LMFDB - Embedded newform 425.4.a.i.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,4,Mod(1,425)] 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("425.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.0758117524$
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} - 2x^{4} - 20x^{3} + 38x^{2} + 69x - 126 $ x^5 - 2x^4 - 20x^3 + 38x^2 + 69x - 126
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.90874$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.14668 q^{2} +1.13153 q^{3} +18.4883 q^{4} +5.82364 q^{6} +26.5778 q^{7} +53.9797 q^{8} -25.7196 q^{9} +65.6224 q^{11} +20.9201 q^{12} -44.3601 q^{13} +136.787 q^{14} +129.910 q^{16} -17.0000 q^{17} -132.371 q^{18} -37.6482 q^{19} +30.0737 q^{21} +337.737 q^{22} -194.150 q^{23} +61.0799 q^{24} -228.307 q^{26} -59.6541 q^{27} +491.377 q^{28} +157.743 q^{29} +287.386 q^{31} +236.766 q^{32} +74.2540 q^{33} -87.4935 q^{34} -475.511 q^{36} +96.6691 q^{37} -193.763 q^{38} -50.1950 q^{39} -106.333 q^{41} +154.779 q^{42} +142.879 q^{43} +1213.24 q^{44} -999.227 q^{46} -275.572 q^{47} +146.997 q^{48} +363.378 q^{49} -19.2361 q^{51} -820.141 q^{52} +180.696 q^{53} -307.020 q^{54} +1434.66 q^{56} -42.6003 q^{57} +811.853 q^{58} +284.982 q^{59} -644.300 q^{61} +1479.08 q^{62} -683.571 q^{63} +179.280 q^{64} +382.161 q^{66} -396.688 q^{67} -314.301 q^{68} -219.687 q^{69} -573.342 q^{71} -1388.34 q^{72} -574.739 q^{73} +497.524 q^{74} -696.050 q^{76} +1744.10 q^{77} -258.337 q^{78} -184.587 q^{79} +626.929 q^{81} -547.263 q^{82} -626.402 q^{83} +556.010 q^{84} +735.351 q^{86} +178.492 q^{87} +3542.28 q^{88} -454.081 q^{89} -1178.99 q^{91} -3589.50 q^{92} +325.187 q^{93} -1418.28 q^{94} +267.909 q^{96} -123.922 q^{97} +1870.19 q^{98} -1687.78 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + q^{3} + 34 q^{4} - 5 q^{6} - 10 q^{7} - 30 q^{8} - 30 q^{9} + 126 q^{11} - 15 q^{12} - 83 q^{13} + 90 q^{14} + 322 q^{16} - 85 q^{17} + 97 q^{18} + 55 q^{19} + 6 q^{21} + 240 q^{22} + 2 q^{23}+ \cdots - 858 q^{99}+O(q^{100}) $ 5 * q - 2 * q^2 + q^3 + 34 * q^4 - 5 * q^6 - 10 * q^7 - 30 * q^8 - 30 * q^9 + 126 * q^11 - 15 * q^12 - 83 * q^13 + 90 * q^14 + 322 * q^16 - 85 * q^17 + 97 * q^18 + 55 * q^19 + 6 * q^21 + 240 * q^22 + 2 * q^23 + 155 * q^24 - 395 * q^26 + 163 * q^27 + 1062 * q^28 + 195 * q^29 + 97 * q^31 - 182 * q^32 + 352 * q^33 + 34 * q^34 - 1437 * q^36 + 476 * q^37 + 149 * q^38 + 373 * q^39 + 298 * q^41 + 18 * q^42 - 168 * q^43 + 1136 * q^44 - 2106 * q^46 - 1095 * q^47 + 1193 * q^48 + 737 * q^49 - 17 * q^51 - 281 * q^52 + 1035 * q^53 - 291 * q^54 - 350 * q^56 + 15 * q^57 + 2199 * q^58 + 1163 * q^59 - 351 * q^61 + 1241 * q^62 - 1584 * q^63 + 2042 * q^64 + 356 * q^66 - 1064 * q^67 - 578 * q^68 + 1298 * q^69 + 1393 * q^71 + 269 * q^72 + 163 * q^73 + 1530 * q^74 - 393 * q^76 + 732 * q^77 + 67 * q^78 + 750 * q^79 + 85 * q^81 - 1144 * q^82 - 270 * q^83 + 2422 * q^84 + 3502 * q^86 + 907 * q^87 + 3216 * q^88 + 1179 * q^89 + 282 * q^91 - 758 * q^92 + 3923 * q^93 + 3473 * q^94 - 2045 * q^96 + 2745 * q^97 - 502 * q^98 - 858 * 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 $

$2$	5.14668	1.81962	0.909812	−	0.415020i	$-0.136225\pi$
$2$	5.14668	1.81962	0.909812	+	0.415020i	$0.136225\pi$
$3$	1.13153	0.217764	0.108882	−	0.994055i	$-0.465273\pi$
$3$	1.13153	0.217764	0.108882	+	0.994055i	$0.465273\pi$
$4$	18.4883	2.31103
$5$	0	0
$5$	0	0
$6$	5.82364	0.396248
$7$	26.5778	1.43506	0.717532	−	0.696525i	$-0.245271\pi$
$7$	26.5778	1.43506	0.717532	+	0.696525i	$0.245271\pi$
$8$	53.9797	2.38559
$9$	−25.7196	−0.952579
$10$	0	0
$11$	65.6224	1.79872	0.899359	−	0.437211i	$-0.144034\pi$
$11$	65.6224	1.79872	0.899359	+	0.437211i	$0.144034\pi$
$12$	20.9201	0.503259
$13$	−44.3601	−0.946406	−0.473203	−	0.880954i	$-0.656902\pi$
$13$	−44.3601	−0.946406	−0.473203	+	0.880954i	$0.656902\pi$
$14$	136.787	2.61128
$15$	0	0
$16$	129.910	2.02984
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	−132.371	−1.73334
$19$	−37.6482	−0.454584	−0.227292	−	0.973827i	$-0.572987\pi$
$19$	−37.6482	−0.454584	−0.227292	+	0.973827i	$0.572987\pi$
$20$	0	0
$21$	30.0737	0.312505
$22$	337.737	3.27299
$23$	−194.150	−1.76013	−0.880067	−	0.474849i	$-0.842503\pi$
$23$	−194.150	−1.76013	−0.880067	+	0.474849i	$0.842503\pi$
$24$	61.0799	0.519495
$25$	0	0
$26$	−228.307	−1.72210
$27$	−59.6541	−0.425201
$28$	491.377	3.31648
$29$	157.743	1.01007	0.505037	−	0.863098i	$-0.331479\pi$
$29$	157.743	1.01007	0.505037	+	0.863098i	$0.331479\pi$
$30$	0	0
$31$	287.386	1.66503	0.832517	−	0.554000i	$-0.186899\pi$
$31$	287.386	1.66503	0.832517	+	0.554000i	$0.186899\pi$
$32$	236.766	1.30796
$33$	74.2540	0.391696
$34$	−87.4935	−0.441324
$35$	0	0
$36$	−475.511	−2.20144
$37$	96.6691	0.429522	0.214761	−	0.976667i	$-0.431103\pi$
$37$	96.6691	0.429522	0.214761	+	0.976667i	$0.431103\pi$
$38$	−193.763	−0.827172
$39$	−50.1950	−0.206093
$40$	0	0
$41$	−106.333	−0.405036	−0.202518	−	0.979279i	$-0.564912\pi$
$41$	−106.333	−0.405036	−0.202518	+	0.979279i	$0.564912\pi$
$42$	154.779	0.568642
$43$	142.879	0.506716	0.253358	−	0.967373i	$-0.418465\pi$
$43$	142.879	0.506716	0.253358	+	0.967373i	$0.418465\pi$
$44$	1213.24	4.15690
$45$	0	0
$46$	−999.227	−3.20278
$47$	−275.572	−0.855241	−0.427621	−	0.903958i	$-0.640648\pi$
$47$	−275.572	−0.855241	−0.427621	+	0.903958i	$0.640648\pi$
$48$	146.997	0.442026
$49$	363.378	1.05941
$50$	0	0
$51$	−19.2361	−0.0528155
$52$	−820.141	−2.18717
$53$	180.696	0.468310	0.234155	−	0.972199i	$-0.424768\pi$
