LMFDB - Embedded newform 175.4.a.i.1.5

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			1.5
Root			$5.04851$ 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)

$f(q)$	$=$	\(q+4.04851 q^{2} -6.52749 q^{3} +8.39045 q^{4} -26.4266 q^{6} -7.00000 q^{7} +1.58074 q^{8} +15.6081 q^{9} +6.78210 q^{11} -54.7685 q^{12} -48.9221 q^{13} -28.3396 q^{14} -60.7239 q^{16} -92.4381 q^{17} +63.1894 q^{18} -125.574 q^{19} +45.6924 q^{21} +27.4574 q^{22} +32.2681 q^{23} -10.3183 q^{24} -198.062 q^{26} +74.3607 q^{27} -58.7332 q^{28} +282.778 q^{29} +205.434 q^{31} -258.488 q^{32} -44.2701 q^{33} -374.237 q^{34} +130.959 q^{36} -190.627 q^{37} -508.388 q^{38} +319.338 q^{39} +123.269 q^{41} +184.986 q^{42} -35.0202 q^{43} +56.9049 q^{44} +130.638 q^{46} -419.030 q^{47} +396.375 q^{48} +49.0000 q^{49} +603.388 q^{51} -410.478 q^{52} +0.365379 q^{53} +301.050 q^{54} -11.0652 q^{56} +819.683 q^{57} +1144.83 q^{58} +328.317 q^{59} -515.707 q^{61} +831.704 q^{62} -109.256 q^{63} -560.699 q^{64} -179.228 q^{66} +828.957 q^{67} -775.597 q^{68} -210.630 q^{69} -496.231 q^{71} +24.6723 q^{72} -701.132 q^{73} -771.757 q^{74} -1053.62 q^{76} -47.4747 q^{77} +1292.84 q^{78} +199.388 q^{79} -906.806 q^{81} +499.056 q^{82} +194.923 q^{83} +383.380 q^{84} -141.780 q^{86} -1845.83 q^{87} +10.7208 q^{88} +137.406 q^{89} +342.454 q^{91} +270.744 q^{92} -1340.97 q^{93} -1696.45 q^{94} +1687.27 q^{96} -220.440 q^{97} +198.377 q^{98} +105.855 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.04851	1.43137	0.715683	−	0.698426i	$-0.246116\pi$
$2$	4.04851	1.43137	0.715683	+	0.698426i	$0.246116\pi$
$3$	−6.52749	−1.25622	−0.628108	−	0.778127i	$-0.716170\pi$
$3$	−6.52749	−1.25622	−0.628108	+	0.778127i	$0.716170\pi$
$4$	8.39045	1.04881
$5$	0	0
$5$	0	0
$6$	−26.4266	−1.79810
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	1.58074	0.0698597
$9$	15.6081	0.578076
$10$	0	0
$11$	6.78210	0.185898	0.0929491	−	0.995671i	$-0.470371\pi$
$11$	6.78210	0.185898	0.0929491	+	0.995671i	$0.470371\pi$
$12$	−54.7685	−1.31753
$13$	−48.9221	−1.04373	−0.521867	−	0.853027i	$-0.674764\pi$
$13$	−48.9221	−1.04373	−0.521867	+	0.853027i	$0.674764\pi$
$14$	−28.3396	−0.541005
$15$	0	0
$16$	−60.7239	−0.948812
$17$	−92.4381	−1.31880	−0.659398	−	0.751794i	$-0.729189\pi$
$17$	−92.4381	−1.31880	−0.659398	+	0.751794i	$0.729189\pi$
$18$	63.1894	0.827438
$19$	−125.574	−1.51625	−0.758123	−	0.652112i	$-0.773883\pi$
$19$	−125.574	−1.51625	−0.758123	+	0.652112i	$0.773883\pi$
$20$	0	0
$21$	45.6924	0.474805
$22$	27.4574	0.266088
$23$	32.2681	0.292538	0.146269	−	0.989245i	$-0.453274\pi$
$23$	32.2681	0.292538	0.146269	+	0.989245i	$0.453274\pi$
$24$	−10.3183	−0.0877588
$25$	0	0
$26$	−198.062	−1.49396
$27$	74.3607	0.530027
$28$	−58.7332	−0.396412
$29$	282.778	1.81071	0.905354	−	0.424659i	$-0.139606\pi$
$29$	282.778	1.81071	0.905354	+	0.424659i	$0.139606\pi$
$30$	0	0
$31$	205.434	1.19023	0.595115	−	0.803641i	$-0.297107\pi$
$31$	205.434	1.19023	0.595115	+	0.803641i	$0.297107\pi$
$32$	−258.488	−1.42796
$33$	−44.2701	−0.233528
$34$	−374.237	−1.88768
$35$	0	0
$36$	130.959	0.606290
$37$	−190.627	−0.846998	−0.423499	−	0.905897i	$-0.639198\pi$
$37$	−190.627	−0.846998	−0.423499	+	0.905897i	$0.639198\pi$
$38$	−508.388	−2.17030
$39$	319.338	1.31115
$40$	0	0
$41$	123.269	0.469546	0.234773	−	0.972050i	$-0.424565\pi$
$41$	123.269	0.469546	0.234773	+	0.972050i	$0.424565\pi$
$42$	184.986	0.679619
$43$	−35.0202	−0.124198	−0.0620991	−	0.998070i	$-0.519779\pi$
$43$	−35.0202	−0.124198	−0.0620991	+	0.998070i	$0.519779\pi$
$44$	56.9049	0.194971
$45$	0	0
$46$	130.638	0.418728
$47$	−419.030	−1.30046	−0.650231	−	0.759736i	$-0.725328\pi$
$47$	−419.030	−1.30046	−0.650231	+	0.759736i	$0.725328\pi$
$48$	396.375	1.19191
$49$	49.0000	0.142857
$50$	0	0
$51$	603.388	1.65669
$52$	−410.478	−1.09467
$53$	0.365379	0.000946956 0	0.000473478	−	1.00000i	$-0.499849\pi$
$53$	0.365379	0.000946956 0	0.000473478		1.00000i	$0.499849\pi$
$54$	301.050	0.758662
$55$	0	0
$56$	−11.0652	−0.0264045
$57$	819.683	1.90473
$58$	1144.83	2.59178
$59$	328.317	0.724461	0.362231	−	0.932088i	$-0.382015\pi$
$59$	328.317	0.724461	0.362231	+	0.932088i	$0.382015\pi$
$60$	0	0
$61$	−515.707	−1.08245	−0.541226	−	0.840877i	$-0.682040\pi$
$61$	−515.707	−1.08245	−0.541226	+	0.840877i	$0.682040\pi$
$62$	831.704	1.70365
$63$	−109.256	−0.218492
$64$	−560.699	−1.09511
$65$	0	0
$66$	−179.228	−0.334264
$67$	828.957	1.51154	0.755770	−	0.654837i	$-0.227263\pi$
$67$	828.957	1.51154	0.755770	+	0.654837i	$0.227263\pi$
$68$	−775.597	−1.38316
$69$	−210.630	−0.367490
$70$	0	0
$71$	−496.231	−0.829462	−0.414731	−	0.909944i	$-0.636124\pi$
$71$	−496.231	−0.829462	−0.414731	+	0.909944i	$0.636124\pi$
$72$	24.6723	0.0403842
$73$	−701.132	−1.12413	−0.562064	−	0.827094i	$-0.689992\pi$
$73$	−701.132	−1.12413	−0.562064	+	0.827094i	$0.689992\pi$
$74$	−771.757	−1.21236
$75$	0	0
$76$	−1053.62	−1.59025
$77$	−47.4747	−0.0702629
$78$	1292.84	1.87674
$79$	199.388	0.283961	0.141981	−	0.989869i	$-0.454653\pi$
$79$	199.388	0.283961	0.141981	+	0.989869i	$0.454653\pi$
$80$	0	0
$81$	−906.806	−1.24390
$82$	499.056	0.672092
$83$	194.923	0.257778	0.128889	−	0.991659i	$-0.458859\pi$
$83$	194.923	0.257778	0.128889	+	0.991659i	$0.458859\pi$
$84$	383.380	0.497978
$85$	0	0
