
$305$	0	0
$306$	−109.638	−0.0204822
$307$	−1665.50	−0.309626	−0.154813	−	0.987944i	$-0.549478\pi$
$307$	−1665.50	−0.309626	−0.154813	+	0.987944i	$0.549478\pi$
$308$	−1165.13	−0.215549
$309$	470.644	0.0866472
$310$	0	0
$311$	2493.66	0.454669	0.227335	−	0.973817i	$-0.426999\pi$
$311$	2493.66	0.454669	0.227335	+	0.973817i	$0.426999\pi$
$312$	9508.53	1.72537
$313$	−2859.53	−0.516390	−0.258195	−	0.966093i	$-0.583128\pi$
$313$	−2859.53	−0.516390	−0.258195	+	0.966093i	$0.583128\pi$
$314$	772.052	0.138756
$315$	0	0
$316$	−5932.46	−1.05610
$317$	8154.58	1.44482	0.722409	−	0.691466i	$-0.243035\pi$
$317$	8154.58	1.44482	0.722409	+	0.691466i	$0.243035\pi$
$318$	629.651	0.111035
$319$	5757.96	1.01061
$320$	0	0
$321$	6395.06	1.11195
$322$	986.571	0.170744
$323$	−1843.28	−0.317532
$324$	3677.73	0.630612
$325$	0	0
$326$	−4614.09	−0.783898
$327$	−1701.82	−0.287801
$328$	5634.27	0.948477
$329$	−1066.09	−0.178649
$330$	0	0
$331$	164.125	0.0272541	0.0136271	−	0.999907i	$-0.495662\pi$
$331$	164.125	0.0272541	0.0136271	+	0.999907i	$0.495662\pi$
$332$	−2811.83	−0.464816
$333$	−244.213	−0.0401885
$334$	−2014.77	−0.330069
$335$	0	0
$336$	−409.743	−0.0665276
$337$	5015.35	0.810692	0.405346	−	0.914163i	$-0.367151\pi$
$337$	5015.35	0.810692	0.405346	+	0.914163i	$0.367151\pi$
$338$	9386.44	1.51052
$339$	−7167.29	−1.14830
$340$	0	0
$341$	−4536.79	−0.720472
$342$	231.063	0.0365335
$343$	343.000	0.0539949
$344$	−8177.00	−1.28161
$345$	0	0
$346$	4969.60	0.772161
$347$	−10679.9	−1.65224	−0.826122	−	0.563492i	$-0.809458\pi$
$347$	−10679.9	−1.65224	−0.826122	+	0.563492i	$0.809458\pi$
$348$	5346.23	0.823529
$349$	−7727.85	−1.18528	−0.592639	−	0.805468i	$-0.701914\pi$
$349$	−7727.85	−1.18528	−0.592639	+	0.805468i	$0.701914\pi$
$350$	0	0
$351$	13227.5	2.01148
$352$	5586.71	0.845946
$353$	−3214.61	−0.484693	−0.242346	−	0.970190i	$-0.577917\pi$
$353$	−3214.61	−0.484693	−0.242346	+	0.970190i	$0.577917\pi$
$354$	1820.19	0.273283
$355$	0	0
$356$	−2692.01	−0.400776
$357$	−1027.12	−0.152271
$358$	−6679.75	−0.986134
$359$	104.088	0.0153023	0.00765116	−	0.999971i	$-0.497565\pi$
$359$	104.088	0.0153023	0.00765116	+	0.999971i	$0.497565\pi$
$360$	0	0
$361$	−2974.25	−0.433628
$362$	−1581.74	−0.229653
$363$	2190.88	0.316781
$364$	−3544.69	−0.510418
$365$	0	0
$366$	−3938.90	−0.562540
$367$	2285.33	0.325049	0.162525	−	0.986704i	$-0.448036\pi$
$367$	2285.33	0.325049	0.162525	+	0.986704i	$0.448036\pi$
$368$	1069.11	0.151444
$369$	635.824	0.0897010
$370$	0	0
$371$	−571.187	−0.0799314
$372$	−4212.38	−0.587101
$373$	5810.11	0.806531	0.403265	−	0.915083i	$-0.367875\pi$
$373$	5810.11	0.806531	0.403265	+	0.915083i	$0.367875\pi$
$374$	1371.74	0.189655
$375$	0	0
$376$	3216.90	0.441221
$377$	17517.5	2.39310
$378$	1587.17	0.215966
$379$	−1457.82	−0.197581	−0.0987904	−	0.995108i	$-0.531497\pi$
$379$	−1457.82	−0.197581	−0.0987904	+	0.995108i	$0.531497\pi$
$380$	0	0
$381$	−8863.79	−1.19188
$382$	−1668.22	−0.223439
$383$	775.614	0.103478	0.0517389	−	0.998661i	$-0.483524\pi$
$383$	775.614	0.103478	0.0517389	+	0.998661i	$0.483524\pi$
$384$	5915.53	0.786134
$385$	0	0
$386$	−2150.54	−0.283575
$387$	−922.769	−0.121207
$388$	−2685.45	−0.351374
$389$	9160.87	1.19402	0.597011	−	0.802233i	$-0.296355\pi$
$389$	9160.87	1.19402	0.597011	+	0.802233i	$0.296355\pi$
$390$	0	0
$391$	2679.99	0.346632
$392$	−1035.00	−0.133355
$393$	4393.41	0.563913
$394$	1451.17	0.185555
$395$	0	0
$396$	396.750	0.0503471
$397$	2306.95	0.291643	0.145822	−	0.989311i	$-0.453417\pi$
$397$	2306.95	0.291643	0.145822	+	0.989311i	$0.453417\pi$
$398$	196.358	0.0247300
$399$	2164.67	0.271601
$400$	0	0
$401$	5554.28	0.691689	0.345845	−	0.938292i	$-0.387592\pi$
$401$	5554.28	0.691689	0.345845	+	0.938292i	$0.387592\pi$
