LMFDB - Embedded newform 175.4.a.j.1.2

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			1.2
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	$-0.595696\pi$
$2$	−1.67516	−0.592259	−0.296130	+	0.955148i	$0.595696\pi$
$3$	−2.49396	−0.479963	−0.239982	−	0.970777i	$-0.577141\pi$
$3$	−2.49396	−0.479963	−0.239982	+	0.970777i	$0.577141\pi$
$4$	−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	$-0.661374\pi$
$13$	−45.5159	−0.971066	−0.485533	+	0.874218i	$0.661374\pi$
$14$	−11.7261	−0.223853
$15$	0	0
$16$	4.52655	0.0707273
$17$	92.0051	1.31262	0.656309	−	0.754492i	$-0.272117\pi$
$17$	92.0051	1.31262	0.656309	+	0.754492i	$0.272117\pi$
$18$	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	$-0.244851\pi$
$23$	158.496	1.43690	0.718451	+	0.695578i	$0.244851\pi$
$24$	−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	$-0.328210\pi$
$37$	231.307	1.02775	0.513874	+	0.857866i	$0.328210\pi$
$38$	−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	$-0.584192\pi$
$43$	−147.428	−0.522849	−0.261425	+	0.965224i	$0.584192\pi$
$44$	299.102	1.02480
$45$	0	0
$46$	−265.507	−0.851018
$47$	−67.0327	−0.208037	−0.104018	−	0.994575i	$-0.533170\pi$
$47$	−67.0327	−0.208037	−0.104018	+	0.994575i	$0.533170\pi$
$48$	−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	$-0.386818\pi$
$53$	268.647	0.696254	0.348127	+	0.937447i	$0.386818\pi$
$54$	−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	$-0.620851\pi$
$67$	−406.498	−0.741218	−0.370609	+	0.928789i	$0.620851\pi$
$68$	−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	$-0.355720\pi$
$73$	546.255	0.875812	0.437906	+	0.899021i	$0.355720\pi$
$74$	−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	$-0.419967\pi$
$83$	376.255	0.497582	0.248791	+	0.968557i	$0.419967\pi$
$84$	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	$-0.664233\pi$
$97$	−942.660	−0.986728	−0.493364	+	0.869823i	$0.664233\pi$
$98$	−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	$-0.545983\pi$
$103$	−300.967	−0.287914	−0.143957	+	0.989584i	$0.545983\pi$
$104$	−1005.98	−0.948509
$105$	0	0
$106$	−450.027	−0.412363
$107$	1511.66	1.36577	0.682886	−	0.730525i	$-0.260724\pi$
$107$	1511.66	1.36577	0.682886	+	0.730525i	$0.260724\pi$
$108$	−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	$-0.356716\pi$
$113$	1045.27	0.870182	0.435091	+	0.900387i	$0.356716\pi$
$114$	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	$-0.470966\pi$
$127$	260.727	0.182171	0.0910857	+	0.995843i	$0.470966\pi$
$128$	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	$-0.422444\pi$
$137$	773.693	0.482490	0.241245	+	0.970464i	$0.422444\pi$
$138$	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	$-0.297416\pi$
$157$	2338.35	1.18867	0.594333	+	0.804219i	$0.297416\pi$
$158$	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	$-0.603145\pi$
$163$	−1325.20	−0.636798	−0.318399	+	0.947957i	$0.603145\pi$
$164$	−880.646	−0.419310
$165$	0	0
$166$	−630.288	−0.294698
$167$	2086.20	0.966675	0.483338	−	0.875434i	$-0.339424\pi$
$167$	2086.20	0.966675	0.483338	+	0.875434i	$0.339424\pi$
$168$	−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	$-0.361505\pi$
$173$	1918.19	0.842990	0.421495	+	0.906831i	$0.361505\pi$
$174$	−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	$-0.224551\pi$
$193$	4082.57	1.52264	0.761321	+	0.648375i	$0.224551\pi$
$194$	1579.11	0.584399
$195$	0	0
