LMFDB - Embedded newform 175.4.a.i.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,4,Mod(1,175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$10.3253342510$
Analytic rank:	$1$
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			1.4
Root			$2.67516$ 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)

$f(q)$	$=$	\(q+1.67516 q^{2} +2.49396 q^{3} -5.19383 q^{4} +4.17779 q^{6} -7.00000 q^{7} -22.1018 q^{8} -20.7802 q^{9} -57.5880 q^{11} -12.9532 q^{12} +45.5159 q^{13} -11.7261 q^{14} +4.52655 q^{16} -92.0051 q^{17} -34.8101 q^{18} +125.177 q^{19} -17.4577 q^{21} -96.4692 q^{22} -158.496 q^{23} -55.1211 q^{24} +76.2466 q^{26} -119.162 q^{27} +36.3568 q^{28} -40.1708 q^{29} +49.5590 q^{31} +184.397 q^{32} -143.622 q^{33} -154.123 q^{34} +107.929 q^{36} -231.307 q^{37} +209.692 q^{38} +113.515 q^{39} +169.556 q^{41} -29.2445 q^{42} +147.428 q^{43} +299.102 q^{44} -265.507 q^{46} +67.0327 q^{47} +11.2890 q^{48} +49.0000 q^{49} -229.457 q^{51} -236.402 q^{52} -268.647 q^{53} -199.615 q^{54} +154.713 q^{56} +312.187 q^{57} -67.2926 q^{58} -240.843 q^{59} +90.4579 q^{61} +83.0194 q^{62} +145.461 q^{63} +272.683 q^{64} -240.591 q^{66} +406.498 q^{67} +477.859 q^{68} -395.283 q^{69} +330.782 q^{71} +459.279 q^{72} -546.255 q^{73} -387.477 q^{74} -650.149 q^{76} +403.116 q^{77} +190.156 q^{78} -25.3087 q^{79} +263.879 q^{81} +284.034 q^{82} -376.255 q^{83} +90.6725 q^{84} +246.965 q^{86} -100.184 q^{87} +1272.80 q^{88} +1026.44 q^{89} -318.612 q^{91} +823.203 q^{92} +123.598 q^{93} +112.291 q^{94} +459.879 q^{96} +942.660 q^{97} +82.0829 q^{98} +1196.69 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.67516	0.592259	0.296130	−	0.955148i	$-0.404304\pi$
$2$	1.67516	0.592259	0.296130	+	0.955148i	$0.404304\pi$
$3$	2.49396	0.479963	0.239982	−	0.970777i	$-0.422859\pi$
$3$	2.49396	0.479963	0.239982	+	0.970777i	$0.422859\pi$
$4$	−5.19383	−0.649229
$5$	0	0
$5$	0	0
$6$	4.17779	0.284263
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−22.1018	−0.976771
$9$	−20.7802	−0.769635
$10$	0	0
$11$	−57.5880	−1.57849	−0.789247	−	0.614076i	$-0.789529\pi$
$11$	−57.5880	−1.57849	−0.789247	+	0.614076i	$0.789529\pi$
$12$	−12.9532	−0.311606
$13$	45.5159	0.971066	0.485533	−	0.874218i	$-0.338626\pi$
$13$	45.5159	0.971066	0.485533	+	0.874218i	$0.338626\pi$
$14$	−11.7261	−0.223853
$15$	0	0
$16$	4.52655	0.0707273
$17$	−92.0051	−1.31262	−0.656309	−	0.754492i	$-0.727883\pi$
$17$	−92.0051	−1.31262	−0.656309	+	0.754492i	$0.727883\pi$
$18$	−34.8101	−0.455824
$19$	125.177	1.51145	0.755726	−	0.654888i	$-0.227284\pi$
$19$	125.177	1.51145	0.755726	+	0.654888i	$0.227284\pi$
$20$	0	0
$21$	−17.4577	−0.181409
$22$	−96.4692	−0.934878
$23$	−158.496	−1.43690	−0.718451	−	0.695578i	$-0.755149\pi$
$23$	−158.496	−1.43690	−0.718451	+	0.695578i	$0.755149\pi$
$24$	−55.1211	−0.468814
$25$	0	0
$26$	76.2466	0.575123
$27$	−119.162	−0.849360
$28$	36.3568	0.245386
$29$	−40.1708	−0.257225	−0.128613	−	0.991695i	$-0.541052\pi$
$29$	−40.1708	−0.257225	−0.128613	+	0.991695i	$0.541052\pi$
$30$	0	0
$31$	49.5590	0.287131	0.143566	−	0.989641i	$-0.454143\pi$
$31$	49.5590	0.287131	0.143566	+	0.989641i	$0.454143\pi$
$32$	184.397	1.01866
$33$	−143.622	−0.757619
$34$	−154.123	−0.777410
$35$	0	0
$36$	107.929	0.499670
$37$	−231.307	−1.02775	−0.513874	−	0.857866i	$-0.671790\pi$
$37$	−231.307	−1.02775	−0.513874	+	0.857866i	$0.671790\pi$
$38$	209.692	0.895171
$39$	113.515	0.466076
$40$	0	0
$41$	169.556	0.645859	0.322929	−	0.946423i	$-0.395332\pi$
$41$	169.556	0.645859	0.322929	+	0.946423i	$0.395332\pi$
$42$	−29.2445	−0.107441
$43$	147.428	0.522849	0.261425	−	0.965224i	$-0.415808\pi$
$43$	147.428	0.522849	0.261425	+	0.965224i	$0.415808\pi$
$44$	299.102	1.02480
$45$	0	0
$46$	−265.507	−0.851018
$47$	67.0327	0.208037	0.104018	−	0.994575i	$-0.466830\pi$
$47$	67.0327	0.208037	0.104018	+	0.994575i	$0.466830\pi$
$48$	11.2890	0.0339465
$49$	49.0000	0.142857
$50$	0	0
$51$	−229.457	−0.630008
$52$	−236.402	−0.630444
$53$	−268.647	−0.696254	−0.348127	−	0.937447i	$-0.613182\pi$
$53$	−268.647	−0.696254	−0.348127	+	0.937447i	$0.613182\pi$
$54$	−199.615	−0.503041
$55$	0	0
$56$	154.713	0.369185
$57$	312.187	0.725441
$58$	−67.2926	−0.152344
$59$	−240.843	−0.531442	−0.265721	−	0.964050i	$-0.585610\pi$
$59$	−240.843	−0.531442	−0.265721	+	0.964050i	$0.585610\pi$
$60$	0	0
$61$	90.4579	0.189868	0.0949340	−	0.995484i	$-0.469736\pi$
$61$	90.4579	0.189868	0.0949340	+	0.995484i	$0.469736\pi$
$62$	83.0194	0.170056
$63$	145.461	0.290895
$64$	272.683	0.532583
$65$	0	0
$66$	−240.591	−0.448707
$67$	406.498	0.741218	0.370609	−	0.928789i	$-0.379149\pi$
$67$	406.498	0.741218	0.370609	+	0.928789i	$0.379149\pi$
$68$	477.859	0.852190
$69$	−395.283	−0.689660
$70$	0	0
$71$	330.782	0.552910	0.276455	−	0.961027i	$-0.410840\pi$
$71$	330.782	0.552910	0.276455	+	0.961027i	$0.410840\pi$
$72$	459.279	0.751758
$73$	−546.255	−0.875812	−0.437906	−	0.899021i	$-0.644280\pi$
$73$	−546.255	−0.875812	−0.437906	+	0.899021i	$0.644280\pi$
$74$	−387.477	−0.608693
$75$	0	0
$76$	−650.149	−0.981279
$77$	403.116	0.596615
$78$	190.156	0.276038
$79$	−25.3087	−0.0360436	−0.0180218	−	0.999838i	$-0.505737\pi$
$79$	−25.3087	−0.0360436	−0.0180218	+	0.999838i	$0.505737\pi$
$80$	0	0
$81$	263.879	0.361974
$82$	284.034	0.382516
$83$	−376.255	−0.497582	−0.248791	−	0.968557i	$-0.580033\pi$