$53$	180.696	0.468310	0.234155	+	0.972199i	$0.424768\pi$
$54$	−307.020	−0.773706
$55$	0	0
$56$	1434.66	3.42347
$57$	−42.6003	−0.0989919
$58$	811.853	1.83796
$59$	284.982	0.628839	0.314420	−	0.949284i	$-0.398190\pi$
$59$	284.982	0.628839	0.314420	+	0.949284i	$0.398190\pi$
$60$	0	0
$61$	−644.300	−1.35236	−0.676182	−	0.736735i	$-0.736367\pi$
$61$	−644.300	−1.35236	−0.676182	+	0.736735i	$0.736367\pi$
$62$	1479.08	3.02974
$63$	−683.571	−1.36701
$64$	179.280	0.350156
$65$	0	0
$66$	382.161	0.712739
$67$	−396.688	−0.723330	−0.361665	−	0.932308i	$-0.617792\pi$
$67$	−396.688	−0.723330	−0.361665	+	0.932308i	$0.617792\pi$
$68$	−314.301	−0.560508
$69$	−219.687	−0.383294
$70$	0	0
$71$	−573.342	−0.958354	−0.479177	−	0.877718i	$-0.659065\pi$
$71$	−573.342	−0.958354	−0.479177	+	0.877718i	$0.659065\pi$
$72$	−1388.34	−2.27246
$73$	−574.739	−0.921481	−0.460741	−	0.887535i	$-0.652416\pi$
$73$	−574.739	−0.921481	−0.460741	+	0.887535i	$0.652416\pi$
$74$	497.524	0.781568
$75$	0	0
$76$	−696.050	−1.05056
$77$	1744.10	2.58128
$78$	−258.337	−0.375012
$79$	−184.587	−0.262882	−0.131441	−	0.991324i	$-0.541960\pi$
$79$	−184.587	−0.262882	−0.131441	+	0.991324i	$0.541960\pi$
$80$	0	0
$81$	626.929	0.859985
$82$	−547.263	−0.737013
$83$	−626.402	−0.828392	−0.414196	−	0.910188i	$-0.635937\pi$
$83$	−626.402	−0.828392	−0.414196	+	0.910188i	$0.635937\pi$
$84$	556.010	0.722210
$85$	0	0
$86$	735.351	0.922034
$87$	178.492	0.219958
$88$	3542.28	4.29100
$89$	−454.081	−0.540815	−0.270407	−	0.962746i	$-0.587158\pi$
$89$	−454.081	−0.540815	−0.270407	+	0.962746i	$0.587158\pi$
$90$	0	0
$91$	−1178.99	−1.35815
$92$	−3589.50	−4.06773
$93$	325.187	0.362584
$94$	−1418.28	−1.55622
$95$	0	0
$96$	267.909	0.284826
$97$	−123.922	−0.129715	−0.0648577	−	0.997895i	$-0.520659\pi$
$97$	−123.922	−0.129715	−0.0648577	+	0.997895i	$0.520659\pi$
$98$	1870.19	1.92773
$99$	−1687.78	−1.71342
$100$	0	0
$101$	−440.028	−0.433509	−0.216755	−	0.976226i	$-0.569547\pi$
$101$	−440.028	−0.433509	−0.216755	+	0.976226i	$0.569547\pi$
$102$	−99.0019	−0.0961044
$103$	−432.014	−0.413278	−0.206639	−	0.978417i	$-0.566253\pi$
$103$	−432.014	−0.413278	−0.206639	+	0.978417i	$0.566253\pi$
$104$	−2394.54	−2.25773
$105$	0	0
$106$	929.981	0.852149
$107$	439.055	0.396683	0.198341	−	0.980133i	$-0.436445\pi$
$107$	439.055	0.396683	0.198341	+	0.980133i	$0.436445\pi$
$108$	−1102.90	−0.982654
$109$	820.559	0.721057	0.360529	−	0.932748i	$-0.382596\pi$
$109$	820.559	0.721057	0.360529	+	0.932748i	$0.382596\pi$
$110$	0	0
$111$	109.384	0.0935343
$112$	3452.71	2.91295
$113$	360.934	0.300476	0.150238	−	0.988650i	$-0.451996\pi$
$113$	360.934	0.300476	0.150238	+	0.988650i	$0.451996\pi$
$114$	−219.250	−0.180128
$115$	0	0
$116$	2916.40	2.33432
$117$	1140.92	0.901526
$118$	1466.71	1.14425
$119$	−451.822	−0.348054
$120$	0	0
$121$	2975.30	2.23539
$122$	−3316.00	−2.46079
$123$	−120.320	−0.0882022
$124$	5313.27	3.84795
$125$	0	0
$126$	−3518.12	−2.48745
$127$	−1217.76	−0.850853	−0.425427	−	0.904993i	$-0.639876\pi$
$127$	−1217.76	−0.850853	−0.425427	+	0.904993i	$0.639876\pi$
$128$	−971.435	−0.670809
$129$	161.672	0.110345
$130$	0	0
$131$	506.759	0.337983	0.168991	−	0.985618i	$-0.445949\pi$
$131$	506.759	0.337983	0.168991	+	0.985618i	$0.445949\pi$
$132$	1372.83	0.905222
$133$	−1000.61	−0.652357
$134$	−2041.62	−1.31619
$135$	0	0
$136$	−917.655	−0.578590
$137$	143.152	0.0892725	0.0446363	−	0.999003i	$-0.485787\pi$
$137$	143.152	0.0892725	0.0446363	+	0.999003i	$0.485787\pi$
$138$	−1130.66	−0.697451
$139$	−1840.90	−1.12333	−0.561667	−	0.827364i	$-0.689840\pi$
$139$	−1840.90	−1.12333	−0.561667	+	0.827364i	$0.689840\pi$
$140$	0	0
$141$	−311.819	−0.186241
$142$	−2950.80	−1.74384
$143$	−2911.02	−1.70232
$144$	−3341.23	−1.93358
$145$	0	0
$146$	−2958.00	−1.67675
$147$	411.175	0.230702
$148$	1787.24	0.992639
$149$	−32.8923	−0.0180849	−0.00904243	−	0.999959i	$-0.502878\pi$
$149$	−32.8923	−0.0180849	−0.00904243	+	0.999959i	$0.502878\pi$
$150$	0	0
$151$	−2682.68	−1.44579	−0.722893	−	0.690960i	$-0.757188\pi$
$151$	−2682.68	−1.44579	−0.722893	+	0.690960i	$0.757188\pi$
$152$	−2032.24	−1.08445
$153$	437.234	0.231034
$154$	8976.30	4.69696
$155$	0	0
$156$	−928.018	−0.476288
$157$	3802.72	1.93306	0.966529	−	0.256557i	$-0.0825883\pi$
$157$	3802.72	1.93306	0.966529	+	0.256557i	$0.0825883\pi$
$158$	−950.012	−0.478347
$159$	204.463	0.101981
$160$	0	0
$161$	−5160.08	−2.52591
$162$	3226.60	1.56485
$163$	−1877.17	−0.902033	−0.451016	−	0.892516i	$-0.648938\pi$
$163$	−1877.17	−0.902033	−0.451016	+	0.892516i	$0.648938\pi$
$164$	−1965.92	−0.936051
$165$	0	0
$166$	−3223.89	−1.50736
$167$	−880.440	−0.407967	−0.203983	−	0.978974i	$-0.565389\pi$
$167$	−880.440	−0.407967	−0.203983	+	0.978974i	$0.565389\pi$
$168$	1623.37	0.745509
$169$	−229.183	−0.104316
$170$	0	0
$171$	968.298	0.433027
$172$	2641.58	1.17104
$173$	478.986	0.210501	0.105250	−	0.994446i	$-0.466436\pi$
$173$	478.986	0.210501	0.105250	+	0.994446i	$0.466436\pi$
$174$	918.639	0.400240
$175$	0	0
$176$	8525.00	3.65111
$177$	322.467	0.136938
$178$	−2337.01	−0.984079
$179$	4787.74	1.99918	0.999588	−	0.0286889i	$-0.00913321\pi$
$179$	4787.74	1.99918	0.999588	+	0.0286889i	$0.00913321\pi$
$180$	0	0
$181$	2740.64	1.12547	0.562735	−	0.826638i	$-0.309749\pi$
$181$	2740.64	1.12547	0.562735	+	0.826638i	$0.309749\pi$
$182$	−6067.89	−2.47133
$183$	−729.047	−0.294496
$184$	−10480.2	−4.19896
$185$	0	0
$186$	1673.63	0.659767
$187$	−1115.58	−0.436253
$188$	−5094.85	−1.97649
$189$	−1585.47	−0.610191