$402$	2680.51	0.332566
$403$	−13802.4	−1.70607
$404$	−5947.51	−0.732425
$405$	0	0
$406$	2101.94	0.256940
$407$	3055.48	0.372125
$408$	3099.31	0.376075
$409$	6860.22	0.829379	0.414690	−	0.909963i	$-0.363890\pi$
$409$	6860.22	0.829379	0.414690	+	0.909963i	$0.363890\pi$
$410$	0	0
$411$	−14979.9	−1.79782
$412$	529.422	0.0633076
$413$	−1651.19	−0.196730
$414$	−335.948	−0.0398815
$415$	0	0
$416$	16996.6	2.00319
$417$	13523.3	1.58810
$418$	−2890.96	−0.338282
$419$	6261.62	0.730072	0.365036	−	0.930993i	$-0.381057\pi$
$419$	6261.62	0.730072	0.365036	+	0.930993i	$0.381057\pi$
$420$	0	0
$421$	−12589.2	−1.45739	−0.728694	−	0.684840i	$-0.759872\pi$
$421$	−12589.2	−1.45739	−0.728694	+	0.684840i	$0.759872\pi$
$422$	−1342.18	−0.154825
$423$	363.025	0.0417279
$424$	1723.55	0.197412
$425$	0	0
$426$	−2446.23	−0.278216
$427$	3573.17	0.404960
$428$	7193.73	0.812434
$429$	−13425.3	−1.51091
$430$	0	0
$431$	13362.1	1.49334	0.746670	−	0.665195i	$-0.231652\pi$
$431$	13362.1	1.49334	0.746670	+	0.665195i	$0.231652\pi$
$432$	1719.96	0.191555
$433$	−13554.8	−1.50439	−0.752194	−	0.658941i	$-0.771005\pi$
$433$	−13554.8	−1.50439	−0.752194	+	0.658941i	$0.771005\pi$
$434$	−1656.15	−0.183175
$435$	0	0
$436$	−1914.36	−0.210278
$437$	−5648.13	−0.618276
$438$	5477.94	0.597593
$439$	−9039.11	−0.982718	−0.491359	−	0.870957i	$-0.663500\pi$
$439$	−9039.11	−0.982718	−0.491359	+	0.870957i	$0.663500\pi$
$440$	0	0
$441$	−116.799	−0.0126119
$442$	4173.27	0.449100
$443$	287.745	0.0308604	0.0154302	−	0.999881i	$-0.495088\pi$
$443$	287.745	0.0308604	0.0154302	+	0.999881i	$0.495088\pi$
$444$	2837.00	0.303239
$445$	0	0
$446$	−7050.37	−0.748530
$447$	−14705.3	−1.55601
$448$	1378.75	0.145401
$449$	672.823	0.0707182	0.0353591	−	0.999375i	$-0.488743\pi$
$449$	672.823	0.0707182	0.0353591	+	0.999375i	$0.488743\pi$
$450$	0	0
$451$	−7955.16	−0.830585
$452$	−8062.39	−0.838989
$453$	−2155.68	−0.223582
$454$	5920.10	0.611991
$455$	0	0
$456$	−6531.85	−0.670794
$457$	−6272.56	−0.642052	−0.321026	−	0.947070i	$-0.604028\pi$
$457$	−6272.56	−0.642052	−0.321026	+	0.947070i	$0.604028\pi$
$458$	−4780.56	−0.487731
$459$	4311.49	0.438439
$460$	0	0
$461$	3967.12	0.400796	0.200398	−	0.979715i	$-0.435776\pi$
$461$	3967.12	0.400796	0.200398	+	0.979715i	$0.435776\pi$
$462$	−1610.91	−0.162221
$463$	14898.3	1.49543	0.747716	−	0.664019i	$-0.231151\pi$
$463$	14898.3	1.49543	0.747716	+	0.664019i	$0.231151\pi$
$464$	2277.80	0.227897
$465$	0	0
$466$	2375.93	0.236187
$467$	−7371.22	−0.730405	−0.365203	−	0.930928i	$-0.619000\pi$
$467$	−7371.22	−0.730405	−0.365203	+	0.930928i	$0.619000\pi$
$468$	1207.04	0.119221
$469$	−2431.62	−0.239407
$470$	0	0
$471$	−2462.93	−0.240946
$472$	4982.43	0.485879
$473$	11545.3	1.12231
$474$	−8202.25	−0.794814
$475$	0	0
$476$	−1155.39	−0.111255
$477$	194.501	0.0186700
$478$	−3664.39	−0.350638
$479$	11934.3	1.13840	0.569198	−	0.822201i	$-0.307254\pi$
$479$	11934.3	1.13840	0.569198	+	0.822201i	$0.307254\pi$
$480$	0	0
$481$	9295.76	0.881185
$482$	−9793.97	−0.925525
$483$	−3147.26	−0.296492
$484$	2464.50	0.231452
$485$	0	0
$486$	−1037.08	−0.0967965
$487$	4462.53	0.415229	0.207614	−	0.978211i	$-0.433430\pi$
$487$	4462.53	0.415229	0.207614	+	0.978211i	$0.433430\pi$
$488$	−10782.0	−1.00016
$489$	14719.4	1.36122
$490$	0	0
$491$	7577.70	0.696491	0.348245	−	0.937403i	$-0.386778\pi$
$491$	7577.70	0.696491	0.348245	+	0.937403i	$0.386778\pi$
$492$	−7386.31	−0.676831
$493$	5709.85	0.521620
$494$	−8795.24	−0.801045
$495$	0	0
$496$	−1794.72	−0.162470
$497$	2219.09	0.200282
$498$	−3887.64	−0.349818
$499$	−11422.5	−1.02474	−0.512368	−	0.858766i	$-0.671231\pi$
$499$	−11422.5	−1.02474	−0.512368	+	0.858766i	$0.671231\pi$
$500$	0	0