$83$	−376.255	−0.497582	−0.248791	+	0.968557i	$0.580033\pi$
$84$	90.6725	0.117776
$85$	0	0
$86$	246.965	0.309662
$87$	−100.184	−0.123459
$88$	1272.80	1.54183
$89$	1026.44	1.22250	0.611248	−	0.791439i	$-0.290668\pi$
$89$	1026.44	1.22250	0.611248	+	0.791439i	$0.290668\pi$
$90$	0	0
$91$	−318.612	−0.367028
$92$	823.203	0.932878
$93$	123.598	0.137812
$94$	112.291	0.123212
$95$	0	0
$96$	459.879	0.488919
$97$	942.660	0.986728	0.493364	−	0.869823i	$-0.335767\pi$
$97$	942.660	0.986728	0.493364	+	0.869823i	$0.335767\pi$
$98$	82.0829	0.0846085
$99$	1196.69	1.21487
$100$	0	0
$101$	−604.617	−0.595659	−0.297830	−	0.954619i	$-0.596263\pi$
$101$	−604.617	−0.595659	−0.297830	+	0.954619i	$0.596263\pi$
$102$	−384.378	−0.373128
$103$	300.967	0.287914	0.143957	−	0.989584i	$-0.454017\pi$
$103$	300.967	0.287914	0.143957	+	0.989584i	$0.454017\pi$
$104$	−1005.98	−0.948509
$105$	0	0
$106$	−450.027	−0.412363
$107$	−1511.66	−1.36577	−0.682886	−	0.730525i	$-0.739276\pi$
$107$	−1511.66	−1.36577	−0.682886	+	0.730525i	$0.739276\pi$
$108$	618.907	0.551429
$109$	−1767.09	−1.55281	−0.776406	−	0.630233i	$-0.782959\pi$
$109$	−1767.09	−1.55281	−0.776406	+	0.630233i	$0.782959\pi$
$110$	0	0
$111$	−576.871	−0.493281
$112$	−31.6858	−0.0267324
$113$	−1045.27	−0.870182	−0.435091	−	0.900387i	$-0.643284\pi$
$113$	−1045.27	−0.870182	−0.435091	+	0.900387i	$0.643284\pi$
$114$	522.963	0.429649
$115$	0	0
$116$	208.640	0.166998
$117$	−945.828	−0.747366
$118$	−403.451	−0.314751
$119$	644.036	0.496123
$120$	0	0
$121$	1985.38	1.49164
$122$	151.532	0.112451
$123$	422.866	0.309988
$124$	−257.401	−0.186414
$125$	0	0
$126$	243.671	0.172285
$127$	−260.727	−0.182171	−0.0910857	−	0.995843i	$-0.529034\pi$
$127$	−260.727	−0.182171	−0.0910857	+	0.995843i	$0.529034\pi$
$128$	−1018.39	−0.703233
$129$	367.679	0.250948
$130$	0	0
$131$	−723.522	−0.482553	−0.241276	−	0.970456i	$-0.577566\pi$
$131$	−723.522	−0.482553	−0.241276	+	0.970456i	$0.577566\pi$
$132$	745.950	0.491868
$133$	−876.240	−0.571275
$134$	680.950	0.438993
$135$	0	0
$136$	2033.48	1.28213
$137$	−773.693	−0.482490	−0.241245	−	0.970464i	$-0.577556\pi$
$137$	−773.693	−0.482490	−0.241245	+	0.970464i	$0.577556\pi$
$138$	−662.164	−0.408457
$139$	−2952.97	−1.80192	−0.900961	−	0.433899i	$-0.857137\pi$
$139$	−2952.97	−1.80192	−0.900961	+	0.433899i	$0.857137\pi$
$140$	0	0
$141$	167.177	0.0998499
$142$	554.114	0.327466
$143$	−2621.17	−1.53282
$144$	−94.0624	−0.0544343
$145$	0	0
$146$	−915.066	−0.518708
$147$	122.204	0.0685662
$148$	1201.37	0.667244
$149$	2514.00	1.38225	0.691124	−	0.722736i	$-0.257116\pi$
$149$	2514.00	1.38225	0.691124	+	0.722736i	$0.257116\pi$
$150$	0	0
$151$	101.052	0.0544605	0.0272302	−	0.999629i	$-0.491331\pi$
$151$	101.052	0.0544605	0.0272302	+	0.999629i	$0.491331\pi$
$152$	−2766.64	−1.47634
$153$	1911.88	1.01024
$154$	675.285	0.353351
$155$	0	0
$156$	−589.578	−0.302590
$157$	−2338.35	−1.18867	−0.594333	−	0.804219i	$-0.702584\pi$
$157$	−2338.35	−1.18867	−0.594333	+	0.804219i	$0.702584\pi$
$158$	−42.3961	−0.0213472
$159$	−669.995	−0.334176
$160$	0	0
$161$	1109.47	0.543098
$162$	442.040	0.214383
$163$	1325.20	0.636798	0.318399	−	0.947957i	$-0.396855\pi$
$163$	1325.20	0.636798	0.318399	+	0.947957i	$0.396855\pi$
$164$	−880.646	−0.419310
$165$	0	0
$166$	−630.288	−0.294698
$167$	−2086.20	−0.966675	−0.483338	−	0.875434i	$-0.660576\pi$
$167$	−2086.20	−0.966675	−0.483338	+	0.875434i	$0.660576\pi$
$168$	385.847	0.177195
$169$	−125.299	−0.0570317
$170$	0	0
$171$	−2601.20	−1.16327
$172$	−765.715	−0.339449
$173$	−1918.19	−0.842990	−0.421495	−	0.906831i	$-0.638495\pi$
$173$	−1918.19	−0.842990	−0.421495	+	0.906831i	$0.638495\pi$
$174$	−167.825	−0.0731195
$175$	0	0
$176$	−260.675	−0.111643
$177$	−600.652	−0.255072
$178$	1719.45	0.724035
$179$	629.046	0.262665	0.131333	−	0.991338i	$-0.458074\pi$
$179$	629.046	0.262665	0.131333	+	0.991338i	$0.458074\pi$
$180$	0	0
$181$	−2800.85	−1.15020	−0.575099	−	0.818084i	$-0.695036\pi$
$181$	−2800.85	−1.15020	−0.575099	+	0.818084i	$0.695036\pi$
$182$	−533.726	−0.217376
$183$	225.599	0.0911296
$184$	3503.05	1.40352
$185$	0	0
$186$	207.047	0.0816207
$187$	5298.39	2.07196
$188$	−348.157	−0.135063
$189$	834.133	0.321028
$190$	0	0
$191$	740.255	0.280434	0.140217	−	0.990121i	$-0.455220\pi$
$191$	740.255	0.280434	0.140217	+	0.990121i	$0.455220\pi$
$192$	680.060	0.255620
$193$	−4082.57	−1.52264	−0.761321	−	0.648375i	$-0.775449\pi$
$193$	−4082.57	−1.52264	−0.761321	+	0.648375i	$0.775449\pi$
$194$	1579.11	0.584399
$195$	0	0
$196$	−254.498	−0.0927470
$197$	−3414.89	−1.23503	−0.617515	−	0.786559i	$-0.711860\pi$
$197$	−3414.89	−1.23503	−0.617515	+	0.786559i	$0.711860\pi$
$198$	2004.65	0.719515
$199$	−3392.44	−1.20846	−0.604231	−	0.796809i	$-0.706520\pi$
$199$	−3392.44	−1.20846	−0.604231	+	0.796809i	$0.706520\pi$
$200$	0	0
$201$	1013.79	0.355757
$202$	−1012.83	−0.352785
$203$	281.196	0.0972220
$204$	1191.76	0.409020
$205$	0	0
$206$	504.169	0.170520
$207$	3293.58	1.10589
$208$	206.030	0.0686809
$209$	−7208.70	−2.38582
$210$	0	0
$211$	3398.04	1.10867	0.554337	−	0.832292i	$-0.312972\pi$
$211$	3398.04	1.10867	0.554337	+	0.832292i	$0.312972\pi$
$212$	1395.31	0.452029
$213$	824.958	0.265376
$214$	−2532.28	−0.808892
$215$	0	0
$216$	2633.69	0.829630
$217$	−346.913	−0.108525
$218$	−2960.16	−0.919667
$219$	−1362.34	−0.420358