$190$	0	0
$191$	4471.93	1.69412	0.847061	−	0.531495i	$-0.178370\pi$
$191$	4471.93	1.69412	0.847061	+	0.531495i	$0.178370\pi$
$192$	202.861	0.0762512
$193$	1752.40	0.653580	0.326790	−	0.945097i	$-0.394033\pi$
$193$	1752.40	0.653580	0.326790	+	0.945097i	$0.394033\pi$
$194$	−637.788	−0.236033
$195$	0	0
$196$	6718.23	2.44834
$197$	832.700	0.301154	0.150577	−	0.988598i	$-0.451887\pi$
$197$	832.700	0.301154	0.150577	+	0.988598i	$0.451887\pi$
$198$	−8686.48	−3.11778
$199$	2142.23	0.763110	0.381555	−	0.924346i	$-0.375389\pi$
$199$	2142.23	0.763110	0.381555	+	0.924346i	$0.375389\pi$
$200$	0	0
$201$	−448.866	−0.157515
$202$	−2264.68	−0.788824
$203$	4192.46	1.44952
$204$	−355.642	−0.122058
$205$	0	0
$206$	−2223.44	−0.752010
$207$	4993.47	1.67667
$208$	−5762.81	−1.92105
$209$	−2470.57	−0.817668
$210$	0	0
$211$	−278.427	−0.0908424	−0.0454212	−	0.998968i	$-0.514463\pi$
$211$	−278.427	−0.0908424	−0.0454212	+	0.998968i	$0.514463\pi$
$212$	3340.75	1.08228
$213$	−648.756	−0.208695
$214$	2259.67	0.721813
$215$	0	0
$216$	−3220.11	−1.01435
$217$	7638.08	2.38943
$218$	4223.15	1.31205
$219$	−650.337	−0.200665
$220$	0	0
$221$	754.121	0.229537
$222$	562.966	0.170197
$223$	222.691	0.0668722	0.0334361	−	0.999441i	$-0.489355\pi$
$223$	222.691	0.0668722	0.0334361	+	0.999441i	$0.489355\pi$
$224$	6292.72	1.87701
$225$	0	0
$226$	1857.61	0.546754
$227$	−6190.22	−1.80995	−0.904977	−	0.425460i	$-0.860112\pi$
$227$	−6190.22	−1.80995	−0.904977	+	0.425460i	$0.860112\pi$
$228$	−787.605	−0.228774
$229$	1590.85	0.459066	0.229533	−	0.973301i	$-0.426280\pi$
$229$	1590.85	0.459066	0.229533	+	0.973301i	$0.426280\pi$
$230$	0	0
$231$	1973.51	0.562109
$232$	8514.93	2.40962
$233$	5133.80	1.44346	0.721731	−	0.692174i	$-0.243347\pi$
$233$	5133.80	1.44346	0.721731	+	0.692174i	$0.243347\pi$
$234$	5871.97	1.64044
$235$	0	0
$236$	5268.82	1.45327
$237$	−208.867	−0.0572463
$238$	−2325.38	−0.633328
$239$	−3291.68	−0.890882	−0.445441	−	0.895311i	$-0.646953\pi$
$239$	−3291.68	−0.890882	−0.445441	+	0.895311i	$0.646953\pi$
$240$	0	0
$241$	178.525	0.0477171	0.0238585	−	0.999715i	$-0.492405\pi$
$241$	178.525	0.0477171	0.0238585	+	0.999715i	$0.492405\pi$
$242$	15312.9	4.06757
$243$	2320.05	0.612475
$244$	−11912.0	−3.12536
$245$	0	0
$246$	−619.247	−0.160495
$247$	1670.08	0.430221
$248$	15513.0	3.97208
$249$	−708.795	−0.180394
$250$	0	0
$251$	3282.42	0.825436	0.412718	−	0.910859i	$-0.364580\pi$
$251$	3282.42	0.825436	0.412718	+	0.910859i	$0.364580\pi$
$252$	−12638.0	−3.15921
$253$	−12740.6	−3.16599
$254$	−6267.39	−1.54823
$255$	0	0
$256$	−6433.90	−1.57078
$257$	−7549.48	−1.83239	−0.916194	−	0.400734i	$-0.868755\pi$
$257$	−7549.48	−1.83239	−0.916194	+	0.400734i	$0.868755\pi$
$258$	832.074	0.200786
$259$	2569.25	0.616391
$260$	0	0
$261$	−4057.09	−0.962176
$262$	2608.12	0.615002
$263$	6855.56	1.60735	0.803673	−	0.595072i	$-0.202876\pi$
$263$	6855.56	1.60735	0.803673	+	0.595072i	$0.202876\pi$
$264$	4008.21	0.934425
$265$	0	0
$266$	−5149.79	−1.18705
$267$	−513.808	−0.117770
$268$	−7334.06	−1.67164
$269$	−7781.23	−1.76368	−0.881840	−	0.471549i	$-0.843695\pi$
$269$	−7781.23	−1.76368	−0.881840	+	0.471549i	$0.843695\pi$
$270$	0	0
$271$	2804.03	0.628535	0.314267	−	0.949335i	$-0.398241\pi$
$271$	2804.03	0.628535	0.314267	+	0.949335i	$0.398241\pi$
$272$	−2208.47	−0.492309
$273$	−1334.07	−0.295757
$274$	736.759	0.162442
$275$	0	0
$276$	−4061.64	−0.885804
$277$	953.981	0.206928	0.103464	−	0.994633i	$-0.467007\pi$
$277$	953.981	0.206928	0.103464	+	0.994633i	$0.467007\pi$
$278$	−9474.53	−2.04404
$279$	−7391.46	−1.58608
$280$	0	0
$281$	939.798	0.199515	0.0997574	−	0.995012i	$-0.468193\pi$
$281$	939.798	0.199515	0.0997574	+	0.995012i	$0.468193\pi$
$282$	−1604.83	−0.338888
$283$	4433.05	0.931157	0.465579	−	0.885007i	$-0.345846\pi$
$283$	4433.05	0.931157	0.465579	+	0.885007i	$0.345846\pi$
$284$	−10600.1	−2.21479
$285$	0	0
$286$	−14982.1	−3.09758
$287$	−2826.10	−0.581253
$288$	−6089.54	−1.24594
$289$	289.000	0.0588235
$290$	0	0
$291$	−140.222	−0.0282473
$292$	−10625.9	−2.12957
$293$	−633.552	−0.126323	−0.0631613	−	0.998003i	$-0.520118\pi$
$293$	−633.552	−0.126323	−0.0631613	+	0.998003i	$0.520118\pi$
$294$	2116.18	0.419790
$295$	0	0
$296$	5218.17	1.02466
$297$	−3914.64	−0.764817
$298$	−169.286	−0.0329076
$299$	8612.51	1.66580
$300$	0	0
$301$	3797.40	0.727171
$302$	−13806.9	−2.63079
$303$	−497.907	−0.0944026
$304$	−4890.87	−0.922733
$305$	0	0
$306$	2250.30	0.420396
$307$	3160.35	0.587527	0.293764	−	0.955878i	$-0.405092\pi$
$307$	3160.35	0.587527	0.293764	+	0.955878i	$0.405092\pi$
$308$	32245.3	5.96542
$309$	−488.839	−0.0899970
$310$	0	0
$311$	−7342.39	−1.33874	−0.669371	−	0.742929i	$-0.733436\pi$
$311$	−7342.39	−1.33874	−0.669371	+	0.742929i	$0.733436\pi$
$312$	−2709.51	−0.491653
$313$	2240.55	0.404612	0.202306	−	0.979322i	$-0.435156\pi$
$313$	2240.55	0.404612	0.202306	+	0.979322i	$0.435156\pi$
$314$	19571.4	3.51744
$315$	0	0
$316$	−3412.70	−0.607530
$317$	−465.566	−0.0824882	−0.0412441	−	0.999149i	$-0.513132\pi$
$317$	−465.566	−0.0824882	−0.0412441	+	0.999149i	$0.513132\pi$
$318$	1052.31	0.185567
$319$	10351.5	1.81684
$320$	0	0
$321$	496.806	0.0863831
$322$	−26557.2	−4.59620
$323$	640.020	0.110253
$324$	11590.8	1.98745
$325$	0	0
$326$	−9661.19	−1.64136
$327$	928.490	0.157020
$328$	−5739.84	−0.966248
$329$	−7324.10	−1.22733
$330$	0	0
$331$	7412.39	1.23088	0.615441	−	0.788183i	$-0.288978\pi$
$331$	7412.39	1.23088	0.615441	+	0.788183i	$0.288978\pi$
$332$	−11581.1	−1.91444