$86$	−141.780	−0.177773
$87$	−1845.83	−2.27464
$88$	10.7208	0.0129868
$89$	137.406	0.163651	0.0818257	−	0.996647i	$-0.473925\pi$
$89$	137.406	0.163651	0.0818257	+	0.996647i	$0.473925\pi$
$90$	0	0
$91$	342.454	0.394494
$92$	270.744	0.306815
$93$	−1340.97	−1.49518
$94$	−1696.45	−1.86144
$95$	0	0
$96$	1687.27	1.79382
$97$	−220.440	−0.230745	−0.115372	−	0.993322i	$-0.536806\pi$
$97$	−220.440	−0.230745	−0.115372	+	0.993322i	$0.536806\pi$
$98$	198.377	0.204481
$99$	105.855	0.107463
$100$	0	0
$101$	−591.358	−0.582597	−0.291298	−	0.956632i	$-0.594087\pi$
$101$	−591.358	−0.582597	−0.291298	+	0.956632i	$0.594087\pi$
$102$	2442.83	2.37133
$103$	476.494	0.455829	0.227914	−	0.973681i	$-0.426809\pi$
$103$	476.494	0.455829	0.227914	+	0.973681i	$0.426809\pi$
$104$	−77.3332	−0.0729149
$105$	0	0
$106$	1.47924	0.00135544
$107$	225.584	0.203813	0.101907	−	0.994794i	$-0.467506\pi$
$107$	225.584	0.203813	0.101907	+	0.994794i	$0.467506\pi$
$108$	623.920	0.555895
$109$	−1627.65	−1.43028	−0.715142	−	0.698979i	$-0.753638\pi$
$109$	−1627.65	−1.43028	−0.715142	+	0.698979i	$0.753638\pi$
$110$	0	0
$111$	1244.32	1.06401
$112$	425.068	0.358617
$113$	−357.040	−0.297235	−0.148617	−	0.988895i	$-0.547482\pi$
$113$	−357.040	−0.297235	−0.148617	+	0.988895i	$0.547482\pi$
$114$	3318.50	2.72636
$115$	0	0
$116$	2372.63	1.89908
$117$	−763.579	−0.603358
$118$	1329.19	1.03697
$119$	647.067	0.498458
$120$	0	0
$121$	−1285.00	−0.965442
$122$	−2087.85	−1.54938
$123$	−804.637	−0.589851
$124$	1723.69	1.24832
$125$	0	0
$126$	−442.326	−0.312742
$127$	1728.25	1.20754	0.603771	−	0.797158i	$-0.293664\pi$
$127$	1728.25	1.20754	0.603771	+	0.797158i	$0.293664\pi$
$128$	−202.094	−0.139553
$129$	228.594	0.156020
$130$	0	0
$131$	1461.19	0.974543	0.487272	−	0.873250i	$-0.337992\pi$
$131$	1461.19	0.974543	0.487272	+	0.873250i	$0.337992\pi$
$132$	−371.446	−0.244926
$133$	879.018	0.573087
$134$	3356.04	2.16357
$135$	0	0
$136$	−146.121	−0.0921306
$137$	−1892.96	−1.18049	−0.590243	−	0.807226i	$-0.700968\pi$
$137$	−1892.96	−1.18049	−0.590243	+	0.807226i	$0.700968\pi$
$138$	−852.737	−0.526013
$139$	−1715.03	−1.04652	−0.523262	−	0.852172i	$-0.675285\pi$
$139$	−1715.03	−1.04652	−0.523262	+	0.852172i	$0.675285\pi$
$140$	0	0
$141$	2735.21	1.63366
$142$	−2009.00	−1.18726
$143$	−331.794	−0.194028
$144$	−947.783	−0.548486
$145$	0	0
$146$	−2838.54	−1.60904
$147$	−319.847	−0.179459
$148$	−1599.45	−0.888337
$149$	−2798.32	−1.53857	−0.769286	−	0.638904i	$-0.779388\pi$
$149$	−2798.32	−1.53857	−0.769286	+	0.638904i	$0.779388\pi$
$150$	0	0
$151$	−2867.49	−1.54538	−0.772692	−	0.634781i	$-0.781090\pi$
$151$	−2867.49	−1.54538	−0.772692	+	0.634781i	$0.781090\pi$
$152$	−198.500	−0.105924
$153$	−1442.78	−0.762365
$154$	−192.202	−0.100572
$155$	0	0
$156$	2679.39	1.37515
$157$	783.696	0.398381	0.199190	−	0.979961i	$-0.436169\pi$
$157$	783.696	0.398381	0.199190	+	0.979961i	$0.436169\pi$
$158$	807.226	0.406452
$159$	−2.38501	−0.00118958
$160$	0	0
$161$	−225.877	−0.110569
$162$	−3671.22	−1.78048
$163$	−2416.93	−1.16140	−0.580702	−	0.814116i	$-0.697222\pi$
$163$	−2416.93	−1.16140	−0.580702	+	0.814116i	$0.697222\pi$
$164$	1034.28	0.492463
$165$	0	0
$166$	789.149	0.368975
$167$	704.424	0.326407	0.163204	−	0.986592i	$-0.447817\pi$
$167$	704.424	0.326407	0.163204	+	0.986592i	$0.447817\pi$
$168$	72.2280	0.0331697
$169$	196.369	0.0893803
$170$	0	0
$171$	−1959.97	−0.876506
$172$	−293.835	−0.130260
$173$	1398.49	0.614598	0.307299	−	0.951613i	$-0.400575\pi$
$173$	1398.49	0.614598	0.307299	+	0.951613i	$0.400575\pi$
$174$	−7472.85	−3.25584
$175$	0	0
$176$	−411.836	−0.176382
$177$	−2143.08	−0.910079
$178$	556.289	0.234245
$179$	368.688	0.153950	0.0769749	−	0.997033i	$-0.475474\pi$
$179$	368.688	0.153950	0.0769749	+	0.997033i	$0.475474\pi$
$180$	0	0
$181$	−315.621	−0.129613	−0.0648064	−	0.997898i	$-0.520643\pi$
$181$	−315.621	−0.129613	−0.0648064	+	0.997898i	$0.520643\pi$
$182$	1386.43	0.564665
$183$	3366.27	1.35979
$184$	51.0076	0.0204366
$185$	0	0
$186$	−5428.94	−2.14016
$187$	−626.924	−0.245162
$188$	−3515.85	−1.36393
$189$	−520.525	−0.200331
$190$	0	0
$191$	−151.629	−0.0574424	−0.0287212	−	0.999587i	$-0.509143\pi$
$191$	−151.629	−0.0574424	−0.0287212	+	0.999587i	$0.509143\pi$
$192$	3659.95	1.37570
$193$	690.689	0.257600	0.128800	−	0.991671i	$-0.458887\pi$
$193$	690.689	0.257600	0.128800	+	0.991671i	$0.458887\pi$
$194$	−892.453	−0.330280
$195$	0	0
$196$	411.132	0.149829
$197$	834.136	0.301674	0.150837	−	0.988559i	$-0.451803\pi$
$197$	834.136	0.301674	0.150837	+	0.988559i	$0.451803\pi$
$198$	428.557	0.153819
$199$	387.269	0.137954	0.0689769	−	0.997618i	$-0.478027\pi$
$199$	387.269	0.137954	0.0689769	+	0.997618i	$0.478027\pi$
$200$	0	0
$201$	−5411.00	−1.89882
$202$	−2394.12	−0.833909
$203$	−1979.44	−0.684383
$204$	5062.70	1.73755
$205$	0	0
$206$	1929.09	0.652457
$207$	503.643	0.169109
$208$	2970.74	0.990307
$209$	−851.656	−0.281867
$210$	0	0
$211$	3070.54	1.00182	0.500912	−	0.865498i	$-0.332998\pi$
$211$	3070.54	1.00182	0.500912	+	0.865498i	$0.332998\pi$
$212$	3.06570	0.000993174 0
$213$	3239.14	1.04198
$214$	913.278	0.291731
$215$	0	0
$216$	117.545	0.0370275
$217$	−1438.04	−0.449865
$218$	−6589.58	−2.04726
$219$	4576.63	1.41215
$220$	0	0
$221$	4522.26	1.37647
$222$	5037.63	1.52299