$220$	0	0
$221$	−4187.70	−1.27464
$222$	−966.353	−0.292150
$223$	−182.611	−0.0548365	−0.0274183	−	0.999624i	$-0.508729\pi$
$223$	−182.611	−0.0548365	−0.0274183	+	0.999624i	$0.508729\pi$
$224$	−1290.78	−0.385017
$225$	0	0
$226$	−1750.99	−0.515373
$227$	3152.33	0.921707	0.460854	−	0.887476i	$-0.347543\pi$
$227$	3152.33	0.921707	0.460854	+	0.887476i	$0.347543\pi$
$228$	−1621.45	−0.470977
$229$	6012.35	1.73497	0.867483	−	0.497466i	$-0.165736\pi$
$229$	6012.35	1.73497	0.867483	+	0.497466i	$0.165736\pi$
$230$	0	0
$231$	1005.36	0.286353
$232$	887.848	0.251250
$233$	940.660	0.264484	0.132242	−	0.991217i	$-0.457782\pi$
$233$	940.660	0.264484	0.132242	+	0.991217i	$0.457782\pi$
$234$	−1584.42	−0.442635
$235$	0	0
$236$	1250.90	0.345027
$237$	−63.1188	−0.0172996
$238$	1078.86	0.293833
$239$	5158.82	1.39622	0.698109	−	0.715991i	$-0.254025\pi$
$239$	5158.82	1.39622	0.698109	+	0.715991i	$0.254025\pi$
$240$	0	0
$241$	−463.836	−0.123976	−0.0619882	−	0.998077i	$-0.519744\pi$
$241$	−463.836	−0.123976	−0.0619882	+	0.998077i	$0.519744\pi$
$242$	3325.83	0.883440
$243$	3875.47	1.02309
$244$	−469.823	−0.123268
$245$	0	0
$246$	708.370	0.183594
$247$	5697.55	1.46772
$248$	−1095.34	−0.280461
$249$	−938.365	−0.238821
$250$	0	0
$251$	2290.25	0.575934	0.287967	−	0.957640i	$-0.407021\pi$
$251$	2290.25	0.575934	0.287967	+	0.957640i	$0.407021\pi$
$252$	−755.501	−0.188857
$253$	9127.48	2.26814
$254$	−436.760	−0.107893
$255$	0	0
$256$	−3887.43	−0.949079
$257$	802.202	0.194708	0.0973541	−	0.995250i	$-0.468962\pi$
$257$	802.202	0.194708	0.0973541	+	0.995250i	$0.468962\pi$
$258$	615.922	0.148627
$259$	1619.15	0.388452
$260$	0	0
$261$	834.756	0.197970
$262$	−1212.02	−0.285796
$263$	286.978	0.0672845	0.0336423	−	0.999434i	$-0.489289\pi$
$263$	286.978	0.0672845	0.0336423	+	0.999434i	$0.489289\pi$
$264$	3174.31	0.740020
$265$	0	0
$266$	−1467.84	−0.338343
$267$	2559.90	0.586753
$268$	−2111.28	−0.481221
$269$	3561.22	0.807180	0.403590	−	0.914940i	$-0.367762\pi$
$269$	3561.22	0.807180	0.403590	+	0.914940i	$0.367762\pi$
$270$	0	0
$271$	1928.81	0.432349	0.216175	−	0.976355i	$-0.430642\pi$
$271$	1928.81	0.432349	0.216175	+	0.976355i	$0.430642\pi$
$272$	−416.466	−0.0928380
$273$	−794.605	−0.176160
$274$	−1296.06	−0.285759
$275$	0	0
$276$	2053.04	0.447747
$277$	−6588.69	−1.42916	−0.714578	−	0.699556i	$-0.753381\pi$
$277$	−6588.69	−1.42916	−0.714578	+	0.699556i	$0.753381\pi$
$278$	−4946.70	−1.06721
$279$	−1029.84	−0.220986
$280$	0	0
$281$	815.552	0.173138	0.0865689	−	0.996246i	$-0.472410\pi$
$281$	815.552	0.173138	0.0865689	+	0.996246i	$0.472410\pi$
$282$	280.049	0.0591371
$283$	6513.49	1.36815	0.684076	−	0.729411i	$-0.260206\pi$
$283$	6513.49	1.36815	0.684076	+	0.729411i	$0.260206\pi$
$284$	−1718.03	−0.358965
$285$	0	0
$286$	−4390.89	−0.907828
$287$	−1186.89	−0.244112
$288$	−3831.80	−0.783997
$289$	3551.94	0.722967
$290$	0	0
$291$	2350.96	0.473593
$292$	2837.16	0.568603
$293$	−435.520	−0.0868373	−0.0434186	−	0.999057i	$-0.513825\pi$
$293$	−435.520	−0.0868373	−0.0434186	+	0.999057i	$0.513825\pi$
$294$	204.712	0.0406089
$295$	0	0
$296$	5112.31	1.00387
$297$	6862.29	1.34071
$298$	4211.36	0.818650
$299$	−7214.10	−1.39533
$300$	0	0
$301$	−1031.99	−0.197618
$302$	169.279	0.0322547
$303$	−1507.89	−0.285895
$304$	566.620	0.106901
$305$	0	0
$306$	3202.71	0.598323
$307$	4915.99	0.913910	0.456955	−	0.889490i	$-0.348940\pi$
$307$	4915.99	0.913910	0.456955	+	0.889490i	$0.348940\pi$
$308$	−2093.72	−0.387340
$309$	750.601	0.138188
$310$	0	0
$311$	−1831.11	−0.333868	−0.166934	−	0.985968i	$-0.553387\pi$
$311$	−1831.11	−0.333868	−0.166934	+	0.985968i	$0.553387\pi$
$312$	−2508.89	−0.455249
$313$	2442.96	0.441163	0.220582	−	0.975369i	$-0.429204\pi$
$313$	2442.96	0.441163	0.220582	+	0.975369i	$0.429204\pi$
$314$	−3917.11	−0.703998
$315$	0	0
$316$	131.449	0.0234006
$317$	−1666.19	−0.295214	−0.147607	−	0.989046i	$-0.547157\pi$
$317$	−1666.19	−0.295214	−0.147607	+	0.989046i	$0.547157\pi$
$318$	−1122.35	−0.197919
$319$	2313.36	0.406029
$320$	0	0
$321$	−3770.02	−0.655521
$322$	1858.55	0.321655
$323$	−11516.9	−1.98396
$324$	−1370.54	−0.235004
$325$	0	0
$326$	2219.93	0.377149
$327$	−4407.05	−0.745292
$328$	−3747.50	−0.630856
$329$	−469.229	−0.0786305
$330$	0	0
$331$	−5466.38	−0.907732	−0.453866	−	0.891070i	$-0.649956\pi$
$331$	−5466.38	−0.907732	−0.453866	+	0.891070i	$0.649956\pi$
$332$	1954.20	0.323045
$333$	4806.60	0.790991
$334$	−3494.72	−0.572522
$335$	0	0
$336$	−79.0233	−0.0128306
$337$	−10650.5	−1.72157	−0.860784	−	0.508970i	$-0.830027\pi$
$337$	−10650.5	−1.72157	−0.860784	+	0.508970i	$0.830027\pi$
$338$	−209.896	−0.0337776
$339$	−2606.86	−0.417655
$340$	0	0
$341$	−2854.01	−0.453235
$342$	−4357.43	−0.688956
$343$	−343.000	−0.0539949
$344$	−3258.42	−0.510704
$345$	0	0
$346$	−3213.28	−0.499269
$347$	−4019.13	−0.621782	−0.310891	−	0.950446i	$-0.600627\pi$
$347$	−4019.13	−0.621782	−0.310891	+	0.950446i	$0.600627\pi$
$348$	520.341	0.0801529
$349$	10544.9	1.61735	0.808674	−	0.588256i	$-0.200185\pi$
$349$	10544.9	1.61735	0.808674	+	0.588256i	$0.200185\pi$
$350$	0	0
$351$	−5423.76	−0.824784
$352$	−10619.1	−1.60795
$353$	−2959.98	−0.446300	−0.223150	−	0.974784i	$-0.571634\pi$
$353$	−2959.98	−0.446300	−0.223150	+	0.974784i	$0.571634\pi$
$354$	−1006.19	−0.151069
$355$	0	0
$356$	−5331.14	−0.793680
$357$	1606.20	0.238121