$196$	−254.498	−0.0927470
$197$	3414.89	1.23503	0.617515	−	0.786559i	$-0.288140\pi$
$197$	3414.89	1.23503	0.617515	+	0.786559i	$0.288140\pi$
$198$	−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	$-0.491271\pi$
$223$	182.611	0.0548365	0.0274183	+	0.999624i	$0.491271\pi$
$224$	−1290.78	−0.385017
$225$	0	0
$226$	−1750.99	−0.515373
$227$	−3152.33	−0.921707	−0.460854	−	0.887476i	$-0.652457\pi$
$227$	−3152.33	−0.921707	−0.460854	+	0.887476i	$0.652457\pi$
$228$	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	$-0.542218\pi$
$233$	−940.660	−0.264484	−0.132242	+	0.991217i	$0.542218\pi$
$234$	−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	$-0.531038\pi$
$257$	−802.202	−0.194708	−0.0973541	+	0.995250i	$0.531038\pi$
$258$	−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	$-0.510711\pi$
$263$	−286.978	−0.0672845	−0.0336423	+	0.999434i	$0.510711\pi$
$264$	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	$-0.246619\pi$
$277$	6588.69	1.42916	0.714578	+	0.699556i	$0.246619\pi$
$278$	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	$-0.739794\pi$
$283$	−6513.49	−1.36815	−0.684076	+	0.729411i	$0.739794\pi$
$284$	−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	$-0.486175\pi$
$293$	435.520	0.0868373	0.0434186	+	0.999057i	$0.486175\pi$
$294$	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	$-0.651060\pi$
$307$	−4915.99	−0.913910	−0.456955	+	0.889490i	$0.651060\pi$
$308$	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	$-0.570796\pi$
$313$	−2442.96	−0.441163	−0.220582	+	0.975369i	$0.570796\pi$
$314$	−3917.11	−0.703998
$315$	0	0
$316$	131.449	0.0234006
$317$	1666.19	0.295214	0.147607	−	0.989046i	$-0.452843\pi$
$317$	1666.19	0.295214	0.147607	+	0.989046i	$0.452843\pi$
$318$	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	$-0.169973\pi$
$337$	10650.5	1.72157	0.860784	+	0.508970i	$0.169973\pi$
$338$	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	$-0.399373\pi$
$347$	4019.13	0.621782	0.310891	+	0.950446i	$0.399373\pi$
$348$	−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	$-0.428366\pi$
$353$	2959.98	0.446300	0.223150	+	0.974784i	$0.428366\pi$
$354$	−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	$-0.528384\pi$
$367$	−1252.20	−0.178105	−0.0890523	+	0.996027i	$0.528384\pi$
$368$	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	$-0.395487\pi$
$373$	4646.02	0.644938	0.322469	+	0.946580i	$0.395487\pi$
$374$	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	$-0.843566\pi$
$383$	−13216.7	−1.76330	−0.881649	+	0.471905i	$0.843566\pi$
$384$	−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	$-0.572264\pi$
$397$	−3560.83	−0.450158	−0.225079	+	0.974341i	$0.572264\pi$
$398$	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	$-0.210461\pi$
$433$	14222.8	1.57853	0.789267	+	0.614051i	$0.210461\pi$
$434$	−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	$-0.431641\pi$
$443$	3974.09	0.426218	0.213109	+	0.977028i	$0.431641\pi$
$444$	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	$-0.685809\pi$
$457$	−10768.9	−1.10229	−0.551145	+	0.834410i	$0.685809\pi$
$458$	−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	$-0.635103\pi$
$463$	−8205.35	−0.823618	−0.411809	+	0.911270i	$0.635103\pi$
$464$	−181.835	−0.0181929
$465$	0	0
$466$	1575.76	0.156643
$467$	−167.628	−0.0166100	−0.00830501	−	0.999966i	$-0.502644\pi$