$333$	−2486.29	−0.409153
$334$	−4531.34	−0.742346
$335$	0	0
$336$	3906.86	0.634336
$337$	−6421.72	−1.03802	−0.519011	−	0.854768i	$-0.673700\pi$
$337$	−6421.72	−1.03802	−0.519011	+	0.854768i	$0.673700\pi$
$338$	−1179.53	−0.189817
$339$	408.409	0.0654329
$340$	0	0
$341$	18859.0	2.99493
$342$	4983.52	0.787947
$343$	541.606	0.0852593
$344$	7712.55	1.20882
$345$	0	0
$346$	2465.18	0.383032
$347$	6678.70	1.03323	0.516616	−	0.856217i	$-0.327192\pi$
$347$	6678.70	1.03323	0.516616	+	0.856217i	$0.327192\pi$
$348$	3300.00	0.508330
$349$	−5086.82	−0.780204	−0.390102	−	0.920772i	$-0.627560\pi$
$349$	−5086.82	−0.780204	−0.390102	+	0.920772i	$0.627560\pi$
$350$	0	0
$351$	2646.26	0.402413
$352$	15537.2	2.35265
$353$	372.107	0.0561055	0.0280528	−	0.999606i	$-0.491069\pi$
$353$	372.107	0.0561055	0.0280528	+	0.999606i	$0.491069\pi$
$354$	1659.63	0.249177
$355$	0	0
$356$	−8395.17	−1.24984
$357$	−511.252	−0.0757937
$358$	24641.0	3.63775
$359$	7167.49	1.05372	0.526860	−	0.849952i	$-0.323369\pi$
$359$	7167.49	1.05372	0.526860	+	0.849952i	$0.323369\pi$
$360$	0	0
$361$	−5441.61	−0.793353
$362$	14105.2	2.04793
$363$	3366.65	0.486787
$364$	−21797.5	−3.13874
$365$	0	0
$366$	−3752.17	−0.535872
$367$	9884.60	1.40592	0.702959	−	0.711230i	$-0.251862\pi$
$367$	9884.60	1.40592	0.702959	+	0.711230i	$0.251862\pi$
$368$	−25222.0	−3.57279
$369$	2734.85	0.385829
$370$	0	0
$371$	4802.49	0.672056
$372$	6012.14	0.837944
$373$	9348.37	1.29770	0.648848	−	0.760918i	$-0.275251\pi$
$373$	9348.37	1.29770	0.648848	+	0.760918i	$0.275251\pi$
$374$	−5741.53	−0.793817
$375$	0	0
$376$	−14875.3	−2.04025
$377$	−6997.50	−0.955940
$378$	−8159.91	−1.11032
$379$	4657.35	0.631219	0.315610	−	0.948889i	$-0.397791\pi$
$379$	4657.35	0.631219	0.315610	+	0.948889i	$0.397791\pi$
$380$	0	0
$381$	−1377.93	−0.185285
$382$	23015.6	3.08267
$383$	−9579.15	−1.27799	−0.638997	−	0.769209i	$-0.720651\pi$
$383$	−9579.15	−1.27799	−0.638997	+	0.769209i	$0.720651\pi$
$384$	−1099.21	−0.146078
$385$	0	0
$386$	9019.06	1.18927
$387$	−3674.79	−0.482687
$388$	−2291.11	−0.299777
$389$	1777.52	0.231681	0.115841	−	0.993268i	$-0.463044\pi$
$389$	1777.52	0.231681	0.115841	+	0.993268i	$0.463044\pi$
$390$	0	0
$391$	3300.55	0.426895
$392$	19615.0	2.52732
$393$	573.415	0.0736004
$394$	4285.64	0.547988
$395$	0	0
$396$	−31204.2	−3.95977
$397$	14297.0	1.80742	0.903710	−	0.428145i	$-0.140833\pi$
$397$	14297.0	1.80742	0.903710	+	0.428145i	$0.140833\pi$
$398$	11025.4	1.38857
$399$	−1132.22	−0.142060
$400$	0	0
$401$	7363.27	0.916967	0.458484	−	0.888703i	$-0.348393\pi$
$401$	7363.27	0.916967	0.458484	+	0.888703i	$0.348393\pi$
$402$	−2310.17	−0.286618
$403$	−12748.5	−1.57580
$404$	−8135.35	−1.00185
$405$	0	0
$406$	21577.2	2.63759
$407$	6343.66	0.772588
$408$	−1038.36	−0.125996
$409$	5342.48	0.645890	0.322945	−	0.946418i	$-0.395327\pi$
$409$	5342.48	0.645890	0.322945	+	0.946418i	$0.395327\pi$
$410$	0	0
$411$	161.982	0.0194403
$412$	−7987.19	−0.955099
$413$	7574.19	0.902425
$414$	25699.8	3.05090
$415$	0	0
$416$	−10503.0	−1.23786
$417$	−2083.04	−0.244621
$418$	−12715.2	−1.48785
$419$	−5069.29	−0.591053	−0.295526	−	0.955335i	$-0.595495\pi$
$419$	−5069.29	−0.591053	−0.295526	+	0.955335i	$0.595495\pi$
$420$	0	0
$421$	4309.77	0.498920	0.249460	−	0.968385i	$-0.419747\pi$
$421$	4309.77	0.498920	0.249460	+	0.968385i	$0.419747\pi$
$422$	−1432.98	−0.165299
$423$	7087.62	0.814685
$424$	9753.89	1.11720
$425$	0	0
$426$	−3338.94	−0.379746
$427$	−17124.1	−1.94073
$428$	8117.36	0.916747
$429$	−3293.91	−0.370703
$430$	0	0
$431$	7402.13	0.827258	0.413629	−	0.910446i	$-0.364261\pi$
$431$	7402.13	0.827258	0.413629	+	0.910446i	$0.364261\pi$
$432$	−7749.65	−0.863091
$433$	7353.85	0.816174	0.408087	−	0.912943i	$-0.366196\pi$
$433$	7353.85	0.816174	0.408087	+	0.912943i	$0.366196\pi$
$434$	39310.7	4.34787
$435$	0	0
$436$	15170.7	1.66639
$437$	7309.41	0.800129
$438$	−3347.07	−0.365135
$439$	−13827.3	−1.50328	−0.751641	−	0.659572i	$-0.770737\pi$
$439$	−13827.3	−1.50328	−0.751641	+	0.659572i	$0.770737\pi$
$440$	0	0
$441$	−9345.95	−1.00917
$442$	3881.22	0.417671
$443$	1253.49	0.134436	0.0672182	−	0.997738i	$-0.478588\pi$
$443$	1253.49	0.134436	0.0672182	+	0.997738i	$0.478588\pi$
$444$	2022.33	0.216161
$445$	0	0
$446$	1146.12	0.121682
$447$	−37.2188	−0.00393823
$448$	4764.85	0.502496
$449$	−2574.10	−0.270555	−0.135278	−	0.990808i	$-0.543193\pi$
$449$	−2574.10	−0.270555	−0.135278	+	0.990808i	$0.543193\pi$
$450$	0	0
$451$	−6977.85	−0.728545
$452$	6673.05	0.694411
$453$	−3035.55	−0.314840
$454$	−31859.1	−3.29344
$455$	0	0
$456$	−2299.55	−0.236154
$457$	−8819.42	−0.902746	−0.451373	−	0.892335i	$-0.649066\pi$
$457$	−8819.42	−0.902746	−0.451373	+	0.892335i	$0.649066\pi$
$458$	8187.58	0.835328
$459$	1014.12	0.103126
$460$	0	0
$461$	5228.41	0.528224	0.264112	−	0.964492i	$-0.414921\pi$
$461$	5228.41	0.528224	0.264112	+	0.964492i	$0.414921\pi$
$462$	10157.0	1.02283
$463$	6194.69	0.621797	0.310898	−	0.950443i	$-0.399370\pi$
$463$	6194.69	0.621797	0.310898	+	0.950443i	$0.399370\pi$
$464$	20492.4	2.05029
$465$	0	0
$466$	26422.0	2.62656
$467$	12457.6	1.23441	0.617205	−	0.786802i	$-0.288265\pi$
$467$	12457.6	1.23441	0.617205	+	0.786802i	$0.288265\pi$
$468$	21093.7	2.08346
$469$	−10543.1	−1.03803
$470$	0	0
$471$	4302.91	0.420950
$472$	15383.2	1.50015
$473$	9376.05	0.911440
$474$	−1074.97	−0.104167
$475$	0	0
$476$	−8353.41	−0.804365
$477$	−4647.42	−0.446102
$478$	−16941.2	−1.62107
$479$	2884.84	0.275181	0.137590	−	0.990489i	$-0.456064\pi$