$501$	6427.31	0.573156
$502$	−5491.87	−0.488275
$503$	6264.74	0.555330	0.277665	−	0.960678i	$-0.410440\pi$
$503$	6264.74	0.555330	0.277665	+	0.960678i	$0.410440\pi$
$504$	352.438	0.0311485
$505$	0	0
$506$	4203.24	0.369282
$507$	−29943.7	−2.62297
$508$	−9970.77	−0.870829
$509$	−6895.92	−0.600504	−0.300252	−	0.953860i	$-0.597071\pi$
$509$	−6895.92	−0.600504	−0.300252	+	0.953860i	$0.597071\pi$
$510$	0	0
$511$	−4969.31	−0.430194
$512$	4203.64	0.362844
$513$	−9086.55	−0.782029
$514$	−3400.26	−0.291788
$515$	0	0
$516$	10719.7	0.914554
$517$	−4542.02	−0.386379
$518$	1115.40	0.0946100
$519$	−15853.5	−1.34083
$520$	0	0
$521$	−12517.6	−1.05260	−0.526302	−	0.850297i	$-0.676422\pi$
$521$	−12517.6	−1.05260	−0.526302	+	0.850297i	$0.676422\pi$
$522$	−715.754	−0.0600148
$523$	9820.31	0.821055	0.410528	−	0.911848i	$-0.365345\pi$
$523$	9820.31	0.821055	0.410528	+	0.911848i	$0.365345\pi$
$524$	4942.09	0.412016
$525$	0	0
$526$	−3782.88	−0.313577
$527$	−4498.88	−0.371868
$528$	−1745.69	−0.143885
$529$	−3955.05	−0.325064
$530$	0	0
$531$	562.263	0.0459513
$532$	2435.01	0.198442
$533$	−24202.1	−1.96681
$534$	−3721.99	−0.301622
$535$	0	0
$536$	7337.37	0.591280
$537$	21309.1	1.71239
$538$	−1594.87	−0.127806
$539$	1461.34	0.116780
$540$	0	0
$541$	−1864.46	−0.148169	−0.0740845	−	0.997252i	$-0.523603\pi$
$541$	−1864.46	−0.148169	−0.0740845	+	0.997252i	$0.523603\pi$
$542$	6003.19	0.475755
$543$	5045.91	0.398786
$544$	5540.04	0.436631
$545$	0	0
$546$	−4900.90	−0.384138
$547$	−13351.8	−1.04366	−0.521830	−	0.853050i	$-0.674750\pi$
$547$	−13351.8	−1.04366	−0.521830	+	0.853050i	$0.674750\pi$
$548$	−16850.7	−1.31355
$549$	−1216.74	−0.0945887
$550$	0	0
$551$	−12033.6	−0.930398
$552$	9496.81	0.732267
$553$	7440.66	0.572168
$554$	−8324.31	−0.638386
$555$	0	0
$556$	15212.2	1.16033
$557$	4423.73	0.336516	0.168258	−	0.985743i	$-0.446186\pi$
$557$	4423.73	0.336516	0.168258	+	0.985743i	$0.446186\pi$
$558$	563.954	0.0427851
$559$	35124.5	2.65761
$560$	0	0
$561$	−4375.99	−0.329330
$562$	2186.55	0.164117
$563$	−2690.20	−0.201383	−0.100691	−	0.994918i	$-0.532105\pi$
$563$	−2690.20	−0.201383	−0.100691	+	0.994918i	$0.532105\pi$
$564$	−4217.24	−0.314854
$565$	0	0
$566$	−2933.63	−0.217861
$567$	−4612.72	−0.341651
$568$	−6696.08	−0.494650
$569$	−14143.3	−1.04203	−0.521016	−	0.853547i	$-0.674447\pi$
$569$	−14143.3	−1.04203	−0.521016	+	0.853547i	$0.674447\pi$
$570$	0	0
$571$	10955.4	0.802924	0.401462	−	0.915876i	$-0.368502\pi$
$571$	10955.4	0.802924	0.401462	+	0.915876i	$0.368502\pi$
$572$	−15102.0	−1.10393
$573$	5321.81	0.387996
$574$	−2904.02	−0.211170
$575$	0	0
$576$	−469.493	−0.0339622
$577$	−469.298	−0.0338599	−0.0169299	−	0.999857i	$-0.505389\pi$
$577$	−469.298	−0.0338599	−0.0169299	+	0.999857i	$0.505389\pi$
$578$	−6280.79	−0.451984
$579$	6860.46	0.492419
$580$	0	0
$581$	3526.67	0.251826
$582$	−3712.92	−0.264442
$583$	−2433.52	−0.172875
$584$	14994.8	1.06248
$585$	0	0
$586$	9743.85	0.686885
$587$	20050.0	1.40980	0.704899	−	0.709308i	$-0.250992\pi$
$587$	20050.0	1.40980	0.704899	+	0.709308i	$0.250992\pi$
$588$	1356.84	0.0951619
$589$	9481.48	0.663289
$590$	0	0
$591$	−4629.37	−0.322211
$592$	1208.72	0.0839160
$593$	26534.2	1.83749	0.918744	−	0.394855i	$-0.129205\pi$
$593$	26534.2	1.83749	0.918744	+	0.394855i	$0.129205\pi$
$594$	6762.06	0.467089
$595$	0	0
$596$	−16541.8	−1.13687
$597$	−626.401	−0.0429429
$598$	12787.6	0.874455
$599$	6886.62	0.469749	0.234875	−	0.972026i	$-0.424532\pi$
$599$	6886.62	0.469749	0.234875	+	0.972026i	$0.424532\pi$
$600$	0	0
$601$	19861.0	1.34800	0.673999	−	0.738733i	$-0.264575\pi$
$601$	19861.0	1.34800	0.673999	+	0.738733i	$0.264575\pi$
$602$	4214.60	0.285339
$603$	828.017	0.0559195
$604$	−2424.89	−0.163357