$467$	−167.628	−0.0166100	−0.00830501	+	0.999966i	$0.502644\pi$
$468$	−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	$-0.910039\pi$
$487$	−20641.6	−1.92065	−0.960327	+	0.278875i	$0.910039\pi$
$488$	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	$-0.832889\pi$
$503$	−19523.7	−1.73065	−0.865326	+	0.501209i	$0.832889\pi$
$504$	−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	$-0.518185\pi$
$523$	−1365.89	−0.114200	−0.0570998	+	0.998368i	$0.518185\pi$
$524$	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	$-0.649117\pi$
$547$	−11552.7	−0.903033	−0.451516	+	0.892263i	$0.649117\pi$
$548$	−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	$-0.714499\pi$
$557$	−16406.2	−1.24803	−0.624014	+	0.781413i	$0.714499\pi$
$558$	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	$-0.329561\pi$
$563$	13631.9	1.02045	0.510227	+	0.860040i	$0.329561\pi$
$564$	−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	$-0.172274\pi$
$577$	23758.4	1.71417	0.857083	+	0.515178i	$0.172274\pi$
$578$	−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	$-0.493320\pi$
$587$	596.893	0.0419701	0.0209850	+	0.999780i	$0.493320\pi$
$588$	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	$-0.264119\pi$
$593$	19496.3	1.35012	0.675058	+	0.737765i	$0.264119\pi$
$594$	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	$-0.287236\pi$
$607$	18536.4	1.23949	0.619745	+	0.784803i	$0.287236\pi$
$608$	−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	$-0.477294\pi$
$613$	2163.47	0.142548	0.0712738	+	0.997457i	$0.477294\pi$
$614$	8235.08	0.541272
$615$	0	0
$616$	−8909.59	−0.582756
$617$	22964.9	1.49843	0.749215	−	0.662327i	$-0.230431\pi$
$617$	22964.9	1.49843	0.749215	+	0.662327i	$0.230431\pi$
$618$	−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	$-0.159672\pi$
$643$	28592.2	1.75360	0.876802	+	0.480851i	$0.159672\pi$
$644$	−5762.42	−0.352595
$645$	0	0
$646$	−19292.7	−1.17502
$647$	−14507.9	−0.881555	−0.440778	−	0.897616i	$-0.645297\pi$
$647$	−14507.9	−0.881555	−0.440778	+	0.897616i	$0.645297\pi$
$648$	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	$-0.432737\pi$
$653$	6999.85	0.419488	0.209744	+	0.977756i	$0.432737\pi$
$654$	−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	$-0.549426\pi$
$673$	−5400.26	−0.309309	−0.154654	+	0.987969i	$0.549426\pi$
$674$	−17841.3	−1.01962
$675$	0	0
$676$	650.781	0.0370267
$677$	6431.09	0.365091	0.182546	−	0.983197i	$-0.441566\pi$
$677$	6431.09	0.365091	0.182546	+	0.983197i	$0.441566\pi$
$678$	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	$-0.698704\pi$
$683$	−20865.8	−1.16897	−0.584486	+	0.811404i	$0.698704\pi$
$684$	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	$-0.129925\pi$
$727$	35983.4	1.83570	0.917849	+	0.396931i	$0.129925\pi$
$728$	−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	$-0.511643\pi$
$733$	−1451.50	−0.0731413	−0.0365706	+	0.999331i	$0.511643\pi$
$734$	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	$-0.441210\pi$
$743$	7438.65	0.367292	0.183646	+	0.982992i	$0.441210\pi$
$744$	−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	$-0.578695\pi$
$757$	−10193.8	−0.489432	−0.244716	+	0.969595i	$0.578695\pi$
$758$	−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	$-0.669303\pi$
$773$	−21799.2	−1.01431	−0.507156	+	0.861854i	$0.669303\pi$
$774$	−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	$-0.683600\pi$