$358$	1053.75	0.155566
$359$	2170.17	0.319045	0.159523	−	0.987194i	$-0.449005\pi$
$359$	2170.17	0.319045	0.159523	+	0.987194i	$0.449005\pi$
$360$	0	0
$361$	8810.30	1.28449
$362$	−4691.88	−0.681215
$363$	4951.46	0.715934
$364$	1654.82	0.238285
$365$	0	0
$366$	377.914	0.0539724
$367$	1252.20	0.178105	0.0890523	−	0.996027i	$-0.471616\pi$
$367$	1252.20	0.178105	0.0890523	+	0.996027i	$0.471616\pi$
$368$	−717.441	−0.101628
$369$	−3523.40	−0.497076
$370$	0	0
$371$	1880.53	0.263159
$372$	−641.949	−0.0894718
$373$	−4646.02	−0.644938	−0.322469	−	0.946580i	$-0.604513\pi$
$373$	−4646.02	−0.644938	−0.322469	+	0.946580i	$0.604513\pi$
$374$	8875.66	1.22714
$375$	0	0
$376$	−1481.54	−0.203204
$377$	−1828.41	−0.249783
$378$	1397.31	0.190132
$379$	1434.84	0.194466	0.0972331	−	0.995262i	$-0.469001\pi$
$379$	1434.84	0.194466	0.0972331	+	0.995262i	$0.469001\pi$
$380$	0	0
$381$	−650.242	−0.0874355
$382$	1240.05	0.166090
$383$	13216.7	1.76330	0.881649	−	0.471905i	$-0.156434\pi$
$383$	13216.7	1.76330	0.881649	+	0.471905i	$0.156434\pi$
$384$	−2539.82	−0.337526
$385$	0	0
$386$	−6838.97	−0.901799
$387$	−3063.57	−0.402403
$388$	−4896.02	−0.640612
$389$	−7755.01	−1.01078	−0.505391	−	0.862890i	$-0.668652\pi$
$389$	−7755.01	−1.01078	−0.505391	+	0.862890i	$0.668652\pi$
$390$	0	0
$391$	14582.5	1.88610
$392$	−1082.99	−0.139539
$393$	−1804.44	−0.231607
$394$	−5720.49	−0.731457
$395$	0	0
$396$	−6215.40	−0.788726
$397$	3560.83	0.450158	0.225079	−	0.974341i	$-0.427736\pi$
$397$	3560.83	0.450158	0.225079	+	0.974341i	$0.427736\pi$
$398$	−5682.89	−0.715723
$399$	−2185.31	−0.274191
$400$	0	0
$401$	−5430.61	−0.676288	−0.338144	−	0.941094i	$-0.609799\pi$
$401$	−5430.61	−0.676288	−0.338144	+	0.941094i	$0.609799\pi$
$402$	1698.26	0.210701
$403$	2255.73	0.278823
$404$	3140.28	0.386719
$405$	0	0
$406$	471.048	0.0575806
$407$	13320.5	1.62229
$408$	5071.42	0.615374
$409$	−9698.79	−1.17255	−0.586277	−	0.810111i	$-0.699407\pi$
$409$	−9698.79	−1.17255	−0.586277	+	0.810111i	$0.699407\pi$
$410$	0	0
$411$	−1929.56	−0.231577
$412$	−1563.17	−0.186922
$413$	1685.90	0.200866
$414$	5517.27	0.654974
$415$	0	0
$416$	8393.01	0.989186
$417$	−7364.58	−0.864856
$418$	−12075.7	−1.41302
$419$	13830.9	1.61261	0.806307	−	0.591498i	$-0.201463\pi$
$419$	13830.9	1.61261	0.806307	+	0.591498i	$0.201463\pi$
$420$	0	0
$421$	16703.0	1.93362	0.966810	−	0.255498i	$-0.0822393\pi$
$421$	16703.0	1.93362	0.966810	+	0.255498i	$0.0822393\pi$
$422$	5692.26	0.656623
$423$	−1392.95	−0.160112
$424$	5937.58	0.680081
$425$	0	0
$426$	1381.94	0.157172
$427$	−633.205	−0.0717634
$428$	7851.31	0.886699
$429$	−6537.10	−0.735698
$430$	0	0
$431$	8174.07	0.913530	0.456765	−	0.889588i	$-0.349008\pi$
$431$	8174.07	0.913530	0.456765	+	0.889588i	$0.349008\pi$
$432$	−539.392	−0.0600729
$433$	−14222.8	−1.57853	−0.789267	−	0.614051i	$-0.789539\pi$
$433$	−14222.8	−1.57853	−0.789267	+	0.614051i	$0.789539\pi$
$434$	−581.136	−0.0642752
$435$	0	0
$436$	9177.96	1.00813
$437$	−19840.1	−2.17181
$438$	−2282.14	−0.248961
$439$	5537.38	0.602016	0.301008	−	0.953622i	$-0.402677\pi$
$439$	5537.38	0.602016	0.301008	+	0.953622i	$0.402677\pi$
$440$	0	0
$441$	−1018.23	−0.109948
$442$	−7015.07	−0.754916
$443$	−3974.09	−0.426218	−0.213109	−	0.977028i	$-0.568359\pi$
$443$	−3974.09	−0.426218	−0.213109	+	0.977028i	$0.568359\pi$
$444$	2996.17	0.320252
$445$	0	0
$446$	−305.903	−0.0324774
$447$	6269.82	0.663428
$448$	−1908.78	−0.201298
$449$	−15243.1	−1.60216	−0.801078	−	0.598559i	$-0.795740\pi$
$449$	−15243.1	−1.60216	−0.801078	+	0.598559i	$0.795740\pi$
$450$	0	0
$451$	−9764.40	−1.01948
$452$	5428.95	0.564947
$453$	252.021	0.0261390
$454$	5280.67	0.545890
$455$	0	0
$456$	−6899.89	−0.708590
$457$	10768.9	1.10229	0.551145	−	0.834410i	$-0.314191\pi$
$457$	10768.9	1.10229	0.551145	+	0.834410i	$0.314191\pi$
$458$	10071.7	1.02755
$459$	10963.5	1.11489
$460$	0	0
$461$	−332.605	−0.0336029	−0.0168015	−	0.999859i	$-0.505348\pi$
$461$	−332.605	−0.0336029	−0.0168015	+	0.999859i	$0.505348\pi$
$462$	1684.13	0.169595
$463$	8205.35	0.823618	0.411809	−	0.911270i	$-0.364897\pi$
$463$	8205.35	0.823618	0.411809	+	0.911270i	$0.364897\pi$
$464$	−181.835	−0.0181929
$465$	0	0
$466$	1575.76	0.156643
$467$	167.628	0.0166100	0.00830501	−	0.999966i	$-0.497356\pi$
$467$	167.628	0.0166100	0.00830501	+	0.999966i	$0.497356\pi$
$468$	4912.47	0.485212
$469$	−2845.49	−0.280154
$470$	0	0
$471$	−5831.75	−0.570515
$472$	5323.06	0.519097
$473$	−8490.07	−0.825315
$474$	−105.734	−0.0102459
$475$	0	0
$476$	−3345.01	−0.322098
$477$	5582.52	0.535862
$478$	8641.86	0.826924
$479$	6628.58	0.632292	0.316146	−	0.948711i	$-0.397611\pi$
$479$	6628.58	0.632292	0.316146	+	0.948711i	$0.397611\pi$
$480$	0	0
$481$	−10528.2	−0.998010
$482$	−777.001	−0.0734262
$483$	2766.98	0.260667
$484$	−10311.7	−0.968419
$485$	0	0
$486$	6492.05	0.605937
$487$	20641.6	1.92065	0.960327	−	0.278875i	$-0.0899614\pi$
$487$	20641.6	1.92065	0.960327	+	0.278875i	$0.0899614\pi$
$488$	−1999.28	−0.185458
$489$	3305.01	0.305640
$490$	0	0
$491$	−16710.8	−1.53594	−0.767972	−	0.640484i	$-0.778734\pi$
$491$	−16710.8	−1.53594	−0.767972	+	0.640484i	$0.778734\pi$
$492$	−2196.30	−0.201253
$493$	3695.92	0.337639
$494$	9544.32	0.869270
$495$	0	0
$496$	224.331	0.0203080
$497$	−2315.47	−0.208980
$498$	−1571.91	−0.141444
$499$	−13728.7	−1.23162	−0.615812	−	0.787893i	$-0.711172\pi$