$223$	−1004.46	−0.301631	−0.150816	−	0.988562i	$-0.548190\pi$
$223$	−1004.46	−0.301631	−0.150816	+	0.988562i	$0.548190\pi$
$224$	1809.41	0.539716
$225$	0	0
$226$	−1445.48	−0.425451
$227$	5374.22	1.57136	0.785681	−	0.618631i	$-0.212313\pi$
$227$	5374.22	1.57136	0.785681	+	0.618631i	$0.212313\pi$
$228$	6877.51	1.99769
$229$	3650.97	1.05355	0.526775	−	0.850005i	$-0.323401\pi$
$229$	3650.97	1.05355	0.526775	+	0.850005i	$0.323401\pi$
$230$	0	0
$231$	309.890	0.0882653
$232$	446.999	0.126495
$233$	−4582.88	−1.28856	−0.644280	−	0.764789i	$-0.722843\pi$
$233$	−4582.88	−1.28856	−0.644280	+	0.764789i	$0.722843\pi$
$234$	−3091.36	−0.863625
$235$	0	0
$236$	2754.73	0.759820
$237$	−1301.50	−0.356716
$238$	2619.66	0.713476
$239$	696.769	0.188578	0.0942892	−	0.995545i	$-0.469942\pi$
$239$	696.769	0.188578	0.0942892	+	0.995545i	$0.469942\pi$
$240$	0	0
$241$	5082.82	1.35856	0.679281	−	0.733878i	$-0.262292\pi$
$241$	5082.82	1.35856	0.679281	+	0.733878i	$0.262292\pi$
$242$	−5202.35	−1.38190
$243$	3911.42	1.03258
$244$	−4327.02	−1.13528
$245$	0	0
$246$	−3257.58	−0.844292
$247$	6143.34	1.58256
$248$	324.739	0.0831490
$249$	−1272.36	−0.323825
$250$	0	0
$251$	−1207.32	−0.303606	−0.151803	−	0.988411i	$-0.548508\pi$
$251$	−1207.32	−0.303606	−0.151803	+	0.988411i	$0.548508\pi$
$252$	−916.711	−0.229156
$253$	218.846	0.0543822
$254$	6996.86	1.72843
$255$	0	0
$256$	3667.41	0.895363
$257$	−510.936	−0.124013	−0.0620064	−	0.998076i	$-0.519750\pi$
$257$	−510.936	−0.124013	−0.0620064	+	0.998076i	$0.519750\pi$
$258$	925.464	0.223321
$259$	1334.39	0.320135
$260$	0	0
$261$	4413.61	1.04673
$262$	5915.67	1.39493
$263$	−2269.04	−0.531997	−0.265998	−	0.963974i	$-0.585702\pi$
$263$	−2269.04	−0.531997	−0.265998	+	0.963974i	$0.585702\pi$
$264$	−69.9796	−0.0163142
$265$	0	0
$266$	3558.72	0.820297
$267$	−896.913	−0.205581
$268$	6955.32	1.58531
$269$	−7836.10	−1.77612	−0.888058	−	0.459731i	$-0.847946\pi$
$269$	−7836.10	−1.77612	−0.888058	+	0.459731i	$0.847946\pi$
$270$	0	0
$271$	1466.28	0.328673	0.164336	−	0.986404i	$-0.447452\pi$
$271$	1466.28	0.328673	0.164336	+	0.986404i	$0.447452\pi$
$272$	5613.21	1.25129
$273$	−2235.37	−0.495570
$274$	−7663.67	−1.68971
$275$	0	0
$276$	−1767.28	−0.385426
$277$	−4154.17	−0.901083	−0.450542	−	0.892755i	$-0.648769\pi$
$277$	−4154.17	−0.901083	−0.450542	+	0.892755i	$0.648769\pi$
$278$	−6943.32	−1.49796
$279$	3206.43	0.688044
$280$	0	0
$281$	−3490.40	−0.740996	−0.370498	−	0.928833i	$-0.620813\pi$
$281$	−3490.40	−0.740996	−0.370498	+	0.928833i	$0.620813\pi$
$282$	11073.5	2.33837
$283$	3125.29	0.656464	0.328232	−	0.944597i	$-0.393547\pi$
$283$	3125.29	0.656464	0.328232	+	0.944597i	$0.393547\pi$
$284$	−4163.60	−0.869945
$285$	0	0
$286$	−1343.27	−0.277725
$287$	−862.883	−0.177472
$288$	−4034.49	−0.825467
$289$	3631.80	0.739223
$290$	0	0
$291$	1438.92	0.289865
$292$	−5882.81	−1.17899
$293$	1447.69	0.288652	0.144326	−	0.989530i	$-0.453899\pi$
$293$	1447.69	0.288652	0.144326	+	0.989530i	$0.453899\pi$
$294$	−1294.90	−0.256872
$295$	0	0
$296$	−301.333	−0.0591710
$297$	504.322	0.0985310
$298$	−11329.0	−2.20226
$299$	−1578.62	−0.305332
$300$	0	0
$301$	245.141	0.0469425
$302$	−11609.1	−2.21201
$303$	3860.08	0.731867
$304$	7625.35	1.43863
$305$	0	0
$306$	−5841.11	−1.09122
$307$	−1591.43	−0.295856	−0.147928	−	0.988998i	$-0.547260\pi$
$307$	−1591.43	−0.295856	−0.147928	+	0.988998i	$0.547260\pi$
$308$	−398.334	−0.0736922
$309$	−3110.31	−0.572619
$310$	0	0
$311$	−8584.92	−1.56529	−0.782647	−	0.622466i	$-0.786131\pi$
$311$	−8584.92	−1.56529	−0.782647	+	0.622466i	$0.786131\pi$
$312$	504.792	0.0915968
$313$	7210.05	1.30203	0.651016	−	0.759064i	$-0.274343\pi$
$313$	7210.05	1.30203	0.651016	+	0.759064i	$0.274343\pi$
$314$	3172.80	0.570228
$315$	0	0
$316$	1672.96	0.297820
$317$	−7787.24	−1.37973	−0.689865	−	0.723938i	$-0.742330\pi$
$317$	−7787.24	−1.37973	−0.689865	+	0.723938i	$0.742330\pi$
$318$	−9.65573	−0.00170272
$319$	1917.83	0.336607
$320$	0	0
$321$	−1472.49	−0.256033
$322$	−914.465	−0.158264
$323$	11607.8	1.99962
$324$	−7608.51	−1.30461
$325$	0	0
$326$	−9784.98	−1.66239
$327$	10624.5	1.79674
$328$	194.857	0.0328023
$329$	2933.21	0.491529
$330$	0	0
$331$	−1729.78	−0.287243	−0.143621	−	0.989633i	$-0.545875\pi$
$331$	−1729.78	−0.287243	−0.143621	+	0.989633i	$0.545875\pi$
$332$	1635.49	0.270359
$333$	−2975.32	−0.489630
$334$	2851.87	0.467208
$335$	0	0
$336$	−2774.62	−0.450500
$337$	7815.06	1.26325	0.631623	−	0.775276i	$-0.282389\pi$
$337$	7815.06	1.26325	0.631623	+	0.775276i	$0.282389\pi$
$338$	795.000	0.127936
$339$	2330.57	0.373391
$340$	0	0
$341$	1393.28	0.221262
$342$	−7934.95	−1.25460
$343$	−343.000	−0.0539949
$344$	−55.3579	−0.00867645
$345$	0	0
$346$	5661.82	0.879714
$347$	2359.77	0.365070	0.182535	−	0.983199i	$-0.441570\pi$
$347$	2359.77	0.365070	0.182535	+	0.983199i	$0.441570\pi$
$348$	−15487.3	−2.38565
$349$	3212.73	0.492761	0.246380	−	0.969173i	$-0.420759\pi$
$349$	3212.73	0.492761	0.246380	+	0.969173i	$0.420759\pi$
$350$	0	0
$351$	−3637.88	−0.553207
$352$	−1753.09	−0.265454
$353$	−5582.04	−0.841649	−0.420824	−	0.907142i	$-0.638259\pi$
$353$	−5582.04	−0.841649	−0.420824	+	0.907142i	$0.638259\pi$
$354$	−8676.30	−1.30266
$355$	0	0
$356$	1152.90	0.171639
$357$	−4223.72	−0.626171
$358$	1492.64	0.220358
$359$	−10630.6	−1.56285	−0.781425	−	0.623999i	$-0.785507\pi$