$605$	0	0
$606$	−8223.06	−0.551220
$607$	−16703.0	−1.11689	−0.558445	−	0.829541i	$-0.688602\pi$
$607$	−16703.0	−1.11689	−0.558445	+	0.829541i	$0.688602\pi$
$608$	−11675.7	−0.778805
$609$	−6705.40	−0.446168
$610$	0	0
$611$	−13818.3	−0.914939
$612$	393.435	0.0259864
$613$	20753.9	1.36745	0.683723	−	0.729742i	$-0.260360\pi$
$613$	20753.9	1.36745	0.683723	+	0.729742i	$0.260360\pi$
$614$	−2590.32	−0.170255
$615$	0	0
$616$	−4409.56	−0.288419
$617$	6769.53	0.441703	0.220852	−	0.975307i	$-0.429116\pi$
$617$	6769.53	0.441703	0.220852	+	0.975307i	$0.429116\pi$
$618$	731.981	0.0476450
$619$	−10307.2	−0.669272	−0.334636	−	0.942347i	$-0.608613\pi$
$619$	−10307.2	−0.669272	−0.334636	+	0.942347i	$0.608613\pi$
$620$	0	0
$621$	13211.1	0.853696
$622$	3878.32	0.250010
$623$	3376.40	0.217131
$624$	−5310.94	−0.340718
$625$	0	0
$626$	−4447.36	−0.283949
$627$	9222.47	0.587416
$628$	−2770.51	−0.176044
$629$	3029.96	0.192070
$630$	0	0
$631$	20158.1	1.27176	0.635879	−	0.771789i	$-0.280638\pi$
$631$	20158.1	1.27176	0.635879	+	0.771789i	$0.280638\pi$
$632$	−22452.1	−1.41313
$633$	4281.69	0.268849
$634$	12682.6	0.794466
$635$	0	0
$636$	−2259.51	−0.140873
$637$	4445.85	0.276532
$638$	8955.21	0.555706
$639$	−755.648	−0.0467809
$640$	0	0
$641$	−9370.22	−0.577381	−0.288691	−	0.957422i	$-0.593220\pi$
$641$	−9370.22	−0.577381	−0.288691	+	0.957422i	$0.593220\pi$
$642$	9946.08	0.611434
$643$	13605.3	0.834434	0.417217	−	0.908807i	$-0.363006\pi$
$643$	13605.3	0.834434	0.417217	+	0.908807i	$0.363006\pi$
$644$	−3540.32	−0.216627
$645$	0	0
$646$	−2866.81	−0.174602
$647$	17228.8	1.04689	0.523443	−	0.852061i	$-0.324647\pi$
$647$	17228.8	1.04689	0.523443	+	0.852061i	$0.324647\pi$
$648$	13918.8	0.843800
$649$	−7034.80	−0.425486
$650$	0	0
$651$	5283.29	0.318078
$652$	16557.7	0.994554
$653$	−24886.7	−1.49141	−0.745706	−	0.666275i	$-0.767888\pi$
$653$	−24886.7	−1.49141	−0.745706	+	0.666275i	$0.767888\pi$
$654$	−2646.80	−0.158254
$655$	0	0
$656$	−3146.99	−0.187301
$657$	1692.15	0.100483
$658$	−1658.06	−0.0982340
$659$	−22993.3	−1.35917	−0.679583	−	0.733598i	$-0.737839\pi$
$659$	−22993.3	−1.35917	−0.679583	+	0.733598i	$0.737839\pi$
$660$	0	0
$661$	−10136.8	−0.596483	−0.298241	−	0.954490i	$-0.596400\pi$
$661$	−10136.8	−0.596483	−0.298241	+	0.954490i	$0.596400\pi$
$662$	255.259	0.0149863
$663$	−13313.1	−0.779849
$664$	−10641.7	−0.621954
$665$	0	0
$666$	−379.818	−0.0220986
$667$	17496.0	1.01566
$668$	7230.01	0.418769
$669$	22491.4	1.29980
$670$	0	0
$671$	15223.3	0.875843
$672$	−6505.97	−0.373472
$673$	21980.8	1.25899	0.629494	−	0.777006i	$-0.283262\pi$
$673$	21980.8	1.25899	0.629494	+	0.777006i	$0.283262\pi$
$674$	7800.25	0.445778
$675$	0	0
$676$	−33683.3	−1.91644
$677$	30956.4	1.75739	0.878695	−	0.477384i	$-0.158415\pi$
$677$	30956.4	1.75739	0.878695	+	0.477384i	$0.158415\pi$
$678$	−11147.1	−0.631419
$679$	3368.17	0.190366
$680$	0	0
$681$	−18885.7	−1.06270
$682$	−7055.96	−0.396168
$683$	−22895.4	−1.28268	−0.641338	−	0.767259i	$-0.721620\pi$
$683$	−22895.4	−1.28268	−0.641338	+	0.767259i	$0.721620\pi$
$684$	−829.172	−0.0463511
$685$	0	0
$686$	533.460	0.0296904
$687$	15250.5	0.846931
$688$	4567.22	0.253087
$689$	−7403.53	−0.409364
$690$	0	0
$691$	−21418.0	−1.17913	−0.589565	−	0.807721i	$-0.700701\pi$
$691$	−21418.0	−1.17913	−0.589565	+	0.807721i	$0.700701\pi$
$692$	−17833.5	−0.979662
$693$	−497.615	−0.0272768
$694$	−16610.2	−0.908524
$695$	0	0
$696$	20233.4	1.10193
$697$	−7888.69	−0.428702
$698$	−12018.9	−0.651753
$699$	−7579.47	−0.410132
$700$	0	0
$701$	−3444.75	−0.185601	−0.0928007	−	0.995685i	$-0.529582\pi$
$701$	−3444.75	−0.185601	−0.0928007	+	0.995685i	$0.529582\pi$
$702$	20572.3	1.10606
$703$	−6385.69	−0.342590