$499$	−13728.7	−1.23162	−0.615812	+	0.787893i	$0.711172\pi$
$500$	0	0
$501$	−5202.90	−0.463969
$502$	3836.54	0.341102
$503$	19523.7	1.73065	0.865326	−	0.501209i	$-0.167111\pi$
$503$	19523.7	1.73065	0.865326	+	0.501209i	$0.167111\pi$
$504$	−3214.95	−0.284138
$505$	0	0
$506$	15290.0	1.34333
$507$	−312.490	−0.0273731
$508$	1354.17	0.118271
$509$	−8688.17	−0.756574	−0.378287	−	0.925688i	$-0.623487\pi$
$509$	−8688.17	−0.756574	−0.378287	+	0.925688i	$0.623487\pi$
$510$	0	0
$511$	3823.78	0.331026
$512$	1635.04	0.141131
$513$	−14916.3	−1.28377
$514$	1343.82	0.115318
$515$	0	0
$516$	−1909.66	−0.162923
$517$	−3860.28	−0.328385
$518$	2712.34	0.230064
$519$	−4783.89	−0.404604
$520$	0	0
$521$	6771.36	0.569402	0.284701	−	0.958616i	$-0.408106\pi$
$521$	6771.36	0.569402	0.284701	+	0.958616i	$0.408106\pi$
$522$	1398.35	0.117249
$523$	1365.89	0.114200	0.0570998	−	0.998368i	$-0.481815\pi$
$523$	1365.89	0.114200	0.0570998	+	0.998368i	$0.481815\pi$
$524$	3757.85	0.313287
$525$	0	0
$526$	480.735	0.0398499
$527$	−4559.68	−0.376894
$528$	−650.113	−0.0535844
$529$	12954.0	1.06469
$530$	0	0
$531$	5004.75	0.409016
$532$	4551.04	0.370888
$533$	7717.51	0.627171
$534$	4288.24	0.347510
$535$	0	0
$536$	−8984.34	−0.724001
$537$	1568.82	0.126070
$538$	5965.62	0.478060
$539$	−2821.81	−0.225499
$540$	0	0
$541$	−23250.1	−1.84769	−0.923844	−	0.382770i	$-0.874970\pi$
$541$	−23250.1	−1.84769	−0.923844	+	0.382770i	$0.874970\pi$
$542$	3231.06	0.256063
$543$	−6985.22	−0.552052
$544$	−16965.5	−1.33711
$545$	0	0
$546$	−1331.09	−0.104332
$547$	11552.7	0.903033	0.451516	−	0.892263i	$-0.350883\pi$
$547$	11552.7	0.903033	0.451516	+	0.892263i	$0.350883\pi$
$548$	4018.43	0.313246
$549$	−1879.73	−0.146129
$550$	0	0
$551$	−5028.47	−0.388784
$552$	8736.48	0.673640
$553$	177.161	0.0136232
$554$	−11037.1	−0.846430
$555$	0	0
$556$	15337.2	1.16986
$557$	16406.2	1.24803	0.624014	−	0.781413i	$-0.285501\pi$
$557$	16406.2	1.24803	0.624014	+	0.781413i	$0.285501\pi$
$558$	−1725.16	−0.130881
$559$	6710.31	0.507721
$560$	0	0
$561$	13214.0	0.994465
$562$	1366.18	0.102542
$563$	−13631.9	−1.02045	−0.510227	−	0.860040i	$-0.670439\pi$
$563$	−13631.9	−1.02045	−0.510227	+	0.860040i	$0.670439\pi$
$564$	−868.289	−0.0648255
$565$	0	0
$566$	10911.2	0.810300
$567$	−1847.15	−0.136813
$568$	−7310.88	−0.540067
$569$	−3086.83	−0.227428	−0.113714	−	0.993514i	$-0.536275\pi$
$569$	−3086.83	−0.227428	−0.113714	+	0.993514i	$0.536275\pi$
$570$	0	0
$571$	−3258.06	−0.238784	−0.119392	−	0.992847i	$-0.538095\pi$
$571$	−3258.06	−0.238784	−0.119392	+	0.992847i	$0.538095\pi$
$572$	13613.9	0.995152
$573$	1846.17	0.134598
$574$	−1988.24	−0.144577
$575$	0	0
$576$	−5666.39	−0.409895
$577$	−23758.4	−1.71417	−0.857083	−	0.515178i	$-0.827726\pi$
$577$	−23758.4	−1.71417	−0.857083	+	0.515178i	$0.827726\pi$
$578$	5950.07	0.428184
$579$	−10181.8	−0.730812
$580$	0	0
$581$	2633.78	0.188068
$582$	3938.23	0.280490
$583$	15470.8	1.09903
$584$	12073.2	0.855468
$585$	0	0
$586$	−729.566	−0.0514302
$587$	−596.893	−0.0419701	−0.0209850	−	0.999780i	$-0.506680\pi$
$587$	−596.893	−0.0419701	−0.0209850	+	0.999780i	$0.506680\pi$
$588$	−634.708	−0.0445151
$589$	6203.66	0.433985
$590$	0	0
$591$	−8516.60	−0.592768
$592$	−1047.02	−0.0726898
$593$	−19496.3	−1.35012	−0.675058	−	0.737765i	$-0.735881\pi$
$593$	−19496.3	−1.35012	−0.675058	+	0.737765i	$0.735881\pi$
$594$	11495.5	0.794047
$595$	0	0
$596$	−13057.3	−0.897396
$597$	−8460.62	−0.580017
$598$	−12084.8	−0.826395
$599$	3797.02	0.259001	0.129501	−	0.991579i	$-0.458663\pi$
$599$	3797.02	0.259001	0.129501	+	0.991579i	$0.458663\pi$
$600$	0	0
$601$	5789.33	0.392931	0.196466	−	0.980511i	$-0.437054\pi$
$601$	5789.33	0.392931	0.196466	+	0.980511i	$0.437054\pi$
$602$	−1728.76	−0.117041
$603$	−8447.09	−0.570468
$604$	−524.850	−0.0353573
$605$	0	0
$606$	−2525.96	−0.169324
$607$	−18536.4	−1.23949	−0.619745	−	0.784803i	$-0.712764\pi$
$607$	−18536.4	−1.23949	−0.619745	+	0.784803i	$0.712764\pi$
$608$	23082.3	1.53966
$609$	701.291	0.0466630
$610$	0	0
$611$	3051.06	0.202017
$612$	−9929.98	−0.655876
$613$	−2163.47	−0.142548	−0.0712738	−	0.997457i	$-0.522706\pi$
$613$	−2163.47	−0.142548	−0.0712738	+	0.997457i	$0.522706\pi$
$614$	8235.08	0.541272
$615$	0	0
$616$	−8909.59	−0.582756
$617$	−22964.9	−1.49843	−0.749215	−	0.662327i	$-0.769569\pi$
$617$	−22964.9	−1.49843	−0.749215	+	0.662327i	$0.769569\pi$
$618$	1257.38	0.0818433
$619$	−1386.67	−0.0900401	−0.0450200	−	0.998986i	$-0.514335\pi$
$619$	−1386.67	−0.0900401	−0.0450200	+	0.998986i	$0.514335\pi$
$620$	0	0
$621$	18886.7	1.22045
$622$	−3067.41	−0.197736
$623$	−7185.06	−0.462060
$624$	513.831	0.0329643
$625$	0	0
$626$	4092.35	0.261283
$627$	−17978.2	−1.14510
$628$	12145.0	0.771716
$629$	21281.4	1.34904
$630$	0	0
$631$	5969.39	0.376605	0.188303	−	0.982111i	$-0.439701\pi$
$631$	5969.39	0.376605	0.188303	+	0.982111i	$0.439701\pi$
$632$	559.367	0.0352064
$633$	8474.57	0.532123
$634$	−2791.14	−0.174843
$635$	0	0
$636$	3479.84	0.216957
$637$	2230.28	0.138724
$638$	3875.25	0.240474
$639$	−6873.70	−0.425539
$640$	0	0
$641$	30367.1	1.87118	0.935592	−	0.353084i	$-0.114867\pi$
$641$	30367.1	1.87118	0.935592	+	0.353084i	$0.114867\pi$
$642$	−6315.40	−0.388238
$643$	−28592.2	−1.75360	−0.876802	−	0.480851i	$-0.840328\pi$
$643$	−28592.2	−1.75360	−0.876802	+	0.480851i	$0.840328\pi$