$359$	−10630.6	−1.56285	−0.781425	+	0.623999i	$0.785507\pi$
$360$	0	0
$361$	8909.84	1.29900
$362$	−1277.80	−0.185523
$363$	8387.84	1.21280
$364$	2873.35	0.413748
$365$	0	0
$366$	13628.4	1.94636
$367$	4514.69	0.642139	0.321070	−	0.947056i	$-0.395958\pi$
$367$	4514.69	0.642139	0.321070	+	0.947056i	$0.395958\pi$
$368$	−1959.45	−0.277563
$369$	1923.99	0.271434
$370$	0	0
$371$	−2.55765	−0.000357916 0
$372$	−11251.3	−1.56816
$373$	11445.8	1.58885	0.794426	−	0.607361i	$-0.207772\pi$
$373$	11445.8	1.58885	0.794426	+	0.607361i	$0.207772\pi$
$374$	−2538.11	−0.350916
$375$	0	0
$376$	−662.378	−0.0908499
$377$	−13834.1	−1.88990
$378$	−2107.35	−0.286747
$379$	8146.48	1.10411	0.552054	−	0.833809i	$-0.313844\pi$
$379$	8146.48	1.10411	0.552054	+	0.833809i	$0.313844\pi$
$380$	0	0
$381$	−11281.2	−1.51693
$382$	−613.872	−0.0822210
$383$	−5261.56	−0.701967	−0.350983	−	0.936382i	$-0.614153\pi$
$383$	−5261.56	−0.701967	−0.350983	+	0.936382i	$0.614153\pi$
$384$	1319.17	0.175308
$385$	0	0
$386$	2796.26	0.368720
$387$	−546.597	−0.0717961
$388$	−1849.59	−0.242007
$389$	13207.1	1.72141	0.860706	−	0.509103i	$-0.170023\pi$
$389$	13207.1	1.72141	0.860706	+	0.509103i	$0.170023\pi$
$390$	0	0
$391$	−2982.80	−0.385798
$392$	77.4564	0.00997995
$393$	−9537.93	−1.22424
$394$	3377.01	0.431805
$395$	0	0
$396$	888.175	0.112708
$397$	−5663.15	−0.715933	−0.357967	−	0.933734i	$-0.616530\pi$
$397$	−5663.15	−0.715933	−0.357967	+	0.933734i	$0.616530\pi$
$398$	1567.86	0.197462
$399$	−5737.78	−0.719920
$400$	0	0
$401$	−11989.0	−1.49302	−0.746512	−	0.665372i	$-0.768273\pi$
$401$	−11989.0	−1.49302	−0.746512	+	0.665372i	$0.768273\pi$
$402$	−21906.5	−2.71790
$403$	−10050.3	−1.24228
$404$	−4961.76	−0.611031
$405$	0	0
$406$	−8013.80	−0.979602
$407$	−1292.85	−0.157455
$408$	953.802	0.115736
$409$	5249.67	0.634669	0.317334	−	0.948314i	$-0.397212\pi$
$409$	5249.67	0.634669	0.317334	+	0.948314i	$0.397212\pi$
$410$	0	0
$411$	12356.3	1.48294
$412$	3998.00	0.478076
$413$	−2298.22	−0.273821
$414$	2039.00	0.242057
$415$	0	0
$416$	12645.7	1.49041
$417$	11194.8	1.31466
$418$	−3447.94	−0.403455
$419$	−14948.9	−1.74297	−0.871484	−	0.490424i	$-0.836842\pi$
$419$	−14948.9	−1.74297	−0.871484	+	0.490424i	$0.836842\pi$
$420$	0	0
$421$	5840.31	0.676103	0.338051	−	0.941128i	$-0.390232\pi$
$421$	5840.31	0.676103	0.338051	+	0.941128i	$0.390232\pi$
$422$	12431.1	1.43398
$423$	−6540.24	−0.751767
$424$	0.577571	6.61540e−5 0
$425$	0	0
$426$	13113.7	1.49146
$427$	3609.95	0.409128
$428$	1892.75	0.213760
$429$	2165.78	0.243741
$430$	0	0
$431$	7439.21	0.831402	0.415701	−	0.909501i	$-0.363536\pi$
$431$	7439.21	0.831402	0.415701	+	0.909501i	$0.363536\pi$
$432$	−4515.48	−0.502896
$433$	−877.657	−0.0974077	−0.0487038	−	0.998813i	$-0.515509\pi$
$433$	−877.657	−0.0974077	−0.0487038	+	0.998813i	$0.515509\pi$
$434$	−5821.93	−0.643920
$435$	0	0
$436$	−13656.8	−1.50009
$437$	−4052.04	−0.443559
$438$	18528.5	2.02130
$439$	−10855.8	−1.18023	−0.590114	−	0.807320i	$-0.700917\pi$
$439$	−10855.8	−1.18023	−0.590114	+	0.807320i	$0.700917\pi$
$440$	0	0
$441$	764.795	0.0825823
$442$	18308.4	1.97023
$443$	−10797.0	−1.15798	−0.578988	−	0.815336i	$-0.696552\pi$
$443$	−10797.0	−1.15798	−0.578988	+	0.815336i	$0.696552\pi$
$444$	10440.4	1.11594
$445$	0	0
$446$	−4066.58	−0.431745
$447$	18266.0	1.93278
$448$	3924.89	0.413914
$449$	8621.70	0.906198	0.453099	−	0.891460i	$-0.350318\pi$
$449$	8621.70	0.906198	0.453099	+	0.891460i	$0.350318\pi$
$450$	0	0
$451$	836.023	0.0872878
$452$	−2995.73	−0.311742
$453$	18717.5	1.94134
$454$	21757.6	2.24919
$455$	0	0
$456$	1295.71	0.133064
$457$	−3785.90	−0.387520	−0.193760	−	0.981049i	$-0.562068\pi$
$457$	−3785.90	−0.387520	−0.193760	+	0.981049i	$0.562068\pi$
$458$	14781.0	1.50801
$459$	−6873.76	−0.698997
$460$	0	0
$461$	5760.18	0.581948	0.290974	−	0.956731i	$-0.406021\pi$
$461$	5760.18	0.581948	0.290974	+	0.956731i	$0.406021\pi$
$462$	1254.59	0.126340
$463$	10760.9	1.08014	0.540068	−	0.841621i	$-0.318398\pi$
$463$	10760.9	1.08014	0.540068	+	0.841621i	$0.318398\pi$
$464$	−17171.4	−1.71802
$465$	0	0
$466$	−18553.9	−1.84440
$467$	2153.58	0.213395	0.106698	−	0.994292i	$-0.465972\pi$
$467$	2153.58	0.213395	0.106698	+	0.994292i	$0.465972\pi$
$468$	−6406.77	−0.632806
$469$	−5802.70	−0.571309
$470$	0	0
$471$	−5115.57	−0.500452
$472$	518.985	0.0506106
$473$	−237.510	−0.0230882
$474$	−5269.15	−0.510591
$475$	0	0
$476$	5429.18	0.522786
$477$	5.70286	0.000547413 0
$478$	2820.88	0.269924
$479$	−6890.26	−0.657253	−0.328626	−	0.944460i	$-0.606586\pi$
$479$	−6890.26	−0.657253	−0.328626	+	0.944460i	$0.606586\pi$
$480$	0	0
$481$	9325.88	0.884041
$482$	20577.9	1.94460
$483$	1474.41	0.138898
$484$	−10781.8	−1.01256
$485$	0	0
$486$	15835.4	1.47801
$487$	−19006.4	−1.76850	−0.884251	−	0.467011i	$-0.845331\pi$
$487$	−19006.4	−1.76850	−0.884251	+	0.467011i	$0.845331\pi$
$488$	−815.201	−0.0756197
$489$	15776.5	1.45897
$490$	0	0
$491$	4530.05	0.416371	0.208186	−	0.978089i	$-0.433244\pi$
$491$	4530.05	0.416371	0.208186	+	0.978089i	$0.433244\pi$
$492$	−6751.27	−0.618639
$493$	−26139.4	−2.38795
$494$	24871.4	2.26522
$495$	0	0
$496$	−12474.8	−1.12930
$497$	3473.62	0.313507
$498$	−5151.16	−0.463512
$499$	3620.18	0.324773	0.162386	−	0.986727i	$-0.448081\pi$
$499$	3620.18	0.324773	0.162386	+	0.986727i	$0.448081\pi$