$479$	2884.84	0.275181	0.137590	+	0.990489i	$0.456064\pi$
$480$	0	0
$481$	−4288.25	−0.406502
$482$	918.811	0.0868271
$483$	−5838.80	−0.550051
$484$	55008.2	5.16605
$485$	0	0
$486$	11940.6	1.11447
$487$	16028.1	1.49138	0.745690	−	0.666294i	$-0.232120\pi$
$487$	16028.1	1.49138	0.745690	+	0.666294i	$0.232120\pi$
$488$	−34779.1	−3.22618
$489$	−2124.08	−0.196430
$490$	0	0
$491$	−16548.5	−1.52102	−0.760511	−	0.649325i	$-0.775051\pi$
$491$	−16548.5	−1.52102	−0.760511	+	0.649325i	$0.775051\pi$
$492$	−2224.50	−0.203838
$493$	−2681.63	−0.244979
$494$	8595.35	0.782840
$495$	0	0
$496$	37334.3	3.37975
$497$	−15238.2	−1.37530
$498$	−3647.94	−0.328249
$499$	−11928.0	−1.07008	−0.535039	−	0.844828i	$-0.679703\pi$
$499$	−11928.0	−1.07008	−0.535039	+	0.844828i	$0.679703\pi$
$500$	0	0
$501$	−996.248	−0.0888404
$502$	16893.5	1.50198
$503$	11152.5	0.988597	0.494298	−	0.869292i	$-0.335425\pi$
$503$	11152.5	0.988597	0.494298	+	0.869292i	$0.335425\pi$
$504$	−36898.9	−3.26113
$505$	0	0
$506$	−65571.7	−5.76091
$507$	−259.328	−0.0227163
$508$	−22514.2	−1.96635
$509$	−3563.51	−0.310314	−0.155157	−	0.987890i	$-0.549588\pi$
$509$	−3563.51	−0.310314	−0.155157	+	0.987890i	$0.549588\pi$
$510$	0	0
$511$	−15275.3	−1.32239
$512$	−25341.7	−2.18741
$513$	2245.87	0.193290
$514$	−38854.7	−3.33426
$515$	0	0
$516$	2989.04	0.255010
$517$	−18083.7	−1.53834
$518$	13223.1	1.12160
$519$	541.989	0.0458394
$520$	0	0
$521$	15600.2	1.31181	0.655907	−	0.754842i	$-0.272286\pi$
$521$	15600.2	1.31181	0.655907	+	0.754842i	$0.272286\pi$
$522$	−20880.5	−1.75080
$523$	6002.01	0.501816	0.250908	−	0.968011i	$-0.419271\pi$
$523$	6002.01	0.501816	0.250908	+	0.968011i	$0.419271\pi$
$524$	9369.10	0.781089
$525$	0	0
$526$	35283.3	2.92476
$527$	−4885.56	−0.403830
$528$	9646.32	0.795080
$529$	25527.3	2.09807
$530$	0	0
$531$	−7329.63	−0.599019
$532$	−18499.5	−1.50762
$533$	4716.95	0.383328
$534$	−2644.40	−0.214297
$535$	0	0
$536$	−21413.1	−1.72557
$537$	5417.49	0.435348
$538$	−40047.4	−3.20923
$539$	23845.7	1.90558
$540$	0	0
$541$	−4717.87	−0.374930	−0.187465	−	0.982271i	$-0.560027\pi$
$541$	−4717.87	−0.374930	−0.187465	+	0.982271i	$0.560027\pi$
$542$	14431.4	1.14370
$543$	3101.12	0.245087
$544$	−4025.02	−0.317227
$545$	0	0
$546$	−6866.03	−0.538166
$547$	17609.0	1.37643	0.688216	−	0.725506i	$-0.258394\pi$
$547$	17609.0	1.37643	0.688216	+	0.725506i	$0.258394\pi$
$548$	2646.64	0.206312
$549$	16571.2	1.28823
$550$	0	0
$551$	−5938.75	−0.459164
$552$	−11858.7	−0.914381
$553$	−4905.92	−0.377253
$554$	4909.83	0.376532
$555$	0	0
$556$	−34035.1	−2.59606
$557$	−6661.09	−0.506714	−0.253357	−	0.967373i	$-0.581535\pi$
$557$	−6661.09	−0.506714	−0.253357	+	0.967373i	$0.581535\pi$
$558$	−38041.4	−2.88606
$559$	−6338.11	−0.479559
$560$	0	0
$561$	−1262.32	−0.0950002
$562$	4836.83	0.363042
$563$	−2807.49	−0.210163	−0.105081	−	0.994464i	$-0.533510\pi$
$563$	−2807.49	−0.210163	−0.105081	+	0.994464i	$0.533510\pi$
$564$	−5765.00	−0.430408
$565$	0	0
$566$	22815.5	1.69436
$567$	16662.4	1.23414
$568$	−30948.8	−2.28624
$569$	−22223.9	−1.63739	−0.818693	−	0.574232i	$-0.805301\pi$
$569$	−22223.9	−1.63739	−0.818693	+	0.574232i	$0.805301\pi$
$570$	0	0
$571$	10261.5	0.752067	0.376033	−	0.926606i	$-0.377288\pi$
$571$	10261.5	0.752067	0.376033	+	0.926606i	$0.377288\pi$
$572$	−53819.6	−3.93411
$573$	5060.14	0.368919
$574$	−14545.0	−1.05766
$575$	0	0
$576$	−4611.01	−0.333551
$577$	6559.87	0.473295	0.236647	−	0.971596i	$-0.423951\pi$
$577$	6559.87	0.473295	0.236647	+	0.971596i	$0.423951\pi$
$578$	1487.39	0.107037
$579$	1982.91	0.142326
$580$	0	0
$581$	−16648.4	−1.18880
$582$	−721.678	−0.0513995
$583$	11857.7	0.842358
$584$	−31024.2	−2.19827
$585$	0	0
$586$	−3260.69	−0.229860
$587$	−10771.7	−0.757406	−0.378703	−	0.925518i	$-0.623630\pi$
$587$	−10771.7	−0.757406	−0.378703	+	0.925518i	$0.623630\pi$
$588$	7601.91	0.533159
$589$	−10819.6	−0.756898
$590$	0	0
$591$	942.228	0.0655805
$592$	12558.3	0.871860
$593$	−16569.7	−1.14745	−0.573724	−	0.819049i	$-0.694502\pi$
$593$	−16569.7	−1.14745	−0.573724	+	0.819049i	$0.694502\pi$
$594$	−20147.4	−1.39168
$595$	0	0
$596$	−608.122	−0.0417947
$597$	2424.01	0.166178
$598$	44325.8	3.03113
$599$	−4532.83	−0.309193	−0.154596	−	0.987978i	$-0.549408\pi$
$599$	−4532.83	−0.309193	−0.154596	+	0.987978i	$0.549408\pi$
$600$	0	0
$601$	13580.6	0.921737	0.460869	−	0.887468i	$-0.347538\pi$
$601$	13580.6	0.921737	0.460869	+	0.887468i	$0.347538\pi$
$602$	19544.0	1.32318
$603$	10202.7	0.689029
$604$	−49598.1	−3.34126
$605$	0	0
$606$	−2562.56	−0.171777
$607$	−4440.40	−0.296920	−0.148460	−	0.988918i	$-0.547432\pi$
$607$	−4440.40	−0.296920	−0.148460	+	0.988918i	$0.547432\pi$
$608$	−8913.82	−0.594578
$609$	4743.91	0.315654
$610$	0	0
$611$	12224.4	0.809405
$612$	8083.69	0.533928
$613$	1138.84	0.0750363	0.0375182	−	0.999296i	$-0.488055\pi$
$613$	1138.84	0.0750363	0.0375182	+	0.999296i	$0.488055\pi$
$614$	16265.3	1.06908
$615$	0	0
$616$	94145.9	6.15786
$617$	6105.12	0.398351	0.199176	−	0.979964i	$-0.436174\pi$
$617$	6105.12	0.398351	0.199176	+	0.979964i	$0.436174\pi$
$618$	−2515.89	−0.163761
$619$	24526.9	1.59260	0.796300	−	0.604901i	$-0.206787\pi$
$619$	24526.9	1.59260	0.796300	+	0.604901i	$0.206787\pi$
$620$	0	0
$621$	11581.8	0.748411
$622$	−37788.9	−2.43601
$623$	−12068.5	−0.776104
$624$	−6520.82	−0.418336
$625$	0	0
$626$	11531.4	0.736242
$627$	−2795.53	−0.178059
$628$	70305.7	4.46736
$629$	−1643.37	−0.104174
$630$	0	0
$631$	−18071.6	−1.14013	−0.570064	−	0.821600i	$-0.693081\pi$