$644$	−5762.42	−0.352595
$645$	0	0
$646$	−19292.7	−1.17502
$647$	14507.9	0.881555	0.440778	−	0.897616i	$-0.354703\pi$
$647$	14507.9	0.881555	0.440778	+	0.897616i	$0.354703\pi$
$648$	−5832.21	−0.353566
$649$	13869.7	0.838877
$650$	0	0
$651$	−865.188	−0.0520882
$652$	−6882.89	−0.413428
$653$	−6999.85	−0.419488	−0.209744	−	0.977756i	$-0.567263\pi$
$653$	−6999.85	−0.419488	−0.209744	+	0.977756i	$0.567263\pi$
$654$	−7382.53	−0.441406
$655$	0	0
$656$	767.504	0.0456799
$657$	11351.3	0.674056
$658$	−786.035	−0.0465696
$659$	7308.92	0.432041	0.216020	−	0.976389i	$-0.430692\pi$
$659$	7308.92	0.432041	0.216020	+	0.976389i	$0.430692\pi$
$660$	0	0
$661$	−30097.2	−1.77102	−0.885512	−	0.464617i	$-0.846192\pi$
$661$	−30097.2	−1.77102	−0.885512	+	0.464617i	$0.846192\pi$
$662$	−9157.07	−0.537613
$663$	−10444.0	−0.611779
$664$	8315.91	0.486024
$665$	0	0
$666$	8051.83	0.468472
$667$	6366.92	0.369608
$668$	10835.4	0.627594
$669$	−455.425	−0.0263195
$670$	0	0
$671$	−5209.29	−0.299706
$672$	−3219.16	−0.184794
$673$	5400.26	0.309309	0.154654	−	0.987969i	$-0.450574\pi$
$673$	5400.26	0.309309	0.154654	+	0.987969i	$0.450574\pi$
$674$	−17841.3	−1.01962
$675$	0	0
$676$	650.781	0.0370267
$677$	−6431.09	−0.365091	−0.182546	−	0.983197i	$-0.558434\pi$
$677$	−6431.09	−0.365091	−0.182546	+	0.983197i	$0.558434\pi$
$678$	−4366.91	−0.247360
$679$	−6598.62	−0.372948
$680$	0	0
$681$	7861.79	0.442385
$682$	−4780.92	−0.268433
$683$	20865.8	1.16897	0.584486	−	0.811404i	$-0.301296\pi$
$683$	20865.8	1.16897	0.584486	+	0.811404i	$0.301296\pi$
$684$	13510.2	0.755227
$685$	0	0
$686$	−574.581	−0.0319790
$687$	14994.6	0.832720
$688$	667.339	0.0369797
$689$	−12227.7	−0.676109
$690$	0	0
$691$	18450.3	1.01575	0.507873	−	0.861432i	$-0.330432\pi$
$691$	18450.3	1.01575	0.507873	+	0.861432i	$0.330432\pi$
$692$	9962.76	0.547294
$693$	−8376.81	−0.459176
$694$	−6732.70	−0.368256
$695$	0	0
$696$	2214.26	0.120591
$697$	−15600.0	−0.847766
$698$	17664.4	0.957890
$699$	2345.97	0.126942
$700$	0	0
$701$	12639.3	0.680996	0.340498	−	0.940245i	$-0.389404\pi$
$701$	12639.3	0.680996	0.340498	+	0.940245i	$0.389404\pi$
$702$	−9085.69	−0.488486
$703$	−28954.4	−1.55339
$704$	−15703.3	−0.840680
$705$	0	0
$706$	−4958.45	−0.264325
$707$	4232.32	0.225138
$708$	3119.69	0.165600
$709$	−23126.8	−1.22503	−0.612514	−	0.790460i	$-0.709842\pi$
$709$	−23126.8	−1.22503	−0.612514	+	0.790460i	$0.709842\pi$
$710$	0	0
$711$	525.918	0.0277404
$712$	−22686.1	−1.19410
$713$	−7854.92	−0.412579
$714$	2690.65	0.141029
$715$	0	0
$716$	−3267.16	−0.170530
$717$	12865.9	0.670134
$718$	3635.39	0.188957
$719$	−24093.1	−1.24968	−0.624841	−	0.780752i	$-0.714836\pi$
$719$	−24093.1	−1.24968	−0.624841	+	0.780752i	$0.714836\pi$
$720$	0	0
$721$	−2106.77	−0.108821
$722$	14758.7	0.760750
$723$	−1156.79	−0.0595041
$724$	14547.2	0.746741
$725$	0	0
$726$	8294.49	0.424019
$727$	−35983.4	−1.83570	−0.917849	−	0.396931i	$-0.870075\pi$
$727$	−35983.4	−1.83570	−0.917849	+	0.396931i	$0.870075\pi$
$728$	7041.89	0.358503
$729$	2540.55	0.129073
$730$	0	0
$731$	−13564.1	−0.686302
$732$	−1171.72	−0.0591640
$733$	1451.50	0.0731413	0.0365706	−	0.999331i	$-0.488357\pi$
$733$	1451.50	0.0731413	0.0365706	+	0.999331i	$0.488357\pi$
$734$	2097.64	0.105484
$735$	0	0
$736$	−29226.3	−1.46371
$737$	−23409.4	−1.17001
$738$	−5902.27	−0.294398
$739$	−5891.67	−0.293273	−0.146636	−	0.989190i	$-0.546845\pi$
$739$	−5891.67	−0.293273	−0.146636	+	0.989190i	$0.546845\pi$
$740$	0	0
$741$	14209.5	0.704451
$742$	3150.19	0.155859
$743$	−7438.65	−0.367292	−0.183646	−	0.982992i	$-0.558790\pi$
$743$	−7438.65	−0.367292	−0.183646	+	0.982992i	$0.558790\pi$
$744$	−2731.75	−0.134611
$745$	0	0
$746$	−7782.84	−0.381970
$747$	7818.64	0.382957
$748$	−27518.9	−1.34518
$749$	10581.6	0.516214
$750$	0	0
$751$	20272.4	0.985018	0.492509	−	0.870307i	$-0.336080\pi$
$751$	20272.4	0.985018	0.492509	+	0.870307i	$0.336080\pi$
$752$	303.427	0.0147139
$753$	5711.80	0.276427
$754$	−3062.89	−0.147936
$755$	0	0
$756$	−4332.35	−0.208421
$757$	10193.8	0.489432	0.244716	−	0.969595i	$-0.421305\pi$
$757$	10193.8	0.489432	0.244716	+	0.969595i	$0.421305\pi$
$758$	2403.59	0.115174
$759$	22763.6	1.08862
$760$	0	0
$761$	−41117.6	−1.95862	−0.979311	−	0.202362i	$-0.935138\pi$
$761$	−41117.6	−1.95862	−0.979311	+	0.202362i	$0.935138\pi$
$762$	−1089.26	−0.0517845
$763$	12369.6	0.586908
$764$	−3844.76	−0.182066
$765$	0	0
$766$	22140.2	1.04433
$767$	−10962.2	−0.516065
$768$	−9695.10	−0.455523
$769$	11486.6	0.538642	0.269321	−	0.963050i	$-0.413201\pi$
$769$	11486.6	0.538642	0.269321	+	0.963050i	$0.413201\pi$
$770$	0	0
$771$	2000.66	0.0934527
$772$	21204.2	0.988544
$773$	21799.2	1.01431	0.507156	−	0.861854i	$-0.330697\pi$
$773$	21799.2	1.01431	0.507156	+	0.861854i	$0.330697\pi$
$774$	−5131.98	−0.238327
$775$	0	0
$776$	−20834.5	−0.963807
$777$	4038.10	0.186443
$778$	−12990.9	−0.598645
$779$	21224.5	0.976185
$780$	0	0
$781$	−19049.1	−0.872765
$782$	24428.0	1.11706
$783$	4786.83	0.218477
$784$	221.801	0.0101039
$785$	0	0
$786$	−3022.72	−0.137172
$787$	24080.3	1.09068	0.545342	−	0.838214i	$-0.316400\pi$
$787$	24080.3	1.09068	0.545342	+	0.838214i	$0.316400\pi$
$788$	17736.4	0.801817
$789$	715.712	0.0322941
$790$	0	0
$791$	7316.87	0.328898
$792$	−26449.0	−1.18665
$793$	4117.28	0.184374
$794$	5964.96	0.266610
$795$	0	0