$704$	5874.10	0.314472
$705$	0	0
$706$	−4999.61	−0.266519
$707$	7459.55	0.396811
$708$	−6531.77	−0.346722
$709$	−5221.70	−0.276594	−0.138297	−	0.990391i	$-0.544163\pi$
$709$	−5221.70	−0.276594	−0.138297	+	0.990391i	$0.544163\pi$
$710$	0	0
$711$	−2533.70	−0.133644
$712$	−10188.2	−0.536264
$713$	−13785.3	−0.724075
$714$	−1597.45	−0.0837298
$715$	0	0
$716$	23970.3	1.25114
$717$	11689.8	0.608874
$718$	161.885	0.00841433
$719$	−13990.0	−0.725645	−0.362822	−	0.931858i	$-0.618187\pi$
$719$	−13990.0	−0.725645	−0.362822	+	0.931858i	$0.618187\pi$
$720$	0	0
$721$	−664.016	−0.0342985
$722$	−4625.78	−0.238440
$723$	31243.7	1.60715
$724$	5676.09	0.291368
$725$	0	0
$726$	3407.43	0.174189
$727$	874.820	0.0446290	0.0223145	−	0.999751i	$-0.492896\pi$
$727$	874.820	0.0446290	0.0223145	+	0.999751i	$0.492896\pi$
$728$	−13415.3	−0.682971
$729$	21100.3	1.07201
$730$	0	0
$731$	11448.8	0.579276
$732$	14134.8	0.713711
$733$	5261.49	0.265126	0.132563	−	0.991175i	$-0.457679\pi$
$733$	5261.49	0.265126	0.132563	+	0.991175i	$0.457679\pi$
$734$	3554.31	0.178736
$735$	0	0
$736$	16975.6	0.850176
$737$	−10359.8	−0.517786
$738$	988.881	0.0493241
$739$	4697.49	0.233829	0.116915	−	0.993142i	$-0.462700\pi$
$739$	4697.49	0.233829	0.116915	+	0.993142i	$0.462700\pi$
$740$	0	0
$741$	28057.7	1.39099
$742$	−888.354	−0.0439521
$743$	−5710.88	−0.281981	−0.140990	−	0.990011i	$-0.545029\pi$
$743$	−5710.88	−0.281981	−0.140990	+	0.990011i	$0.545029\pi$
$744$	−15942.2	−0.785579
$745$	0	0
$746$	9036.32	0.443490
$747$	−1200.91	−0.0588204
$748$	−4922.49	−0.240621
$749$	−9022.58	−0.440158
$750$	0	0
$751$	−22319.1	−1.08447	−0.542234	−	0.840228i	$-0.682421\pi$
$751$	−22319.1	−1.08447	−0.542234	+	0.840228i	$0.682421\pi$
$752$	−1796.79	−0.0871304
$753$	17519.6	0.847875
$754$	27244.6	1.31590
$755$	0	0
$756$	−5695.56	−0.274002
$757$	−18686.6	−0.897192	−0.448596	−	0.893735i	$-0.648076\pi$
$757$	−18686.6	−0.897192	−0.448596	+	0.893735i	$0.648076\pi$
$758$	−2267.31	−0.108644
$759$	−13408.8	−0.641248
$760$	0	0
$761$	−10187.1	−0.485259	−0.242630	−	0.970119i	$-0.578010\pi$
$761$	−10187.1	−0.485259	−0.242630	+	0.970119i	$0.578010\pi$
$762$	−13785.6	−0.655382
$763$	2401.04	0.113923
$764$	5986.44	0.283484
$765$	0	0
$766$	1206.29	0.0568997
$767$	−21402.1	−1.00754
$768$	17018.2	0.799596
$769$	3239.30	0.151901	0.0759506	−	0.997112i	$-0.475801\pi$
$769$	3239.30	0.151901	0.0759506	+	0.997112i	$0.475801\pi$
$770$	0	0
$771$	10847.2	0.506681
$772$	7717.24	0.359779
$773$	−7855.19	−0.365500	−0.182750	−	0.983159i	$-0.558500\pi$
$773$	−7855.19	−0.365500	−0.182750	+	0.983159i	$0.558500\pi$
$774$	−1435.16	−0.0666483
$775$	0	0
$776$	−10163.4	−0.470161
$777$	−3558.25	−0.164288
$778$	14247.7	0.656560
$779$	16625.6	0.764663
$780$	0	0
$781$	9454.35	0.433167
$782$	4168.12	0.190603
$783$	28147.0	1.28466
$784$	578.092	0.0263344
$785$	0	0
$786$	6832.96	0.310081
$787$	15718.5	0.711948	0.355974	−	0.934496i	$-0.384149\pi$
$787$	15718.5	0.711948	0.355974	+	0.934496i	$0.384149\pi$
$788$	−5207.52	−0.235419
$789$	12067.8	0.544517
$790$	0	0
$791$	10112.1	0.454544
$792$	1501.55	0.0673676
$793$	46314.2	2.07398
$794$	3587.94	0.160367
$795$	0	0
$796$	−704.631	−0.0313756
$797$	−9616.93	−0.427414	−0.213707	−	0.976898i	$-0.568554\pi$
$797$	−9616.93	−0.427414	−0.213707	+	0.976898i	$0.568554\pi$
$798$	3366.66	0.149346
$799$	−4504.07	−0.199428
$800$	0	0
$801$	−1149.73	−0.0507164
$802$	8638.43	0.380341
$803$	−21171.5	−0.930418
$804$	−9619.01	−0.421936
$805$	0	0
$806$	−21466.5	−0.938119
$807$	5087.78	0.221931
$808$	−22509.1	−0.980032
$809$	−3533.48	−0.153561	−0.0767803	−	0.997048i	$-0.524464\pi$
$809$	−3533.48	−0.153561	−0.0767803	+	0.997048i	$0.524464\pi$
$810$	0	0
$811$	−19156.3	−0.829432	−0.414716	−	0.909951i	$-0.636119\pi$