$787$	−24080.3	−1.09068	−0.545342	+	0.838214i	$0.683600\pi$
$788$	−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	$-0.624796\pi$
$797$	−17194.3	−0.764184	−0.382092	+	0.924124i	$0.624796\pi$
$798$	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	$-0.202598\pi$
$823$	37974.3	1.60838	0.804192	+	0.594369i	$0.202598\pi$
$824$	−6651.92	−0.281226
$825$	0	0
$826$	2824.15	0.118965
$827$	15796.2	0.664193	0.332096	−	0.943245i	$-0.392244\pi$
$827$	15796.2	0.664193	0.332096	+	0.943245i	$0.392244\pi$
$828$	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	$-0.758613\pi$
$853$	−36172.5	−1.45196	−0.725980	+	0.687716i	$0.758613\pi$
$854$	−1060.72	−0.0425025
$855$	0	0
$856$	33410.4	1.33405
$857$	−32039.9	−1.27709	−0.638544	−	0.769586i	$-0.720463\pi$
$857$	−32039.9	−1.27709	−0.638544	+	0.769586i	$0.720463\pi$
$858$	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	$-0.702927\pi$
$863$	−30179.3	−1.19040	−0.595200	+	0.803577i	$0.702927\pi$
$864$	−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	$-0.510423\pi$
$877$	−1700.51	−0.0654758	−0.0327379	+	0.999464i	$0.510423\pi$
$878$	−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	$-0.437203\pi$
$883$	10285.8	0.392009	0.196005	+	0.980603i	$0.437203\pi$
$884$	21750.2	0.827532
$885$	0	0
$886$	−6657.24	−0.252432
$887$	−12167.5	−0.460593	−0.230296	−	0.973121i	$-0.573970\pi$
$887$	−12167.5	−0.460593	−0.230296	+	0.973121i	$0.573970\pi$
$888$	−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	$-0.879545\pi$
$907$	−50766.0	−1.85850	−0.929250	+	0.369452i	$0.879545\pi$
$908$	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	$-0.377063\pi$
$937$	21608.3	0.753375	0.376688	+	0.926340i	$0.377063\pi$
$938$	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	$-0.422940\pi$
$947$	13972.8	0.479466	0.239733	+	0.970839i	$0.422940\pi$
$948$	−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	$-0.560719\pi$
$953$	−11155.9	−0.379197	−0.189598	+	0.981862i	$0.560719\pi$
$954$	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	$-0.506418\pi$
$967$	−1212.55	−0.0403238	−0.0201619	+	0.999797i	$0.506418\pi$
$968$	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	$-0.468998\pi$
$977$	5939.19	0.194485	0.0972424	+	0.995261i	$0.468998\pi$
$978$	−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	$-0.615417\pi$
$983$	−21863.5	−0.709398	−0.354699	+	0.934980i	$0.615417\pi$
$984$	−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	$-0.705764\pi$
$997$	−37923.9	−1.20468	−0.602338	+	0.798241i	$0.705764\pi$
$998$	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.j.1.2		5
3.2	odd	2		1575.4.a.bn.1.4		5
5.2	odd	4		35.4.b.a.29.4	✓	10
5.3	odd	4		35.4.b.a.29.7	yes	10
5.4	even	2		175.4.a.i.1.4		5
7.6	odd	2		1225.4.a.bh.1.2		5
15.2	even	4		315.4.d.c.64.7		10
15.8	even	4		315.4.d.c.64.4		10
15.14	odd	2		1575.4.a.bq.1.2		5
20.3	even	4		560.4.g.f.449.7		10
20.7	even	4		560.4.g.f.449.4		10
35.2	odd	12		245.4.j.e.214.4		20
35.3	even	12		245.4.j.f.79.4		20
35.12	even	12		245.4.j.f.214.4		20
35.13	even	4		245.4.b.d.99.7		10
35.17	even	12		245.4.j.f.79.7		20
35.18	odd	12		245.4.j.e.79.4		20
35.23	odd	12		245.4.j.e.214.7		20
35.27	even	4		245.4.b.d.99.4		10
35.32	odd	12		245.4.j.e.79.7		20
35.33	even	12		245.4.j.f.214.7		20
35.34	odd	2		1225.4.a.be.1.4		5

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

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