$500$	0	0
$501$	−4598.12	−0.410038
$502$	−4887.84	−0.434571
$503$	11761.8	1.04261	0.521303	−	0.853372i	$-0.325446\pi$
$503$	11761.8	1.04261	0.521303	+	0.853372i	$0.325446\pi$
$504$	−172.706	−0.0152638
$505$	0	0
$506$	885.999	0.0778408
$507$	−1281.79	−0.112281
$508$	14500.8	1.26648
$509$	−5254.47	−0.457564	−0.228782	−	0.973478i	$-0.573474\pi$
$509$	−5254.47	−0.457564	−0.228782	+	0.973478i	$0.573474\pi$
$510$	0	0
$511$	4907.92	0.424880
$512$	16464.3	1.42114
$513$	−9337.77	−0.803651
$514$	−2068.53	−0.177508
$515$	0	0
$516$	1918.00	0.163634
$517$	−2841.90	−0.241754
$518$	5402.30	0.458230
$519$	−9128.64	−0.772067
$520$	0	0
$521$	−15511.8	−1.30439	−0.652193	−	0.758053i	$-0.726151\pi$
$521$	−15511.8	−1.30439	−0.652193	+	0.758053i	$0.726151\pi$
$522$	17868.6	1.49825
$523$	3814.73	0.318942	0.159471	−	0.987203i	$-0.449021\pi$
$523$	3814.73	0.318942	0.159471	+	0.987203i	$0.449021\pi$
$524$	12260.1	1.02211
$525$	0	0
$526$	−9186.24	−0.761481
$527$	−18990.0	−1.56967
$528$	2688.25	0.221574
$529$	−11125.8	−0.914422
$530$	0	0
$531$	5124.39	0.418794
$532$	7375.36	0.601057
$533$	−6030.58	−0.490081
$534$	−3631.17	−0.294262
$535$	0	0
$536$	1310.37	0.105596
$537$	−2406.60	−0.193394
$538$	−31724.5	−2.54227
$539$	332.323	0.0265569
$540$	0	0
$541$	−4573.77	−0.363478	−0.181739	−	0.983347i	$-0.558173\pi$
$541$	−4573.77	−0.363478	−0.181739	+	0.983347i	$0.558173\pi$
$542$	5936.26	0.470451
$543$	2060.21	0.162822
$544$	23894.1	1.88318
$545$	0	0
$546$	−9049.91	−0.709341
$547$	13327.6	1.04177	0.520885	−	0.853627i	$-0.325602\pi$
$547$	13327.6	1.04177	0.520885	+	0.853627i	$0.325602\pi$
$548$	−15882.8	−1.23810
$549$	−8049.19	−0.625740
$550$	0	0
$551$	−35509.5	−2.74548
$552$	−332.952	−0.0256728
$553$	−1395.72	−0.107327
$554$	−16818.2	−1.28978
$555$	0	0
$556$	−14389.9	−1.09760
$557$	12096.1	0.920155	0.460077	−	0.887879i	$-0.347822\pi$
$557$	12096.1	0.920155	0.460077	+	0.887879i	$0.347822\pi$
$558$	12981.3	0.984842
$559$	1713.26	0.129630
$560$	0	0
$561$	4092.24	0.307976
$562$	−14130.9	−1.06064
$563$	22943.6	1.71751	0.858755	−	0.512387i	$-0.171239\pi$
$563$	22943.6	1.71751	0.858755	+	0.512387i	$0.171239\pi$
$564$	22949.6	1.71339
$565$	0	0
$566$	12652.8	0.939639
$567$	6347.64	0.470152
$568$	−784.414	−0.0579459
$569$	485.307	0.0357560	0.0178780	−	0.999840i	$-0.494309\pi$
$569$	485.307	0.0357560	0.0178780	+	0.999840i	$0.494309\pi$
$570$	0	0
$571$	12271.8	0.899401	0.449701	−	0.893179i	$-0.351531\pi$
$571$	12271.8	0.899401	0.449701	+	0.893179i	$0.351531\pi$
$572$	−2783.90	−0.203498
$573$	989.756	0.0721600
$574$	−3493.39	−0.254027
$575$	0	0
$576$	−8751.42	−0.633060
$577$	−14122.8	−1.01896	−0.509478	−	0.860484i	$-0.670162\pi$
$577$	−14122.8	−1.01896	−0.509478	+	0.860484i	$0.670162\pi$
$578$	14703.4	1.05810
$579$	−4508.46	−0.323601
$580$	0	0
$581$	−1364.46	−0.0974310
$582$	5825.47	0.414903
$583$	2.47804	0.000176037 0
$584$	−1108.31	−0.0785312
$585$	0	0
$586$	5861.00	0.413167
$587$	11005.3	0.773826	0.386913	−	0.922116i	$-0.373541\pi$
$587$	11005.3	0.773826	0.386913	+	0.922116i	$0.373541\pi$
$588$	−2683.66	−0.188218
$589$	−25797.2	−1.80468
$590$	0	0
$591$	−5444.81	−0.378967
$592$	11575.6	0.803642
$593$	−11233.1	−0.777889	−0.388944	−	0.921261i	$-0.627160\pi$
$593$	−11233.1	−0.777889	−0.388944	+	0.921261i	$0.627160\pi$
$594$	2041.75	0.141034
$595$	0	0
$596$	−23479.2	−1.61366
$597$	−2527.89	−0.173300
$598$	−6391.08	−0.437041
$599$	14855.4	1.01331	0.506656	−	0.862149i	$-0.330882\pi$
$599$	14855.4	1.01331	0.506656	+	0.862149i	$0.330882\pi$
$600$	0	0
$601$	25358.8	1.72115	0.860573	−	0.509327i	$-0.170106\pi$
$601$	25358.8	1.72115	0.860573	+	0.509327i	$0.170106\pi$
$602$	992.457	0.0671919
$603$	12938.4	0.873786
$604$	−24059.5	−1.62081
$605$	0	0
$606$	15627.6	1.04757
$607$	14393.9	0.962487	0.481243	−	0.876587i	$-0.340185\pi$
$607$	14393.9	0.962487	0.481243	+	0.876587i	$0.340185\pi$
$608$	32459.3	2.16513
$609$	12920.8	0.859732
$610$	0	0
$611$	20499.8	1.35734
$612$	−12105.6	−0.799573
$613$	4769.94	0.314284	0.157142	−	0.987576i	$-0.449772\pi$
$613$	4769.94	0.314284	0.157142	+	0.987576i	$0.449772\pi$
$614$	−6442.92	−0.423477
$615$	0	0
$616$	−75.0453	−0.00490854
$617$	−7769.86	−0.506974	−0.253487	−	0.967339i	$-0.581577\pi$
$617$	−7769.86	−0.506974	−0.253487	+	0.967339i	$0.581577\pi$
$618$	−12592.1	−0.819627
$619$	−5680.75	−0.368867	−0.184433	−	0.982845i	$-0.559045\pi$
$619$	−5680.75	−0.368867	−0.184433	+	0.982845i	$0.559045\pi$
$620$	0	0
$621$	2399.48	0.155053
$622$	−34756.2	−2.24051
$623$	−961.840	−0.0618544
$624$	−19391.5	−1.24404
$625$	0	0
$626$	29190.0	1.86368
$627$	5559.17	0.354086
$628$	6575.56	0.417824
$629$	17621.2	1.11702
$630$	0	0
$631$	−10395.8	−0.655864	−0.327932	−	0.944701i	$-0.606352\pi$
$631$	−10395.8	−0.655864	−0.327932	+	0.944701i	$0.606352\pi$
$632$	315.182	0.0198374
$633$	−20042.9	−1.25851
$634$	−31526.7	−1.97490
$635$	0	0
$636$	−20.0113	−0.00124764
$637$	−2397.18	−0.149105
$638$	7764.34	0.481808
$639$	−7745.21	−0.479492
$640$	0	0
$641$	−8194.28	−0.504921	−0.252461	−	0.967607i	$-0.581240\pi$
$641$	−8194.28	−0.504921	−0.252461	+	0.967607i	$0.581240\pi$
$642$	−5961.41	−0.366477
$643$	−32118.2	−1.96985	−0.984927	−	0.172970i	$-0.944664\pi$
$643$	−32118.2	−1.96985	−0.984927	+	0.172970i	$0.944664\pi$
$644$	−1895.21	−0.115965