$811$	−19156.3	−0.829432	−0.414716	+	0.909951i	$0.636119\pi$
$812$	−7542.82	−0.325987
$813$	−19150.8	−0.826135
$814$	4752.12	0.204621
$815$	0	0
$816$	−1731.10	−0.0742657
$817$	−24128.6	−1.03324
$818$	10669.5	0.456053
$819$	−1513.90	−0.0645911
$820$	0	0
$821$	10117.5	0.430090	0.215045	−	0.976604i	$-0.431010\pi$
$821$	10117.5	0.430090	0.215045	+	0.976604i	$0.431010\pi$
$822$	−23297.9	−0.988573
$823$	−45130.4	−1.91148	−0.955739	−	0.294217i	$-0.904941\pi$
$823$	−45130.4	−1.91148	−0.955739	+	0.294217i	$0.904941\pi$
$824$	2003.66	0.0847096
$825$	0	0
$826$	−2568.05	−0.108177
$827$	−33833.7	−1.42263	−0.711313	−	0.702875i	$-0.751899\pi$
$827$	−33833.7	−1.42263	−0.711313	+	0.702875i	$0.751899\pi$
$828$	1205.55	0.0505988
$829$	9630.51	0.403476	0.201738	−	0.979440i	$-0.435341\pi$
$829$	9630.51	0.403476	0.201738	+	0.979440i	$0.435341\pi$
$830$	0	0
$831$	26555.4	1.10854
$832$	17870.9	0.744665
$833$	1449.13	0.0602752
$834$	21032.5	0.873255
$835$	0	0
$836$	10374.2	0.429188
$837$	−22177.5	−0.915849
$838$	9738.55	0.401447
$839$	−21003.2	−0.864258	−0.432129	−	0.901812i	$-0.642237\pi$
$839$	−21003.2	−0.864258	−0.432129	+	0.901812i	$0.642237\pi$
$840$	0	0
$841$	12887.0	0.528393
$842$	−19579.7	−0.801378
$843$	−6975.31	−0.284985
$844$	4816.42	0.196431
$845$	0	0
$846$	564.605	0.0229450
$847$	−3091.05	−0.125395
$848$	−962.679	−0.0389841
$849$	9358.57	0.378310
$850$	0	0
$851$	9284.30	0.373986
$852$	8778.31	0.352981
$853$	−35952.4	−1.44313	−0.721563	−	0.692349i	$-0.756576\pi$
$853$	−35952.4	−1.44313	−0.721563	+	0.692349i	$0.756576\pi$
$854$	5557.27	0.222677
$855$	0	0
$856$	27225.5	1.08709
$857$	−11294.5	−0.450191	−0.225095	−	0.974337i	$-0.572269\pi$
$857$	−11294.5	−0.450191	−0.225095	+	0.974337i	$0.572269\pi$
$858$	−20880.1	−0.830808
$859$	−16077.2	−0.638586	−0.319293	−	0.947656i	$-0.603445\pi$
$859$	−16077.2	−0.638586	−0.319293	+	0.947656i	$0.603445\pi$
$860$	0	0
$861$	9264.13	0.366691
$862$	20781.7	0.821147
$863$	−9154.92	−0.361109	−0.180555	−	0.983565i	$-0.557789\pi$
$863$	−9154.92	−0.361109	−0.180555	+	0.983565i	$0.557789\pi$
$864$	27309.9	1.07535
$865$	0	0
$866$	−21081.4	−0.827223
$867$	20036.4	0.784856
$868$	5943.11	0.232399
$869$	31700.6	1.23748
$870$	0	0
$871$	−31517.8	−1.22611
$872$	−7245.11	−0.281365
$873$	−1146.93	−0.0444648
$874$	−8784.40	−0.339973
$875$	0	0
$876$	−19657.6	−0.758184
$877$	17118.2	0.659110	0.329555	−	0.944136i	$-0.393101\pi$
$877$	17118.2	0.659110	0.329555	+	0.944136i	$0.393101\pi$
$878$	−14058.3	−0.540370
$879$	−31083.9	−1.19276
$880$	0	0
$881$	−38904.8	−1.48778	−0.743891	−	0.668301i	$-0.767022\pi$
$881$	−38904.8	−1.48778	−0.743891	+	0.668301i	$0.767022\pi$
$882$	−181.654	−0.00693494
$883$	−22859.4	−0.871211	−0.435605	−	0.900138i	$-0.643466\pi$
$883$	−22859.4	−0.871211	−0.435605	+	0.900138i	$0.643466\pi$
$884$	−14975.8	−0.569786
$885$	0	0
$886$	447.523	0.0169693
$887$	−8653.76	−0.327582	−0.163791	−	0.986495i	$-0.552372\pi$
$887$	−8653.76	−0.327582	−0.163791	+	0.986495i	$0.552372\pi$
$888$	10737.0	0.405753
$889$	12505.6	0.471794
$890$	0	0
$891$	−19652.3	−0.738918
$892$	25300.3	0.949682
$893$	9492.41	0.355713
$894$	−22870.7	−0.855606
$895$	0	0
$896$	−8346.03	−0.311184
$897$	−40793.7	−1.51847
$898$	1046.42	0.0388860
$899$	−29370.3	−1.08961
$900$	0	0
$901$	−2413.18	−0.0892285
$902$	−12372.5	−0.456716
$903$	−13445.0	−0.495484
$904$	−30513.1	−1.12262
$905$	0	0
$906$	−3352.67	−0.122942
$907$	−34173.4	−1.25106	−0.625529	−	0.780201i	$-0.715117\pi$
$907$	−34173.4	−1.25106	−0.625529	+	0.780201i	$0.715117\pi$
$908$	−21244.3	−0.776451
$909$	−2540.13	−0.0926852
$910$	0	0
$911$	−18578.2	−0.675657	−0.337829	−	0.941208i	$-0.609692\pi$
$911$	−18578.2	−0.675657	−0.337829	+	0.941208i	$0.609692\pi$