$631$	−18071.6	−1.14013	−0.570064	+	0.821600i	$0.693081\pi$
$632$	−9963.97	−0.627129
$633$	−315.050	−0.0197822
$634$	−2396.11	−0.150098
$635$	0	0
$636$	3780.17	0.235682
$637$	−16119.5	−1.00263
$638$	53275.7	3.30597
$639$	14746.1	0.912908
$640$	0	0
$641$	14761.2	0.909569	0.454784	−	0.890602i	$-0.349716\pi$
$641$	14761.2	0.909569	0.454784	+	0.890602i	$0.349716\pi$
$642$	2556.90	0.157185
$643$	−5798.96	−0.355659	−0.177829	−	0.984061i	$-0.556908\pi$
$643$	−5798.96	−0.355659	−0.177829	+	0.984061i	$0.556908\pi$
$644$	−95400.9	−5.83746
$645$	0	0
$646$	3293.97	0.200619
$647$	−19687.2	−1.19627	−0.598134	−	0.801396i	$-0.704091\pi$
$647$	−19687.2	−1.19627	−0.598134	+	0.801396i	$0.704091\pi$
$648$	33841.5	2.05157
$649$	18701.2	1.13110
$650$	0	0
$651$	8642.75	0.520332
$652$	−34705.6	−2.08463
$653$	−32380.8	−1.94052	−0.970260	−	0.242067i	$-0.922175\pi$
$653$	−32380.8	−1.94052	−0.970260	+	0.242067i	$0.922175\pi$
$654$	4778.64	0.285718
$655$	0	0
$656$	−13813.7	−0.822158
$657$	14782.1	0.877784
$658$	−37694.7	−2.23327
$659$	−8565.95	−0.506346	−0.253173	−	0.967421i	$-0.581474\pi$
$659$	−8565.95	−0.506346	−0.253173	+	0.967421i	$0.581474\pi$
$660$	0	0
$661$	−4061.88	−0.239015	−0.119507	−	0.992833i	$-0.538131\pi$
$661$	−4061.88	−0.239015	−0.119507	+	0.992833i	$0.538131\pi$
$662$	38149.2	2.23974
$663$	853.314	0.0499849
$664$	−33813.0	−1.97620
$665$	0	0
$666$	−12796.1	−0.744505
$667$	−30625.8	−1.77787
$668$	−16277.8	−0.942825
$669$	251.982	0.0145623
$670$	0	0
$671$	−42280.5	−2.43252
$672$	7120.42	0.408745
$673$	1784.75	0.102224	0.0511121	−	0.998693i	$-0.483723\pi$
$673$	1784.75	0.102224	0.0511121	+	0.998693i	$0.483723\pi$
$674$	−33050.5	−1.88881
$675$	0	0
$676$	−4237.20	−0.241079
$677$	13057.7	0.741284	0.370642	−	0.928776i	$-0.379138\pi$
$677$	13057.7	0.741284	0.370642	+	0.928776i	$0.379138\pi$
$678$	2101.95	0.119063
$679$	−3293.58	−0.186150
$680$	0	0
$681$	−7004.45	−0.394143
$682$	97060.9	5.44964
$683$	12956.6	0.725875	0.362937	−	0.931814i	$-0.381774\pi$
$683$	12956.6	0.725875	0.362937	+	0.931814i	$0.381774\pi$
$684$	17902.2	1.00074
$685$	0	0
$686$	2787.47	0.155140
$687$	1800.10	0.0999681
$688$	18561.4	1.02855
$689$	−8015.67	−0.443211
$690$	0	0
$691$	−26271.1	−1.44631	−0.723156	−	0.690685i	$-0.757309\pi$
$691$	−26271.1	−1.44631	−0.723156	+	0.690685i	$0.757309\pi$
$692$	8855.61	0.486474
$693$	−44857.5	−2.45887
$694$	34373.1	1.88009
$695$	0	0
$696$	9634.93	0.524729
$697$	1807.67	0.0982356
$698$	−26180.2	−1.41968
$699$	5809.07	0.314334
$700$	0	0
$701$	−24449.3	−1.31732	−0.658658	−	0.752442i	$-0.728876\pi$
$701$	−24449.3	−1.31732	−0.658658	+	0.752442i	$0.728876\pi$
$702$	13619.4	0.732240
$703$	−3639.42	−0.195254
$704$	11764.8	0.629831
$705$	0	0
$706$	1915.11	0.102091
$707$	−11695.0	−0.622114
$708$	5961.86	0.316469
$709$	−36036.5	−1.90886	−0.954428	−	0.298441i	$-0.903533\pi$
$709$	−36036.5	−1.90886	−0.954428	+	0.298441i	$0.903533\pi$
$710$	0	0
$711$	4747.52	0.250416
$712$	−24511.2	−1.29016
$713$	−55796.0	−2.93068
$714$	−2631.25	−0.137916
$715$	0	0
$716$	88517.0	4.62016
$717$	−3724.64	−0.194002
$718$	36888.7	1.91738
$719$	−23679.1	−1.22821	−0.614103	−	0.789226i	$-0.710482\pi$
$719$	−23679.1	−1.22821	−0.614103	+	0.789226i	$0.710482\pi$
$720$	0	0
$721$	−11482.0	−0.593080
$722$	−28006.2	−1.44361
$723$	202.007	0.0103911
$724$	50669.6	2.60100
$725$	0	0
$726$	17327.1	0.885769
$727$	1248.96	0.0637159	0.0318579	−	0.999492i	$-0.489858\pi$
$727$	1248.96	0.0637159	0.0318579	+	0.999492i	$0.489858\pi$
$728$	−63641.6	−3.23999
$729$	−14301.9	−0.726611
$730$	0	0
$731$	−2428.94	−0.122897
$732$	−13478.8	−0.680590
$733$	−15813.6	−0.796846	−0.398423	−	0.917202i	$-0.630442\pi$
$733$	−15813.6	−0.796846	−0.398423	+	0.917202i	$0.630442\pi$
$734$	50872.8	2.55824
$735$	0	0
$736$	−45968.2	−2.30219
$737$	−26031.6	−1.30107
$738$	14075.4	0.702063
$739$	28793.6	1.43327	0.716636	−	0.697447i	$-0.245681\pi$
$739$	28793.6	1.43327	0.716636	+	0.697447i	$0.245681\pi$
$740$	0	0
$741$	1889.75	0.0936865
$742$	24716.8	1.22289
$743$	−2077.59	−0.102583	−0.0512917	−	0.998684i	$-0.516334\pi$
$743$	−2077.59	−0.102583	−0.0512917	+	0.998684i	$0.516334\pi$
$744$	17553.5	0.864976
$745$	0	0
$746$	48113.0	2.36132
$747$	16110.8	0.789109
$748$	−20625.2	−1.00820
$749$	11669.1	0.569265
$750$	0	0
$751$	−9719.10	−0.472244	−0.236122	−	0.971723i	$-0.575876\pi$
$751$	−9719.10	−0.472244	−0.236122	+	0.971723i	$0.575876\pi$
$752$	−35799.5	−1.73600
$753$	3714.17	0.179750
$754$	−36013.8	−1.73945
$755$	0	0
$756$	−29312.6	−1.41017
$757$	27467.9	1.31881	0.659403	−	0.751790i	$-0.270809\pi$
$757$	27467.9	1.31881	0.659403	+	0.751790i	$0.270809\pi$
$758$	23969.9	1.14858
$759$	−14416.4	−0.689437
$760$	0	0
$761$	29176.3	1.38980	0.694902	−	0.719104i	$-0.255448\pi$
$761$	29176.3	1.38980	0.694902	+	0.719104i	$0.255448\pi$
$762$	−7091.77	−0.337149
$763$	21808.6	1.03476
$764$	82678.2	3.91517
$765$	0	0
$766$	−49300.8	−2.32547
$767$	−12641.8	−0.595137
$768$	−7280.17	−0.342058
$769$	16183.3	0.758887	0.379444	−	0.925215i	$-0.376115\pi$
$769$	16183.3	0.758887	0.379444	+	0.925215i	$0.376115\pi$
$770$	0	0
$771$	−8542.50	−0.399028
$772$	32398.9	1.51044
$773$	−30864.9	−1.43614	−0.718069	−	0.695972i	$-0.754974\pi$
$773$	−30864.9	−1.43614	−0.718069	+	0.695972i	$0.754974\pi$
$774$	−18912.9	−0.878310
$775$	0	0
$776$	−6689.28	−0.309448
$777$	2907.19	0.134228
$778$	9148.33	0.421573
$779$	4003.26	0.184123
$780$	0	0
$781$	−37624.1	−1.72381
$782$	16986.9	0.776789
$783$	−9410.02	−0.429485
$784$	47206.4	2.15044