$796$	17619.8	0.784569
$797$	17194.3	0.764184	0.382092	−	0.924124i	$-0.375204\pi$
$797$	17194.3	0.764184	0.382092	+	0.924124i	$0.375204\pi$
$798$	−3660.74	−0.162392
$799$	−6167.35	−0.273073
$800$	0	0
$801$	−21329.5	−0.940876
$802$	−9097.15	−0.400538
$803$	31457.7	1.38246
$804$	−5265.46	−0.230968
$805$	0	0
$806$	3778.71	0.165136
$807$	8881.55	0.387417
$808$	13363.1	0.581823
$809$	33349.1	1.44931	0.724656	−	0.689111i	$-0.241999\pi$
$809$	33349.1	1.44931	0.724656	+	0.689111i	$0.241999\pi$
$810$	0	0
$811$	4577.87	0.198213	0.0991066	−	0.995077i	$-0.468402\pi$
$811$	4577.87	0.198213	0.0991066	+	0.995077i	$0.468402\pi$
$812$	−1460.48	−0.0631194
$813$	4810.37	0.207512
$814$	22314.0	0.960819
$815$	0	0
$816$	−1038.65	−0.0445588
$817$	18454.6	0.790262
$818$	−16247.0	−0.694455
$819$	6620.80	0.282478
$820$	0	0
$821$	−5832.52	−0.247937	−0.123969	−	0.992286i	$-0.539562\pi$
$821$	−5832.52	−0.247937	−0.123969	+	0.992286i	$0.539562\pi$
$822$	−3232.33	−0.137154
$823$	−37974.3	−1.60838	−0.804192	−	0.594369i	$-0.797402\pi$
$823$	−37974.3	−1.60838	−0.804192	+	0.594369i	$0.797402\pi$
$824$	−6651.92	−0.281226
$825$	0	0
$826$	2824.15	0.118965
$827$	−15796.2	−0.664193	−0.332096	−	0.943245i	$-0.607756\pi$
$827$	−15796.2	−0.664193	−0.332096	+	0.943245i	$0.607756\pi$
$828$	−17106.3	−0.717976
$829$	−12714.1	−0.532666	−0.266333	−	0.963881i	$-0.585812\pi$
$829$	−12714.1	−0.532666	−0.266333	+	0.963881i	$0.585812\pi$
$830$	0	0
$831$	−16431.9	−0.685942
$832$	12411.4	0.517173
$833$	−4508.25	−0.187517
$834$	−12336.9	−0.512219
$835$	0	0
$836$	37440.8	1.54894
$837$	−5905.55	−0.243878
$838$	23169.0	0.955085
$839$	−42924.2	−1.76628	−0.883140	−	0.469109i	$-0.844575\pi$
$839$	−42924.2	−1.76628	−0.883140	+	0.469109i	$0.844575\pi$
$840$	0	0
$841$	−22775.3	−0.933835
$842$	27980.2	1.14520
$843$	2033.95	0.0830998
$844$	−17648.8	−0.719784
$845$	0	0
$846$	−2333.42	−0.0948281
$847$	−13897.6	−0.563788
$848$	−1216.04	−0.0492442
$849$	16244.4	0.656662
$850$	0	0
$851$	36661.3	1.47677
$852$	−4284.69	−0.172290
$853$	36172.5	1.45196	0.725980	−	0.687716i	$-0.241387\pi$
$853$	36172.5	1.45196	0.725980	+	0.687716i	$0.241387\pi$
$854$	−1060.72	−0.0425025
$855$	0	0
$856$	33410.4	1.33405
$857$	32039.9	1.27709	0.638544	−	0.769586i	$-0.279537\pi$
$857$	32039.9	1.27709	0.638544	+	0.769586i	$0.279537\pi$
$858$	−10950.7	−0.435724
$859$	−6798.07	−0.270020	−0.135010	−	0.990844i	$-0.543107\pi$
$859$	−6798.07	−0.270020	−0.135010	+	0.990844i	$0.543107\pi$
$860$	0	0
$861$	−2960.06	−0.117165
$862$	13692.9	0.541046
$863$	30179.3	1.19040	0.595200	−	0.803577i	$-0.297073\pi$
$863$	30179.3	1.19040	0.595200	+	0.803577i	$0.297073\pi$
$864$	−21973.1	−0.865209
$865$	0	0
$866$	−23825.5	−0.934901
$867$	8858.39	0.346997
$868$	1801.81	0.0704578
$869$	1457.47	0.0568946
$870$	0	0
$871$	18502.1	0.719772
$872$	39055.9	1.51674
$873$	−19588.6	−0.759421
$874$	−33235.4	−1.28627
$875$	0	0
$876$	7075.76	0.272908
$877$	1700.51	0.0654758	0.0327379	−	0.999464i	$-0.489577\pi$
$877$	1700.51	0.0654758	0.0327379	+	0.999464i	$0.489577\pi$
$878$	9276.02	0.356549
$879$	−1086.17	−0.0416787
$880$	0	0
$881$	1678.46	0.0641869	0.0320935	−	0.999485i	$-0.489783\pi$
$881$	1678.46	0.0641869	0.0320935	+	0.999485i	$0.489783\pi$
$882$	−1705.70	−0.0651177
$883$	−10285.8	−0.392009	−0.196005	−	0.980603i	$-0.562797\pi$
$883$	−10285.8	−0.392009	−0.196005	+	0.980603i	$0.562797\pi$
$884$	21750.2	0.827532
$885$	0	0
$886$	−6657.24	−0.252432
$887$	12167.5	0.460593	0.230296	−	0.973121i	$-0.426030\pi$
$887$	12167.5	0.460593	0.230296	+	0.973121i	$0.426030\pi$
$888$	12749.9	0.481823
$889$	1825.09	0.0688543
$890$	0	0
$891$	−15196.3	−0.571374
$892$	948.452	0.0356015
$893$	8390.96	0.314438
$894$	10503.0	0.392922
$895$	0	0
$896$	7128.73	0.265797
$897$	−17991.7	−0.669705
$898$	−25534.7	−0.948892
$899$	−1990.83	−0.0738574
$900$	0	0
$901$	24716.9	0.913916
$902$	−16356.9	−0.603799
$903$	−2573.75	−0.0948496
$904$	23102.3	0.849968
$905$	0	0
$906$	422.176	0.0154811
$907$	50766.0	1.85850	0.929250	−	0.369452i	$-0.120455\pi$
$907$	50766.0	1.85850	0.929250	+	0.369452i	$0.120455\pi$
$908$	−16372.7	−0.598399
$909$	12564.0	0.458441
$910$	0	0
$911$	18451.1	0.671033	0.335517	−	0.942034i	$-0.391089\pi$
$911$	18451.1	0.671033	0.335517	+	0.942034i	$0.391089\pi$
$912$	1413.13	0.0513085
$913$	21667.8	0.785431
$914$	18039.6	0.652841
$915$	0	0
$916$	−31227.1	−1.12639
$917$	5064.65	0.182388
$918$	18365.6	0.660301
$919$	393.861	0.0141374	0.00706870	−	0.999975i	$-0.497750\pi$
$919$	393.861	0.0141374	0.00706870	+	0.999975i	$0.497750\pi$
$920$	0	0
$921$	12260.3	0.438643
$922$	−557.167	−0.0199016
$923$	15055.9	0.536912
$924$	−5221.65	−0.185909
$925$	0	0
$926$	13745.3	0.487795
$927$	−6254.15	−0.221589
$928$	−7407.39	−0.262025
$929$	27452.3	0.969517	0.484759	−	0.874648i	$-0.338907\pi$
$929$	27452.3	0.969517	0.484759	+	0.874648i	$0.338907\pi$
$930$	0	0
$931$	6133.68	0.215922
$932$	−4885.63	−0.171710
$933$	−4566.72	−0.160244
$934$	280.803	0.00983744
$935$	0	0
$936$	20904.5	0.730006
$937$	−21608.3	−0.753375	−0.376688	−	0.926340i	$-0.622937\pi$
$937$	−21608.3	−0.753375	−0.376688	+	0.926340i	$0.622937\pi$
$938$	−4766.65	−0.165924
$939$	6092.64	0.211742
$940$	0	0
$941$	−16708.5	−0.578832	−0.289416	−	0.957203i	$-0.593461\pi$
$941$	−16708.5	−0.578832	−0.289416	+	0.957203i	$0.593461\pi$
$942$	−9769.12	−0.337893