$912$	3648.33	0.132465
$913$	15025.2	0.544647
$914$	−9755.56	−0.353047
$915$	0	0
$916$	17155.1	0.618799
$917$	−6198.51	−0.223220
$918$	6705.56	0.241086
$919$	21712.3	0.779349	0.389674	−	0.920953i	$-0.372588\pi$
$919$	21712.3	0.779349	0.389674	+	0.920953i	$0.372588\pi$
$920$	0	0
$921$	8263.37	0.295643
$922$	6169.96	0.220387
$923$	28763.1	1.02573
$924$	5780.76	0.205815
$925$	0	0
$926$	23171.0	0.822298
$927$	226.111	0.00801130
$928$	36167.4	1.27937
$929$	−21430.5	−0.756848	−0.378424	−	0.925632i	$-0.623534\pi$
$929$	−21430.5	−0.756848	−0.378424	+	0.925632i	$0.623534\pi$
$930$	0	0
$931$	−3054.06	−0.107511
$932$	−8526.06	−0.299657
$933$	−12372.2	−0.434136
$934$	−11464.3	−0.401630
$935$	0	0
$936$	4568.18	0.159525
$937$	−20091.8	−0.700501	−0.350250	−	0.936656i	$-0.613904\pi$
$937$	−20091.8	−0.700501	−0.350250	+	0.936656i	$0.613904\pi$
$938$	−3781.84	−0.131643
$939$	14187.5	0.493069
$940$	0	0
$941$	−1000.61	−0.0346640	−0.0173320	−	0.999850i	$-0.505517\pi$
$941$	−1000.61	−0.0346640	−0.0173320	+	0.999850i	$0.505517\pi$
$942$	−3830.53	−0.132490
$943$	−24172.3	−0.834738
$944$	−2782.91	−0.0959492
$945$	0	0
$946$	17956.1	0.617129
$947$	19867.8	0.681751	0.340875	−	0.940108i	$-0.389277\pi$
$947$	19867.8	0.681751	0.340875	+	0.940108i	$0.389277\pi$
$948$	29433.8	1.00840
$949$	−64410.4	−2.20322
$950$	0	0
$951$	−40458.9	−1.37957
$952$	−4372.72	−0.148866
$953$	1557.33	0.0529347	0.0264673	−	0.999650i	$-0.491574\pi$
$953$	1557.33	0.0529347	0.0264673	+	0.999650i	$0.491574\pi$
$954$	302.503	0.0102661
$955$	0	0
$956$	13149.7	0.444865
$957$	−28568.0	−0.964967
$958$	18561.1	0.625973
$959$	21134.7	0.711651
$960$	0	0
$961$	−6649.63	−0.223209
$962$	14457.5	0.484540
$963$	3072.38	0.102810
$964$	35145.7	1.17424
$965$	0	0
$966$	−4894.86	−0.163033
$967$	45866.9	1.52532	0.762658	−	0.646802i	$-0.223894\pi$
$967$	45866.9	1.52532	0.762658	+	0.646802i	$0.223894\pi$
$968$	9327.18	0.309697
$969$	9145.42	0.303192
$970$	0	0
$971$	−26313.5	−0.869661	−0.434830	−	0.900512i	$-0.643192\pi$
$971$	−26313.5	−0.869661	−0.434830	+	0.900512i	$0.643192\pi$
$972$	3721.59	0.122809
$973$	−19079.6	−0.628637
$974$	6940.46	0.228323
$975$	0	0
$976$	6022.23	0.197507
$977$	46807.1	1.53274	0.766372	−	0.642397i	$-0.222060\pi$
$977$	46807.1	1.53274	0.766372	+	0.642397i	$0.222060\pi$
$978$	22892.8	0.748496
$979$	14385.0	0.469608
$980$	0	0
$981$	−817.605	−0.0266097
$982$	11785.4	0.382981
$983$	43413.8	1.40863	0.704316	−	0.709887i	$-0.251254\pi$
$983$	43413.8	1.40863	0.704316	+	0.709887i	$0.251254\pi$
$984$	−27954.4	−0.905643
$985$	0	0
$986$	8880.39	0.286825
$987$	5289.38	0.170580
$988$	31561.8	1.01631
$989$	35081.2	1.12792
$990$	0	0
$991$	52226.1	1.67408	0.837042	−	0.547139i	$-0.184283\pi$
$991$	52226.1	1.67408	0.837042	+	0.547139i	$0.184283\pi$
$992$	−28496.9	−0.912073
$993$	−814.303	−0.0260233
$994$	3451.30	0.110129
$995$	0	0
$996$	13950.8	0.443824
$997$	−23852.0	−0.757673	−0.378836	−	0.925464i	$-0.623676\pi$
$997$	−23852.0	−0.757673	−0.378836	+	0.925464i	$0.623676\pi$
$998$	−17765.2	−0.563474
$999$	14936.3	0.473037

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.a.j.1.3		5
3.2	odd	2		1575.4.a.bn.1.3		5
5.2	odd	4		35.4.b.a.29.6	yes	10
5.3	odd	4		35.4.b.a.29.5	✓	10
5.4	even	2		175.4.a.i.1.3		5
7.6	odd	2		1225.4.a.bh.1.3		5
15.2	even	4		315.4.d.c.64.5		10
15.8	even	4		315.4.d.c.64.6		10
15.14	odd	2		1575.4.a.bq.1.3		5
20.3	even	4		560.4.g.f.449.8		10
20.7	even	4		560.4.g.f.449.3		10
35.2	odd	12		245.4.j.e.214.6		20
35.3	even	12		245.4.j.f.79.6		20
35.12	even	12		245.4.j.f.214.6		20
35.13	even	4		245.4.b.d.99.5		10
35.17	even	12		245.4.j.f.79.5		20
35.18	odd	12		245.4.j.e.79.6		20
35.23	odd	12		245.4.j.e.214.5		20