$785$	0	0
$786$	2951.18	0.133925
$787$	−8093.40	−0.366580	−0.183290	−	0.983059i	$-0.558675\pi$
$787$	−8093.40	−0.366580	−0.183290	+	0.983059i	$0.558675\pi$
$788$	15395.2	0.695978
$789$	7757.30	0.350022
$790$	0	0
$791$	9592.83	0.431203
$792$	−91106.1	−4.08752
$793$	28581.2	1.27988
$794$	73582.0	3.28883
$795$	0	0
$796$	39606.2	1.76357
$797$	−879.467	−0.0390870	−0.0195435	−	0.999809i	$-0.506221\pi$
$797$	−879.467	−0.0390870	−0.0195435	+	0.999809i	$0.506221\pi$
$798$	−5827.17	−0.258496
$799$	4684.73	0.207426
$800$	0	0
$801$	11678.8	0.515168
$802$	37896.3	1.66854
$803$	−37715.8	−1.65749
$804$	−8298.75	−0.364023
$805$	0	0
$806$	−65612.2	−2.86736
$807$	−8804.72	−0.384066
$808$	−23752.6	−1.03417
$809$	−29010.8	−1.26077	−0.630387	−	0.776281i	$-0.717104\pi$
$809$	−29010.8	−1.26077	−0.630387	+	0.776281i	$0.717104\pi$
$810$	0	0
$811$	−40609.1	−1.75830	−0.879149	−	0.476548i	$-0.841888\pi$
$811$	−40609.1	−1.75830	−0.879149	+	0.476548i	$0.841888\pi$
$812$	77511.3	3.34989
$813$	3172.86	0.136872
$814$	32648.7	1.40582
$815$	0	0
$816$	−2498.96	−0.107207
$817$	−5379.13	−0.230345
$818$	27496.0	1.17528
$819$	30323.2	1.29375
$820$	0	0
$821$	−24338.7	−1.03463	−0.517313	−	0.855796i	$-0.673068\pi$
$821$	−24338.7	−1.03463	−0.517313	+	0.855796i	$0.673068\pi$
$822$	833.668	0.0353741
$823$	33785.6	1.43097	0.715487	−	0.698626i	$-0.246205\pi$
$823$	33785.6	1.43097	0.715487	+	0.698626i	$0.246205\pi$
$824$	−23320.0	−0.985910
$825$	0	0
$826$	38981.9	1.64207
$827$	−27334.1	−1.14933	−0.574667	−	0.818388i	$-0.694868\pi$
$827$	−27334.1	−1.14933	−0.574667	+	0.818388i	$0.694868\pi$
$828$	92320.6	3.87483
$829$	26231.5	1.09898	0.549491	−	0.835499i	$-0.314822\pi$
$829$	26231.5	1.09898	0.549491	+	0.835499i	$0.314822\pi$
$830$	0	0
$831$	1079.46	0.0450615
$832$	−7952.86	−0.331389
$833$	−6177.43	−0.256945
$834$	−10720.8	−0.445119
$835$	0	0
$836$	−45676.5	−1.88966
$837$	−17143.7	−0.707974
$838$	−26090.0	−1.07549
$839$	−15856.1	−0.652458	−0.326229	−	0.945291i	$-0.605778\pi$
$839$	−15856.1	−0.652458	−0.326229	+	0.945291i	$0.605778\pi$
$840$	0	0
$841$	493.893	0.0202507
$842$	22181.0	0.907847
$843$	1063.41	0.0434471
$844$	−5147.64	−0.209940
$845$	0	0
$846$	36477.7	1.48242
$847$	79076.9	3.20793
$848$	23474.1	0.950595
$849$	5016.15	0.202772
$850$	0	0
$851$	−18768.3	−0.756016
$852$	−11994.4	−0.482301
$853$	−4337.51	−0.174107	−0.0870537	−	0.996204i	$-0.527745\pi$
$853$	−4337.51	−0.174107	−0.0870537	+	0.996204i	$0.527745\pi$
$854$	−88132.0	−3.53140
$855$	0	0
$856$	23700.0	0.946321
$857$	−21521.9	−0.857844	−0.428922	−	0.903341i	$-0.641107\pi$
$857$	−21521.9	−0.857844	−0.428922	+	0.903341i	$0.641107\pi$
$858$	−16952.7	−0.674541
$859$	−1918.52	−0.0762038	−0.0381019	−	0.999274i	$-0.512131\pi$
$859$	−1918.52	−0.0762038	−0.0381019	+	0.999274i	$0.512131\pi$
$860$	0	0
$861$	−3197.83	−0.126576
$862$	38096.4	1.50530
$863$	34932.6	1.37789	0.688946	−	0.724813i	$-0.258074\pi$
$863$	34932.6	1.37789	0.688946	+	0.724813i	$0.258074\pi$
$864$	−14124.1	−0.556146
$865$	0	0
$866$	37847.9	1.48513
$867$	327.013	0.0128096
$868$	141215.	5.52206
$869$	−12113.1	−0.472851
$870$	0	0
$871$	17597.1	0.684563
$872$	44293.5	1.72015
$873$	3187.23	0.123564
$874$	37619.1	1.45593
$875$	0	0
$876$	−12023.6	−0.463744
$877$	25793.1	0.993124	0.496562	−	0.868001i	$-0.334595\pi$
$877$	25793.1	0.993124	0.496562	+	0.868001i	$0.334595\pi$
$878$	−71164.6	−2.73541
$879$	−716.886	−0.0275085
$880$	0	0
$881$	26596.2	1.01708	0.508541	−	0.861038i	$-0.330185\pi$
$881$	26596.2	1.01708	0.508541	+	0.861038i	$0.330185\pi$
$882$	−48100.6	−1.83632
$883$	16304.3	0.621386	0.310693	−	0.950510i	$-0.399439\pi$
$883$	16304.3	0.621386	0.310693	+	0.950510i	$0.399439\pi$
$884$	13942.4	0.530468
$885$	0	0
$886$	6451.33	0.244624
$887$	−22972.6	−0.869611	−0.434806	−	0.900524i	$-0.643183\pi$
$887$	−22972.6	−0.869611	−0.434806	+	0.900524i	$0.643183\pi$
$888$	5904.53	0.223134
$889$	−32365.2	−1.22103
$890$	0	0
$891$	41140.6	1.54687
$892$	4117.17	0.154544
$893$	10374.8	0.388779
$894$	−191.553	−0.00716609
$895$	0	0
$896$	−25818.6	−0.962654
$897$	9745.35	0.362751
$898$	−13248.1	−0.492309
$899$	45333.2	1.68181
$900$	0	0
$901$	−3071.82	−0.113582
$902$	−35912.7	−1.32568
$903$	4296.89	0.158352
$904$	19483.1	0.716813
$905$	0	0
$906$	−15623.0	−0.572890
$907$	12166.9	0.445419	0.222710	−	0.974885i	$-0.428510\pi$
$907$	12166.9	0.445419	0.222710	+	0.974885i	$0.428510\pi$
$908$	−114447.	−4.18286
$909$	11317.4	0.412952
$910$	0	0
$911$	−50725.0	−1.84478	−0.922390	−	0.386259i	$-0.873767\pi$
$911$	−50725.0	−1.84478	−0.922390	+	0.386259i	$0.873767\pi$
$912$	−5534.19	−0.200938
$913$	−41106.0	−1.49004
$914$	−45390.7	−1.64266
$915$	0	0
$916$	29412.0	1.06092
$917$	13468.5	0.485027
$918$	5219.34	0.187651
$919$	13152.5	0.472100	0.236050	−	0.971741i	$-0.424147\pi$
$919$	13152.5	0.472100	0.236050	+	0.971741i	$0.424147\pi$
$920$	0	0
$921$	3576.05	0.127942
$922$	26908.9	0.961170
$923$	25433.5	0.906992
$924$	36486.7	1.29905
$925$	0	0
$926$	31882.1	1.13144
$927$	11111.2	0.393680
$928$	37348.2	1.32114
$929$	−45107.7	−1.59304	−0.796521	−	0.604611i	$-0.793329\pi$
$929$	−45107.7	−1.59304	−0.796521	+	0.604611i	$0.793329\pi$
$930$	0	0
$931$	−13680.5	−0.481591
$932$	94915.1	3.33589
$933$	−8308.16	−0.291529
$934$	64115.3	2.24616
$935$	0	0
$936$	61586.8	2.15067
$937$	27517.9	0.959413	0.479707	−	0.877429i	$-0.340743\pi$
$937$	27517.9	0.959413	0.479707	+	0.877429i	$0.340743\pi$
$938$	−54261.8	−1.88882
$939$	2535.26	0.0881099
$940$	0	0