$645$	0	0
$646$	46994.4	2.86218
$647$	−22299.0	−1.35496	−0.677482	−	0.735539i	$-0.736929\pi$
$647$	−22299.0	−1.35496	−0.677482	+	0.735539i	$0.736929\pi$
$648$	−1433.43	−0.0868987
$649$	2226.68	0.134676
$650$	0	0
$651$	9386.79	0.565127
$652$	−20279.1	−1.21809
$653$	920.410	0.0551584	0.0275792	−	0.999620i	$-0.491220\pi$
$653$	920.410	0.0551584	0.0275792	+	0.999620i	$0.491220\pi$
$654$	43013.4	2.57180
$655$	0	0
$656$	−7485.38	−0.445511
$657$	−10943.3	−0.649832
$658$	11875.1	0.703557
$659$	17824.3	1.05362	0.526812	−	0.849982i	$-0.323387\pi$
$659$	17824.3	1.05362	0.526812	+	0.849982i	$0.323387\pi$
$660$	0	0
$661$	11343.8	0.667510	0.333755	−	0.942660i	$-0.391684\pi$
$661$	11343.8	0.667510	0.333755	+	0.942660i	$0.391684\pi$
$662$	−7003.03	−0.411149
$663$	−29519.0	−1.72915
$664$	308.123	0.0180083
$665$	0	0
$666$	−12045.6	−0.700839
$667$	9124.71	0.529700
$668$	5910.44	0.342338
$669$	6556.61	0.378914
$670$	0	0
$671$	−3497.58	−0.201226
$672$	−11810.9	−0.678000
$673$	−13422.9	−0.768816	−0.384408	−	0.923163i	$-0.625594\pi$
$673$	−13422.9	−0.768816	−0.384408	+	0.923163i	$0.625594\pi$
$674$	31639.4	1.80817
$675$	0	0
$676$	1647.62	0.0937426
$677$	−1066.77	−0.0605602	−0.0302801	−	0.999541i	$-0.509640\pi$
$677$	−1066.77	−0.0605602	−0.0302801	+	0.999541i	$0.509640\pi$
$678$	9435.36	0.534458
$679$	1543.08	0.0872134
$680$	0	0
$681$	−35080.1	−1.97397
$682$	5640.70	0.316706
$683$	−19090.6	−1.06952	−0.534759	−	0.845005i	$-0.679598\pi$
$683$	−19090.6	−1.06952	−0.534759	+	0.845005i	$0.679598\pi$
$684$	−16445.0	−0.919285
$685$	0	0
$686$	−1388.64	−0.0772865
$687$	−23831.6	−1.32348
$688$	2126.56	0.117841
$689$	−17.8751	−0.000988370 0
$690$	0	0
$691$	−16878.4	−0.929208	−0.464604	−	0.885518i	$-0.653803\pi$
$691$	−16878.4	−0.929208	−0.464604	+	0.885518i	$0.653803\pi$
$692$	11734.0	0.644594
$693$	−740.988	−0.0406173
$694$	9553.57	0.522549
$695$	0	0
$696$	−2917.78	−0.158905
$697$	−11394.8	−0.619236
$698$	13006.8	0.705321
$699$	29914.7	1.61871
$700$	0	0
$701$	30272.6	1.63107	0.815535	−	0.578707i	$-0.196443\pi$
$701$	30272.6	1.63107	0.815535	+	0.578707i	$0.196443\pi$
$702$	−14728.0	−0.791841
$703$	23937.8	1.28426
$704$	−3802.71	−0.203580
$705$	0	0
$706$	−22599.0	−1.20471
$707$	4139.50	0.220201
$708$	−17981.4	−0.954497
$709$	−6593.32	−0.349248	−0.174624	−	0.984635i	$-0.555871\pi$
$709$	−6593.32	−0.349248	−0.174624	+	0.984635i	$0.555871\pi$
$710$	0	0
$711$	3112.06	0.164151
$712$	217.203	0.0114326
$713$	6628.99	0.348187
$714$	−17099.8	−0.896279
$715$	0	0
$716$	3093.46	0.161464
$717$	−4548.15	−0.236895
$718$	−43038.2	−2.23701
$719$	3293.72	0.170842	0.0854208	−	0.996345i	$-0.472777\pi$
$719$	3293.72	0.170842	0.0854208	+	0.996345i	$0.472777\pi$
$720$	0	0
$721$	−3335.46	−0.172287
$722$	36071.6	1.85934
$723$	−33178.1	−1.70665
$724$	−2648.20	−0.135939
$725$	0	0
$726$	33958.3	1.73596
$727$	27757.8	1.41606	0.708032	−	0.706181i	$-0.249583\pi$
$727$	27757.8	1.41606	0.708032	+	0.706181i	$0.249583\pi$
$728$	541.333	0.0275592
$729$	−1048.00	−0.0532439
$730$	0	0
$731$	3237.20	0.163792
$732$	28244.5	1.42616
$733$	38324.6	1.93117	0.965587	−	0.260080i	$-0.0837489\pi$
$733$	38324.6	1.93117	0.965587	+	0.260080i	$0.0837489\pi$
$734$	18277.8	0.919136
$735$	0	0
$736$	−8340.91	−0.417731
$737$	5622.07	0.280993
$738$	7789.30	0.388521
$739$	21957.3	1.09298	0.546490	−	0.837466i	$-0.315964\pi$
$739$	21957.3	1.09298	0.546490	+	0.837466i	$0.315964\pi$
$740$	0	0
$741$	−40100.6	−1.98803
$742$	−10.3547	−0.000512308 0
$743$	−14695.0	−0.725580	−0.362790	−	0.931871i	$-0.618176\pi$
$743$	−14695.0	−0.725580	−0.362790	+	0.931871i	$0.618176\pi$
$744$	−2119.73	−0.104453
$745$	0	0
$746$	46338.5	2.27423
$747$	3042.37	0.149016
$748$	−5260.18	−0.257127
$749$	−1579.09	−0.0770341
$750$	0	0
$751$	−21439.1	−1.04171	−0.520855	−	0.853645i	$-0.674387\pi$
$751$	−21439.1	−1.04171	−0.520855	+	0.853645i	$0.674387\pi$
$752$	25445.1	1.23389
$753$	7880.74	0.381395
$754$	−56007.4	−2.70513
$755$	0	0
$756$	−4367.44	−0.210109
$757$	−23896.8	−1.14735	−0.573675	−	0.819083i	$-0.694483\pi$
$757$	−23896.8	−1.14735	−0.573675	+	0.819083i	$0.694483\pi$
$758$	32981.1	1.58038
$759$	−1428.51	−0.0683158
$760$	0	0
$761$	−24436.9	−1.16404	−0.582022	−	0.813173i	$-0.697738\pi$
$761$	−24436.9	−1.16404	−0.582022	+	0.813173i	$0.697738\pi$
$762$	−45671.9	−2.17128
$763$	11393.6	0.540597
$764$	−1272.24	−0.0602459
$765$	0	0
$766$	−21301.5	−1.00477
$767$	−16061.9	−0.756145
$768$	−23938.9	−1.12477
$769$	−30689.3	−1.43912	−0.719560	−	0.694430i	$-0.755657\pi$
$769$	−30689.3	−1.43912	−0.719560	+	0.694430i	$0.755657\pi$
$770$	0	0
$771$	3335.13	0.155787
$772$	5795.19	0.270173
$773$	−5110.74	−0.237801	−0.118901	−	0.992906i	$-0.537937\pi$
$773$	−5110.74	−0.237801	−0.118901	+	0.992906i	$0.537937\pi$
$774$	−2212.90	−0.102766
$775$	0	0
$776$	−348.459	−0.0161198
$777$	−8710.22	−0.402159
$778$	53469.3	2.46397
$779$	−15479.4	−0.711947
$780$	0	0
$781$	−3365.49	−0.154195
$782$	−12075.9	−0.552217
$783$	21027.6	0.959723
$784$	−2975.47	−0.135545
$785$	0	0
$786$	−38614.4	−1.75233
$787$	6991.30	0.316662	0.158331	−	0.987386i	$-0.449389\pi$
$787$	6991.30	0.316662	0.158331	+	0.987386i	$0.449389\pi$
$788$	6998.78	0.316397
$789$	14811.1	0.668302
$790$	0	0
$791$	2499.28	0.112344
$792$	167.330	0.00750735
$793$	25229.5	1.12979