35.27	even	4		245.4.b.d.99.6		10
35.32	odd	12		245.4.j.e.79.5		20
35.33	even	12		245.4.j.f.214.5		20
35.34	odd	2		1225.4.a.be.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.5	✓	10	5.3	odd	4
35.4.b.a.29.6	yes	10	5.2	odd	4
175.4.a.i.1.3		5	5.4	even	2
175.4.a.j.1.3		5	1.1	even	1	trivial
245.4.b.d.99.5		10	35.13	even	4
245.4.b.d.99.6		10	35.27	even	4
245.4.j.e.79.5		20	35.32	odd	12
245.4.j.e.79.6		20	35.18	odd	12
245.4.j.e.214.5		20	35.23	odd	12
245.4.j.e.214.6		20	35.2	odd	12
245.4.j.f.79.5		20	35.17	even	12
245.4.j.f.79.6		20	35.3	even	12
245.4.j.f.214.5		20	35.33	even	12
245.4.j.f.214.6		20	35.12	even	12
315.4.d.c.64.5		10	15.2	even	4
315.4.d.c.64.6		10	15.8	even	4
560.4.g.f.449.3		10	20.7	even	4
560.4.g.f.449.8		10	20.3	even	4
1225.4.a.be.1.3		5	35.34	odd	2
1225.4.a.bh.1.3		5	7.6	odd	2
1575.4.a.bn.1.3		5	3.2	odd	2
1575.4.a.bq.1.3		5	15.14	odd	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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)

\(f(q)\)	\(=\)	\(q+1.55528 q^{2} -4.96149 q^{3} -5.58112 q^{4} -7.71648 q^{6} +7.00000 q^{7} -21.1224 q^{8} -2.38365 q^{9} +29.8232 q^{11} +27.6906 q^{12} +90.7316 q^{13} +10.8869 q^{14} +11.7978 q^{16} +29.5740 q^{17} -3.70723 q^{18} -62.3278 q^{19} -34.7304 q^{21} +46.3833 q^{22} +90.6198 q^{23} +104.798 q^{24} +141.113 q^{26} +145.787 q^{27} -39.0678 q^{28} +193.070 q^{29} -152.123 q^{31} +187.328 q^{32} -147.967 q^{33} +45.9957 q^{34} +13.3034 q^{36} +102.453 q^{37} -96.9368 q^{38} -450.164 q^{39} -266.744 q^{41} -54.0154 q^{42} +387.125 q^{43} -166.447 q^{44} +140.939 q^{46} -152.298 q^{47} -58.5346 q^{48} +49.0000 q^{49} -146.731 q^{51} -506.384 q^{52} -81.5982 q^{53} +226.738 q^{54} -147.857 q^{56} +309.238 q^{57} +300.277 q^{58} -235.884 q^{59} +510.453 q^{61} -236.593 q^{62} -16.6855 q^{63} +196.964 q^{64} -230.130 q^{66} -347.374 q^{67} -165.056 q^{68} -449.609 q^{69} +317.014 q^{71} +50.3483 q^{72} -709.901 q^{73} +159.343 q^{74} +347.858 q^{76} +208.762 q^{77} -700.129 q^{78} +1062.95 q^{79} -658.960 q^{81} -414.861 q^{82} +503.810 q^{83} +193.834 q^{84} +602.086 q^{86} -957.914 q^{87} -629.937 q^{88} +482.342 q^{89} +635.121 q^{91} -505.760 q^{92} +754.756 q^{93} -236.866 q^{94} -929.425 q^{96} +481.167 q^{97} +76.2085 q^{98} -71.0879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 10 q^{3} + 18 q^{4} + 6 q^{6} + 35 q^{7} + 42 q^{8} + 23 q^{9} + 42 q^{11} + 136 q^{12} + 34 q^{13} + 28 q^{14} + 74 q^{16} + 238 q^{17} - 2 q^{18} - 36 q^{19} + 70 q^{21} + 358 q^{22} + 152 q^{23}+ \cdots + 2652 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 + 10 * q^3 + 18 * q^4 + 6 * q^6 + 35 * q^7 + 42 * q^8 + 23 * q^9 + 42 * q^11 + 136 * q^12 + 34 * q^13 + 28 * q^14 + 74 * q^16 + 238 * q^17 - 2 * q^18 - 36 * q^19 + 70 * q^21 + 358 * q^22 + 152 * q^23 - 36 * q^24 - 310 * q^26 + 334 * q^27 + 126 * q^28 - 44 * q^29 + 60 * q^31 + 710 * q^32 + 426 * q^33 - 482 * q^34 - 210 * q^36 + 312 * q^37 + 280 * q^38 - 106 * q^39 - 426 * q^41 + 42 * q^42 + 304 * q^43 + 712 * q^44 + 88 * q^46 + 370 * q^47 - 696 * q^48 + 245 * q^49 + 638 * q^51 - 1156 * q^52 + 976 * q^53 + 498 * q^54 + 294 * q^56 - 588 * q^57 - 2722 * q^58 - 432 * q^59 - 442 * q^61 - 956 * q^62 + 161 * q^63 + 1362 * q^64 + 574 * q^66 - 804 * q^67 - 420 * q^68 - 2404 * q^69 + 440 * q^71 - 3150 * q^72 + 564 * q^73 - 1512 * q^74 - 1336 * q^76 + 294 * q^77 - 2742 * q^78 + 1790 * q^79 - 151 * q^81 - 3480 * q^82 + 1656 * q^83 + 952 * q^84 + 1216 * q^86 - 1674 * q^87 - 1092 * q^88 + 746 * q^89 + 238 * q^91 + 572 * q^92 + 676 * q^93 - 826 * q^94 + 2040 * q^96 + 518 * q^97 + 196 * q^98 + 2652 * q^99

Introduction

L-functions

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