$941$	−8306.17	−0.287751	−0.143875	−	0.989596i	$-0.545956\pi$
$941$	−8306.17	−0.287751	−0.143875	+	0.989596i	$0.545956\pi$
$942$	22145.7	0.765971
$943$	20644.6	0.712917
$944$	37022.0	1.27644
$945$	0	0
$946$	48255.5	1.65848
$947$	−8181.39	−0.280739	−0.140369	−	0.990099i	$-0.544829\pi$
$947$	−8181.39	−0.280739	−0.140369	+	0.990099i	$0.544829\pi$
$948$	−3861.59	−0.132298
$949$	25495.5	0.872095
$950$	0	0
$951$	−526.803	−0.0179630
$952$	−24389.2	−0.830314
$953$	37847.9	1.28648	0.643239	−	0.765665i	$-0.277590\pi$
$953$	37847.9	1.28648	0.643239	+	0.765665i	$0.277590\pi$
$954$	−23918.8	−0.811739
$955$	0	0
$956$	−60857.4	−2.05886
$957$	11713.1	0.395642
$958$	14847.3	0.500725
$959$	3804.67	0.128112
$960$	0	0
$961$	52799.7	1.77234
$962$	−22070.2	−0.739680
$963$	−11292.3	−0.377871
$964$	3300.62	0.110276
$965$	0	0
$966$	−30050.4	−1.00089
$967$	−40403.1	−1.34362	−0.671808	−	0.740726i	$-0.734482\pi$
$967$	−40403.1	−1.34362	−0.671808	+	0.740726i	$0.734482\pi$
$968$	160606.	5.33271
$969$	724.204	0.0240091
$970$	0	0
$971$	−3181.11	−0.105136	−0.0525679	−	0.998617i	$-0.516741\pi$
$971$	−3181.11	−0.105136	−0.0525679	+	0.998617i	$0.516741\pi$
$972$	42893.7	1.41545
$973$	−48927.1	−1.61206
$974$	82491.3	2.71375
$975$	0	0
$976$	−83700.9	−2.74508
$977$	18934.0	0.620011	0.310006	−	0.950735i	$-0.399669\pi$
$977$	18934.0	0.620011	0.310006	+	0.950735i	$0.399669\pi$
$978$	−10932.0	−0.357429
$979$	−29797.9	−0.972773
$980$	0	0
$981$	−21104.5	−0.686864
$982$	−85169.5	−2.76769
$983$	−11459.8	−0.371833	−0.185916	−	0.982566i	$-0.559525\pi$
$983$	−11459.8	−0.371833	−0.185916	+	0.982566i	$0.559525\pi$
$984$	−6494.82	−0.210414
$985$	0	0
$986$	−13801.5	−0.445770
$987$	−8287.47	−0.267267
$988$	30876.8	0.994254
$989$	−27739.9	−0.891889
$990$	0	0
$991$	19430.0	0.622820	0.311410	−	0.950276i	$-0.399199\pi$
$991$	19430.0	0.622820	0.311410	+	0.950276i	$0.399199\pi$
$992$	68043.3	2.17780
$993$	8387.37	0.268042
$994$	−78425.8	−2.50253
$995$	0	0
$996$	−13104.4	−0.416896
$997$	−43572.9	−1.38412	−0.692061	−	0.721839i	$-0.743297\pi$
$997$	−43572.9	−1.38412	−0.692061	+	0.721839i	$0.743297\pi$
$998$	−61389.3	−1.94714
$999$	−5766.70	−0.182633

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.4.a.i.1.5		5
5.2	odd	4		425.4.b.i.324.9		10
5.3	odd	4		425.4.b.i.324.2		10
5.4	even	2		85.4.a.g.1.1	✓	5
15.14	odd	2		765.4.a.m.1.5		5
20.19	odd	2		1360.4.a.w.1.4		5
85.84	even	2		1445.4.a.l.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.a.g.1.1	✓	5	5.4	even	2
425.4.a.i.1.5		5	1.1	even	1	trivial
425.4.b.i.324.2		10	5.3	odd	4
425.4.b.i.324.9		10	5.2	odd	4
765.4.a.m.1.5		5	15.14	odd	2
1360.4.a.w.1.4		5	20.19	odd	2
1445.4.a.l.1.1		5	85.84	even	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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q+5.14668 q^{2} +1.13153 q^{3} +18.4883 q^{4} +5.82364 q^{6} +26.5778 q^{7} +53.9797 q^{8} -25.7196 q^{9} +65.6224 q^{11} +20.9201 q^{12} -44.3601 q^{13} +136.787 q^{14} +129.910 q^{16} -17.0000 q^{17} -132.371 q^{18} -37.6482 q^{19} +30.0737 q^{21} +337.737 q^{22} -194.150 q^{23} +61.0799 q^{24} -228.307 q^{26} -59.6541 q^{27} +491.377 q^{28} +157.743 q^{29} +287.386 q^{31} +236.766 q^{32} +74.2540 q^{33} -87.4935 q^{34} -475.511 q^{36} +96.6691 q^{37} -193.763 q^{38} -50.1950 q^{39} -106.333 q^{41} +154.779 q^{42} +142.879 q^{43} +1213.24 q^{44} -999.227 q^{46} -275.572 q^{47} +146.997 q^{48} +363.378 q^{49} -19.2361 q^{51} -820.141 q^{52} +180.696 q^{53} -307.020 q^{54} +1434.66 q^{56} -42.6003 q^{57} +811.853 q^{58} +284.982 q^{59} -644.300 q^{61} +1479.08 q^{62} -683.571 q^{63} +179.280 q^{64} +382.161 q^{66} -396.688 q^{67} -314.301 q^{68} -219.687 q^{69} -573.342 q^{71} -1388.34 q^{72} -574.739 q^{73} +497.524 q^{74} -696.050 q^{76} +1744.10 q^{77} -258.337 q^{78} -184.587 q^{79} +626.929 q^{81} -547.263 q^{82} -626.402 q^{83} +556.010 q^{84} +735.351 q^{86} +178.492 q^{87} +3542.28 q^{88} -454.081 q^{89} -1178.99 q^{91} -3589.50 q^{92} +325.187 q^{93} -1418.28 q^{94} +267.909 q^{96} -123.922 q^{97} +1870.19 q^{98} -1687.78 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + q^{3} + 34 q^{4} - 5 q^{6} - 10 q^{7} - 30 q^{8} - 30 q^{9} + 126 q^{11} - 15 q^{12} - 83 q^{13} + 90 q^{14} + 322 q^{16} - 85 q^{17} + 97 q^{18} + 55 q^{19} + 6 q^{21} + 240 q^{22} + 2 q^{23}+ \cdots - 858 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 + q^3 + 34 * q^4 - 5 * q^6 - 10 * q^7 - 30 * q^8 - 30 * q^9 + 126 * q^11 - 15 * q^12 - 83 * q^13 + 90 * q^14 + 322 * q^16 - 85 * q^17 + 97 * q^18 + 55 * q^19 + 6 * q^21 + 240 * q^22 + 2 * q^23 + 155 * q^24 - 395 * q^26 + 163 * q^27 + 1062 * q^28 + 195 * q^29 + 97 * q^31 - 182 * q^32 + 352 * q^33 + 34 * q^34 - 1437 * q^36 + 476 * q^37 + 149 * q^38 + 373 * q^39 + 298 * q^41 + 18 * q^42 - 168 * q^43 + 1136 * q^44 - 2106 * q^46 - 1095 * q^47 + 1193 * q^48 + 737 * q^49 - 17 * q^51 - 281 * q^52 + 1035 * q^53 - 291 * q^54 - 350 * q^56 + 15 * q^57 + 2199 * q^58 + 1163 * q^59 - 351 * q^61 + 1241 * q^62 - 1584 * q^63 + 2042 * q^64 + 356 * q^66 - 1064 * q^67 - 578 * q^68 + 1298 * q^69 + 1393 * q^71 + 269 * q^72 + 163 * q^73 + 1530 * q^74 - 393 * q^76 + 732 * q^77 + 67 * q^78 + 750 * q^79 + 85 * q^81 - 1144 * q^82 - 270 * q^83 + 2422 * q^84 + 3502 * q^86 + 907 * q^87 + 3216 * q^88 + 1179 * q^89 + 282 * q^91 - 758 * q^92 + 3923 * q^93 + 3473 * q^94 - 2045 * q^96 + 2745 * q^97 - 502 * q^98 - 858 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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