$794$	−22927.3	−1.02476
$795$	0	0
$796$	3249.36	0.144687
$797$	4020.31	0.178678	0.0893392	−	0.996001i	$-0.471524\pi$
$797$	4020.31	0.178678	0.0893392	+	0.996001i	$0.471524\pi$
$798$	−23229.5	−1.03047
$799$	38734.3	1.71505
$800$	0	0
$801$	2144.64	0.0946030
$802$	−48537.7	−2.13706
$803$	−4755.15	−0.208973
$804$	−45400.8	−1.99149
$805$	0	0
$806$	−40688.7	−1.77816
$807$	51150.0	2.23118
$808$	−934.785	−0.0407000
$809$	−41608.1	−1.80824	−0.904119	−	0.427281i	$-0.859472\pi$
$809$	−41608.1	−1.80824	−0.904119	+	0.427281i	$0.859472\pi$
$810$	0	0
$811$	42271.3	1.83027	0.915133	−	0.403152i	$-0.132086\pi$
$811$	42271.3	1.83027	0.915133	+	0.403152i	$0.132086\pi$
$812$	−16608.4	−0.717785
$813$	−9571.13	−0.412884
$814$	−5234.13	−0.225376
$815$	0	0
$816$	−36640.1	−1.57189
$817$	4397.62	0.188315
$818$	21253.4	0.908443
$819$	5345.05	0.228048
$820$	0	0
$821$	27184.7	1.15561	0.577804	−	0.816176i	$-0.303910\pi$
$821$	27184.7	1.15561	0.577804	+	0.816176i	$0.303910\pi$
$822$	50024.5	2.12263
$823$	−12967.9	−0.549250	−0.274625	−	0.961551i	$-0.588554\pi$
$823$	−12967.9	−0.549250	−0.274625	+	0.961551i	$0.588554\pi$
$824$	753.215	0.0318440
$825$	0	0
$826$	−9304.36	−0.391937
$827$	−33111.2	−1.39225	−0.696124	−	0.717921i	$-0.745094\pi$
$827$	−33111.2	−1.39225	−0.696124	+	0.717921i	$0.745094\pi$
$828$	4225.79	0.177363
$829$	3715.75	0.155673	0.0778367	−	0.996966i	$-0.475199\pi$
$829$	3715.75	0.155673	0.0778367	+	0.996966i	$0.475199\pi$
$830$	0	0
$831$	27116.3	1.13195
$832$	27430.5	1.14301
$833$	−4529.47	−0.188399
$834$	45322.4	1.88176
$835$	0	0
$836$	−7145.77	−0.295624
$837$	15276.3	0.630854
$838$	−60521.0	−2.49482
$839$	−1460.56	−0.0601005	−0.0300502	−	0.999548i	$-0.509567\pi$
$839$	−1460.56	−0.0601005	−0.0300502	+	0.999548i	$0.509567\pi$
$840$	0	0
$841$	55574.2	2.27866
$842$	23644.6	0.967750
$843$	22783.6	0.930851
$844$	25763.2	1.05072
$845$	0	0
$846$	−26478.2	−1.07605
$847$	8995.02	0.364903
$848$	−22.1873	−0.000898483 0
$849$	−20400.3	−0.824660
$850$	0	0
$851$	−6151.19	−0.247779
$852$	27177.9	1.09284
$853$	−36027.4	−1.44613	−0.723067	−	0.690777i	$-0.757268\pi$
$853$	−36027.4	−1.44613	−0.723067	+	0.690777i	$0.757268\pi$
$854$	14614.9	0.585612
$855$	0	0
$856$	356.590	0.0142383
$857$	23500.7	0.936719	0.468359	−	0.883538i	$-0.344845\pi$
$857$	23500.7	0.936719	0.468359	+	0.883538i	$0.344845\pi$
$858$	8768.20	0.348883
$859$	5551.11	0.220491	0.110245	−	0.993904i	$-0.464836\pi$
$859$	5551.11	0.220491	0.110245	+	0.993904i	$0.464836\pi$
$860$	0	0
$861$	5632.46	0.222943
$862$	30117.7	1.19004
$863$	27721.4	1.09345	0.546725	−	0.837312i	$-0.315874\pi$
$863$	27721.4	1.09345	0.546725	+	0.837312i	$0.315874\pi$
$864$	−19221.3	−0.756855
$865$	0	0
$866$	−3553.21	−0.139426
$867$	−23706.6	−0.928624
$868$	−12065.8	−0.471821
$869$	1352.27	0.0527878
$870$	0	0
$871$	−40554.3	−1.57765
$872$	−2572.90	−0.0999192
$873$	−3440.64	−0.133388
$874$	−16404.7	−0.634895
$875$	0	0
$876$	38400.0	1.48107
$877$	47255.6	1.81951	0.909754	−	0.415149i	$-0.136270\pi$
$877$	47255.6	1.81951	0.909754	+	0.415149i	$0.136270\pi$
$878$	−43950.0	−1.68934
$879$	−9449.79	−0.362609
$880$	0	0
$881$	32267.0	1.23394	0.616972	−	0.786985i	$-0.288359\pi$
$881$	32267.0	1.23394	0.616972	+	0.786985i	$0.288359\pi$
$882$	3096.28	0.118205
$883$	5062.08	0.192925	0.0964623	−	0.995337i	$-0.469247\pi$
$883$	5062.08	0.192925	0.0964623	+	0.995337i	$0.469247\pi$
$884$	37943.8	1.44365
$885$	0	0
$886$	−43712.0	−1.65749
$887$	7626.08	0.288679	0.144340	−	0.989528i	$-0.453894\pi$
$887$	7626.08	0.288679	0.144340	+	0.989528i	$0.453894\pi$
$888$	1966.95	0.0743315
$889$	−12097.8	−0.456408
$890$	0	0
$891$	−6150.05	−0.231239
$892$	−8427.89	−0.316353
$893$	52619.2	1.97182
$894$	73950.1	2.76651
$895$	0	0
$896$	1414.66	0.0527460
$897$	10304.4	0.383562
$898$	34905.0	1.29710
$899$	58092.3	2.15516
$900$	0	0
$901$	−33.7750	−0.00124884
$902$	3384.65	0.124941
$903$	−1600.16	−0.0589699
$904$	−564.389	−0.0207647
$905$	0	0
$906$	75778.0	2.77876
$907$	−8546.31	−0.312873	−0.156436	−	0.987688i	$-0.550001\pi$
$907$	−8546.31	−0.312873	−0.156436	+	0.987688i	$0.550001\pi$
$908$	45092.1	1.64806
$909$	−9229.95	−0.336786
$910$	0	0
$911$	8778.92	0.319274	0.159637	−	0.987176i	$-0.448968\pi$
$911$	8778.92	0.319274	0.159637	+	0.987176i	$0.448968\pi$
$912$	−49774.4	−1.80723
$913$	1321.99	0.0479205
$914$	−15327.3	−0.554683
$915$	0	0
$916$	30633.3	1.10497
$917$	−10228.4	−0.368343
$918$	−27828.5	−1.00052
$919$	21626.0	0.776253	0.388126	−	0.921606i	$-0.373122\pi$
$919$	21626.0	0.776253	0.388126	+	0.921606i	$0.373122\pi$
$920$	0	0
$921$	10388.0	0.371658
$922$	23320.1	0.832981
$923$	24276.7	0.865738
$924$	2600.12	0.0925732
$925$	0	0
$926$	43565.8	1.54607
$927$	7437.15	0.263504
$928$	−73094.5	−2.58561
$929$	12262.6	0.433073	0.216536	−	0.976275i	$-0.430524\pi$
$929$	12262.6	0.433073	0.216536	+	0.976275i	$0.430524\pi$
$930$	0	0
$931$	−6153.13	−0.216606
$932$	−38452.5	−1.35145
$933$	56037.9	1.96634
$934$	8718.78	0.305447
$935$	0	0
$936$	−1207.02	−0.0421504
$937$	−13362.6	−0.465889	−0.232945	−	0.972490i	$-0.574836\pi$
$937$	−13362.6	−0.465889	−0.232945	+	0.972490i	$0.574836\pi$
$938$	−23492.3	−0.817751
$939$	−47063.5	−1.63563
$940$	0	0
$941$	−46330.1	−1.60501	−0.802506	−	0.596643i	$-0.796501\pi$
$941$	−46330.1	−1.60501	−0.802506	+	0.596643i	$0.796501\pi$