$943$	−26874.0	−0.928036
$944$	−1090.19	−0.0375874
$945$	0	0
$946$	−14222.2	−0.488800
$947$	−13972.8	−0.479466	−0.239733	−	0.970839i	$-0.577060\pi$
$947$	−13972.8	−0.479466	−0.239733	+	0.970839i	$0.577060\pi$
$948$	327.828	0.0112314
$949$	−24863.3	−0.850471
$950$	0	0
$951$	−4155.42	−0.141692
$952$	−14234.4	−0.484599
$953$	11155.9	0.379197	0.189598	−	0.981862i	$-0.439281\pi$
$953$	11155.9	0.379197	0.189598	+	0.981862i	$0.439281\pi$
$954$	9351.63	0.317369
$955$	0	0
$956$	−26794.0	−0.906466
$957$	5769.42	0.194879
$958$	11104.0	0.374481
$959$	5415.85	0.182364
$960$	0	0
$961$	−27334.9	−0.917556
$962$	−17636.4	−0.591081
$963$	31412.5	1.05115
$964$	2409.09	0.0804891
$965$	0	0
$966$	4635.15	0.154382
$967$	1212.55	0.0403238	0.0201619	−	0.999797i	$-0.493582\pi$
$967$	1212.55	0.0403238	0.0201619	+	0.999797i	$0.493582\pi$
$968$	−43880.4	−1.45699
$969$	−28722.8	−0.952227
$970$	0	0
$971$	−48832.2	−1.61390	−0.806952	−	0.590617i	$-0.798885\pi$
$971$	−48832.2	−1.61390	−0.806952	+	0.590617i	$0.798885\pi$
$972$	−20128.6	−0.664222
$973$	20670.8	0.681063
$974$	34578.0	1.13753
$975$	0	0
$976$	409.462	0.0134289
$977$	−5939.19	−0.194485	−0.0972424	−	0.995261i	$-0.531002\pi$
$977$	−5939.19	−0.194485	−0.0972424	+	0.995261i	$0.531002\pi$
$978$	5536.43	0.181018
$979$	−59110.5	−1.92970
$980$	0	0
$981$	36720.4	1.19510
$982$	−27993.3	−0.909677
$983$	21863.5	0.709398	0.354699	−	0.934980i	$-0.384583\pi$
$983$	21863.5	0.709398	0.354699	+	0.934980i	$0.384583\pi$
$984$	−9346.11	−0.302788
$985$	0	0
$986$	6191.26	0.199970
$987$	−1170.24	−0.0377397
$988$	−29592.1	−0.952886
$989$	−23366.7	−0.751283
$990$	0	0
$991$	−44618.5	−1.43023	−0.715113	−	0.699009i	$-0.753625\pi$
$991$	−44618.5	−1.43023	−0.715113	+	0.699009i	$0.753625\pi$
$992$	9138.55	0.292489
$993$	−13632.9	−0.435678
$994$	−3878.80	−0.123771
$995$	0	0
$996$	4873.71	0.155050
$997$	37923.9	1.20468	0.602338	−	0.798241i	$-0.294236\pi$
$997$	37923.9	1.20468	0.602338	+	0.798241i	$0.294236\pi$
$998$	−22997.8	−0.729441
$999$	27563.0	0.872928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.a.i.1.4		5
3.2	odd	2		1575.4.a.bq.1.2		5
5.2	odd	4		35.4.b.a.29.7	yes	10
5.3	odd	4		35.4.b.a.29.4	✓	10
5.4	even	2		175.4.a.j.1.2		5
7.6	odd	2		1225.4.a.be.1.4		5
15.2	even	4		315.4.d.c.64.4		10
15.8	even	4		315.4.d.c.64.7		10
15.14	odd	2		1575.4.a.bn.1.4		5
20.3	even	4		560.4.g.f.449.4		10
20.7	even	4		560.4.g.f.449.7		10
35.2	odd	12		245.4.j.e.214.7		20
35.3	even	12		245.4.j.f.79.7		20
35.12	even	12		245.4.j.f.214.7		20
35.13	even	4		245.4.b.d.99.4		10
35.17	even	12		245.4.j.f.79.4		20
35.18	odd	12		245.4.j.e.79.7		20
35.23	odd	12		245.4.j.e.214.4		20
35.27	even	4		245.4.b.d.99.7		10
35.32	odd	12		245.4.j.e.79.4		20
35.33	even	12		245.4.j.f.214.4		20
35.34	odd	2		1225.4.a.bh.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.4	✓	10	5.3	odd	4
35.4.b.a.29.7	yes	10	5.2	odd	4
175.4.a.i.1.4		5	1.1	even	1	trivial
175.4.a.j.1.2		5	5.4	even	2
245.4.b.d.99.4		10	35.13	even	4
245.4.b.d.99.7		10	35.27	even	4
245.4.j.e.79.4		20	35.32	odd	12
245.4.j.e.79.7		20	35.18	odd	12
245.4.j.e.214.4		20	35.23	odd	12
245.4.j.e.214.7		20	35.2	odd	12
245.4.j.f.79.4		20	35.17	even	12
245.4.j.f.79.7		20	35.3	even	12
245.4.j.f.214.4		20	35.33	even	12
245.4.j.f.214.7		20	35.12	even	12
315.4.d.c.64.4		10	15.2	even	4
315.4.d.c.64.7		10	15.8	even	4
560.4.g.f.449.4		10	20.3	even	4
560.4.g.f.449.7		10	20.7	even	4
1225.4.a.be.1.4		5	7.6	odd	2
1225.4.a.bh.1.2		5	35.34	odd	2
1575.4.a.bn.1.4		5	15.14	odd	2
1575.4.a.bq.1.2		5	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)

\(f(q)\)	\(=\)	\(q+1.67516 q^{2} +2.49396 q^{3} -5.19383 q^{4} +4.17779 q^{6} -7.00000 q^{7} -22.1018 q^{8} -20.7802 q^{9} -57.5880 q^{11} -12.9532 q^{12} +45.5159 q^{13} -11.7261 q^{14} +4.52655 q^{16} -92.0051 q^{17} -34.8101 q^{18} +125.177 q^{19} -17.4577 q^{21} -96.4692 q^{22} -158.496 q^{23} -55.1211 q^{24} +76.2466 q^{26} -119.162 q^{27} +36.3568 q^{28} -40.1708 q^{29} +49.5590 q^{31} +184.397 q^{32} -143.622 q^{33} -154.123 q^{34} +107.929 q^{36} -231.307 q^{37} +209.692 q^{38} +113.515 q^{39} +169.556 q^{41} -29.2445 q^{42} +147.428 q^{43} +299.102 q^{44} -265.507 q^{46} +67.0327 q^{47} +11.2890 q^{48} +49.0000 q^{49} -229.457 q^{51} -236.402 q^{52} -268.647 q^{53} -199.615 q^{54} +154.713 q^{56} +312.187 q^{57} -67.2926 q^{58} -240.843 q^{59} +90.4579 q^{61} +83.0194 q^{62} +145.461 q^{63} +272.683 q^{64} -240.591 q^{66} +406.498 q^{67} +477.859 q^{68} -395.283 q^{69} +330.782 q^{71} +459.279 q^{72} -546.255 q^{73} -387.477 q^{74} -650.149 q^{76} +403.116 q^{77} +190.156 q^{78} -25.3087 q^{79} +263.879 q^{81} +284.034 q^{82} -376.255 q^{83} +90.6725 q^{84} +246.965 q^{86} -100.184 q^{87} +1272.80 q^{88} +1026.44 q^{89} -318.612 q^{91} +823.203 q^{92} +123.598 q^{93} +112.291 q^{94} +459.879 q^{96} +942.660 q^{97} +82.0829 q^{98} +1196.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} - 10 q^{3} + 18 q^{4} + 6 q^{6} - 35 q^{7} - 42 q^{8} + 23 q^{9} + 42 q^{11} - 136 q^{12} - 34 q^{13} + 28 q^{14} + 74 q^{16} - 238 q^{17} + 2 q^{18} - 36 q^{19} + 70 q^{21} - 358 q^{22} - 152 q^{23}+ \cdots + 2652 q^{99}+O(q^{100}) \) 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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Database

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