$942$	−20710.4	−0.716329
$943$	3977.66	0.137360
$944$	−19936.7	−0.687377
$945$	0	0
$946$	−961.563	−0.0330477
$947$	−17387.6	−0.596643	−0.298321	−	0.954465i	$-0.596427\pi$
$947$	−17387.6	−0.596643	−0.298321	+	0.954465i	$0.596427\pi$
$948$	−10920.2	−0.374126
$949$	34300.8	1.17329
$950$	0	0
$951$	50831.1	1.73324
$952$	1022.85	0.0348221
$953$	−42384.6	−1.44068	−0.720342	−	0.693619i	$-0.756015\pi$
$953$	−42384.6	−1.44068	−0.720342	+	0.693619i	$0.756015\pi$
$954$	23.0881	0.000783548 0
$955$	0	0
$956$	5846.20	0.197782
$957$	−12518.6	−0.422851
$958$	−27895.3	−0.940769
$959$	13250.7	0.446182
$960$	0	0
$961$	12412.3	0.416647
$962$	37755.9	1.26539
$963$	3520.92	0.117820
$964$	42647.2	1.42487
$965$	0	0
$966$	5969.16	0.198814
$967$	−15771.6	−0.524489	−0.262245	−	0.965001i	$-0.584463\pi$
$967$	−15771.6	−0.524489	−0.262245	+	0.965001i	$0.584463\pi$
$968$	−2031.26	−0.0674454
$969$	−75769.9	−2.51195
$970$	0	0
$971$	36370.5	1.20204	0.601022	−	0.799232i	$-0.294760\pi$
$971$	36370.5	1.20204	0.601022	+	0.799232i	$0.294760\pi$
$972$	32818.6	1.08298
$973$	12005.2	0.395549
$974$	−76947.5	−2.53137
$975$	0	0
$976$	31315.8	1.02704
$977$	−35122.8	−1.15013	−0.575065	−	0.818108i	$-0.695023\pi$
$977$	−35122.8	−1.15013	−0.575065	+	0.818108i	$0.695023\pi$
$978$	63871.3	2.08832
$979$	931.899	0.0304225
$980$	0	0
$981$	−25404.5	−0.826814
$982$	18340.0	0.595979
$983$	−12798.9	−0.415282	−0.207641	−	0.978205i	$-0.566579\pi$
$983$	−12798.9	−0.415282	−0.207641	+	0.978205i	$0.566579\pi$
$984$	−1271.92	−0.0412068
$985$	0	0
$986$	−105826.	−3.41803
$987$	−19146.5	−0.617466
$988$	51545.4	1.65980
$989$	−1130.03	−0.0363327
$990$	0	0
$991$	−46594.9	−1.49358	−0.746789	−	0.665061i	$-0.768406\pi$
$991$	−46594.9	−1.49358	−0.746789	+	0.665061i	$0.768406\pi$
$992$	−53102.3	−1.69960
$993$	11291.1	0.360838
$994$	14063.0	0.448743
$995$	0	0
$996$	−10675.7	−0.339630
$997$	47534.1	1.50995	0.754975	−	0.655753i	$-0.227649\pi$
$997$	47534.1	1.50995	0.754975	+	0.655753i	$0.227649\pi$
$998$	14656.4	0.464869
$999$	−14175.2	−0.448932

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.a.i.1.5		5
3.2	odd	2		1575.4.a.bq.1.1		5
5.2	odd	4		35.4.b.a.29.9	yes	10
5.3	odd	4		35.4.b.a.29.2	✓	10
5.4	even	2		175.4.a.j.1.1		5
7.6	odd	2		1225.4.a.be.1.5		5
15.2	even	4		315.4.d.c.64.2		10
15.8	even	4		315.4.d.c.64.9		10
15.14	odd	2		1575.4.a.bn.1.5		5
20.3	even	4		560.4.g.f.449.9		10
20.7	even	4		560.4.g.f.449.2		10
35.2	odd	12		245.4.j.e.214.9		20
35.3	even	12		245.4.j.f.79.9		20
35.12	even	12		245.4.j.f.214.9		20
35.13	even	4		245.4.b.d.99.2		10
35.17	even	12		245.4.j.f.79.2		20
35.18	odd	12		245.4.j.e.79.9		20
35.23	odd	12		245.4.j.e.214.2		20
35.27	even	4		245.4.b.d.99.9		10
35.32	odd	12		245.4.j.e.79.2		20
35.33	even	12		245.4.j.f.214.2		20
35.34	odd	2		1225.4.a.bh.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.2	✓	10	5.3	odd	4
35.4.b.a.29.9	yes	10	5.2	odd	4
175.4.a.i.1.5		5	1.1	even	1	trivial
175.4.a.j.1.1		5	5.4	even	2
245.4.b.d.99.2		10	35.13	even	4
245.4.b.d.99.9		10	35.27	even	4
245.4.j.e.79.2		20	35.32	odd	12
245.4.j.e.79.9		20	35.18	odd	12
245.4.j.e.214.2		20	35.23	odd	12
245.4.j.e.214.9		20	35.2	odd	12
245.4.j.f.79.2		20	35.17	even	12
245.4.j.f.79.9		20	35.3	even	12
245.4.j.f.214.2		20	35.33	even	12
245.4.j.f.214.9		20	35.12	even	12
315.4.d.c.64.2		10	15.2	even	4
315.4.d.c.64.9		10	15.8	even	4
560.4.g.f.449.2		10	20.7	even	4
560.4.g.f.449.9		10	20.3	even	4
1225.4.a.be.1.5		5	7.6	odd	2
1225.4.a.bh.1.1		5	35.34	odd	2
1575.4.a.bn.1.5		5	15.14	odd	2
1575.4.a.bq.1.1		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)

\(f(q)\)	\(=\)	\(q+4.04851 q^{2} -6.52749 q^{3} +8.39045 q^{4} -26.4266 q^{6} -7.00000 q^{7} +1.58074 q^{8} +15.6081 q^{9} +6.78210 q^{11} -54.7685 q^{12} -48.9221 q^{13} -28.3396 q^{14} -60.7239 q^{16} -92.4381 q^{17} +63.1894 q^{18} -125.574 q^{19} +45.6924 q^{21} +27.4574 q^{22} +32.2681 q^{23} -10.3183 q^{24} -198.062 q^{26} +74.3607 q^{27} -58.7332 q^{28} +282.778 q^{29} +205.434 q^{31} -258.488 q^{32} -44.2701 q^{33} -374.237 q^{34} +130.959 q^{36} -190.627 q^{37} -508.388 q^{38} +319.338 q^{39} +123.269 q^{41} +184.986 q^{42} -35.0202 q^{43} +56.9049 q^{44} +130.638 q^{46} -419.030 q^{47} +396.375 q^{48} +49.0000 q^{49} +603.388 q^{51} -410.478 q^{52} +0.365379 q^{53} +301.050 q^{54} -11.0652 q^{56} +819.683 q^{57} +1144.83 q^{58} +328.317 q^{59} -515.707 q^{61} +831.704 q^{62} -109.256 q^{63} -560.699 q^{64} -179.228 q^{66} +828.957 q^{67} -775.597 q^{68} -210.630 q^{69} -496.231 q^{71} +24.6723 q^{72} -701.132 q^{73} -771.757 q^{74} -1053.62 q^{76} -47.4747 q^{77} +1292.84 q^{78} +199.388 q^{79} -906.806 q^{81} +499.056 q^{82} +194.923 q^{83} +383.380 q^{84} -141.780 q^{86} -1845.83 q^{87} +10.7208 q^{88} +137.406 q^{89} +342.454 q^{91} +270.744 q^{92} -1340.97 q^{93} -1696.45 q^{94} +1687.27 q^{96} -220.440 q^{97} +198.377 q^{98} +105.855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} - 10 q^{3} + 18 q^{4} + 6 q^{6} - 35 q^{7} - 42 q^{8} + 23 q^{9} + 42 q^{11} - 136 q^{12} - 34 q^{13} + 28 q^{14} + 74 q^{16} - 238 q^{17} + 2 q^{18} - 36 q^{19} + 70 q^{21} - 358 q^{22} - 152 q^{23}+ \cdots + 2652 q^{99}+O(q^{100}) \) 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

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