LMFDB - Embedded newform 2175.4.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,4,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2175.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2175.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:	$128.329154262$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 49x^{4} + 27x^{3} + 692x^{2} - 82x - 2588 $ x^6 - x^5 - 49x^4 + 27x^3 + 692x^2 - 82x - 2588
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.39070$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.39070 q^{2} -3.00000 q^{3} +11.2783 q^{4} +13.1721 q^{6} -31.5336 q^{7} -14.3939 q^{8} +9.00000 q^{9} -5.28853 q^{11} -33.8348 q^{12} +5.98731 q^{13} +138.455 q^{14} -27.0267 q^{16} -84.6758 q^{17} -39.5163 q^{18} +57.3480 q^{19} +94.6008 q^{21} +23.2204 q^{22} -31.0898 q^{23} +43.1818 q^{24} -26.2885 q^{26} -27.0000 q^{27} -355.645 q^{28} +29.0000 q^{29} -14.6873 q^{31} +233.818 q^{32} +15.8656 q^{33} +371.786 q^{34} +101.504 q^{36} +7.76174 q^{37} -251.798 q^{38} -17.9619 q^{39} -399.867 q^{41} -415.364 q^{42} +17.9967 q^{43} -59.6455 q^{44} +136.506 q^{46} +262.798 q^{47} +81.0800 q^{48} +651.368 q^{49} +254.027 q^{51} +67.5265 q^{52} +64.9925 q^{53} +118.549 q^{54} +453.893 q^{56} -172.044 q^{57} -127.330 q^{58} +122.788 q^{59} +264.384 q^{61} +64.4877 q^{62} -283.802 q^{63} -810.411 q^{64} -69.6611 q^{66} -622.712 q^{67} -954.997 q^{68} +93.2694 q^{69} +327.717 q^{71} -129.545 q^{72} +833.518 q^{73} -34.0795 q^{74} +646.787 q^{76} +166.767 q^{77} +78.8655 q^{78} -666.457 q^{79} +81.0000 q^{81} +1755.70 q^{82} +1320.72 q^{83} +1066.93 q^{84} -79.0180 q^{86} -87.0000 q^{87} +76.1229 q^{88} -189.059 q^{89} -188.801 q^{91} -350.640 q^{92} +44.0620 q^{93} -1153.87 q^{94} -701.453 q^{96} +137.630 q^{97} -2859.97 q^{98} -47.5968 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} - 18 q^{3} + 51 q^{4} + 3 q^{6} - 47 q^{7} - 51 q^{8} + 54 q^{9} + 81 q^{11} - 153 q^{12} - 169 q^{13} - 30 q^{14} + 131 q^{16} + q^{17} - 9 q^{18} + 116 q^{19} + 141 q^{21} - 90 q^{22} + 52 q^{23}+ \cdots + 729 q^{99}+O(q^{100}) $ 6 * q - q^2 - 18 * q^3 + 51 * q^4 + 3 * q^6 - 47 * q^7 - 51 * q^8 + 54 * q^9 + 81 * q^11 - 153 * q^12 - 169 * q^13 - 30 * q^14 + 131 * q^16 + q^17 - 9 * q^18 + 116 * q^19 + 141 * q^21 - 90 * q^22 + 52 * q^23 + 153 * q^24 + 294 * q^26 - 162 * q^27 - 344 * q^28 + 174 * q^29 + 340 * q^31 - 499 * q^32 - 243 * q^33 + 920 * q^34 + 459 * q^36 - 332 * q^37 + 378 * q^38 + 507 * q^39 - 616 * q^41 + 90 * q^42 - 334 * q^43 - 52 * q^44 - 158 * q^46 + 85 * q^47 - 393 * q^48 + 879 * q^49 - 3 * q^51 - 2220 * q^52 + 850 * q^53 + 27 * q^54 - 624 * q^56 - 348 * q^57 - 29 * q^58 - 758 * q^59 - 36 * q^61 + 152 * q^62 - 423 * q^63 + 1795 * q^64 + 270 * q^66 - 939 * q^67 + 186 * q^68 - 156 * q^69 - 1388 * q^71 - 459 * q^72 - 1708 * q^73 - 814 * q^74 + 566 * q^76 - 2585 * q^77 - 882 * q^78 + 1250 * q^79 + 486 * q^81 - 1372 * q^82 + 748 * q^83 + 1032 * q^84 - 800 * q^86 - 522 * q^87 - 536 * q^88 + 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 1020 * q^93 - 4542 * q^94 + 1497 * q^96 + 22 * q^97 + 1433 * q^98 + 729 * 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 $

$2$	−4.39070	−1.55235	−0.776174	−	0.630519i	$-0.782842\pi$
$2$	−4.39070	−1.55235	−0.776174	+	0.630519i	$0.782842\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	11.2783	1.40978
$5$	0	0
$5$	0	0
$6$	13.1721	0.896249
$7$	−31.5336	−1.70265	−0.851327	−	0.524635i	$-0.824202\pi$
$7$	−31.5336	−1.70265	−0.851327	+	0.524635i	$0.824202\pi$
$8$	−14.3939	−0.636128
$9$	9.00000	0.333333
$10$	0	0
$11$	−5.28853	−0.144959	−0.0724797	−	0.997370i	$-0.523091\pi$
$11$	−5.28853	−0.144959	−0.0724797	+	0.997370i	$0.523091\pi$
$12$	−33.8348	−0.813940
$13$	5.98731	0.127737	0.0638685	−	0.997958i	$-0.479656\pi$
$13$	5.98731	0.127737	0.0638685	+	0.997958i	$0.479656\pi$
$14$	138.455	2.64311
$15$	0	0
$16$	−27.0267	−0.422292
$17$	−84.6758	−1.20805	−0.604026	−	0.796965i	$-0.706438\pi$
$17$	−84.6758	−1.20805	−0.604026	+	0.796965i	$0.706438\pi$
$18$	−39.5163	−0.517449
$19$	57.3480	0.692449	0.346225	−	0.938152i	$-0.387464\pi$
$19$	57.3480	0.692449	0.346225	+	0.938152i	$0.387464\pi$
$20$	0	0
$21$	94.6008	0.983028
$22$	23.2204	0.225027
$23$	−31.0898	−0.281855	−0.140928	−	0.990020i	$-0.545009\pi$
$23$	−31.0898	−0.281855	−0.140928	+	0.990020i	$0.545009\pi$
$24$	43.1818	0.367269
$25$	0	0
$26$	−26.2885	−0.198292
$27$	−27.0000	−0.192450
$28$	−355.645	−2.40038
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−14.6873	−0.0850942	−0.0425471	−	0.999094i	$-0.513547\pi$
$31$	−14.6873	−0.0850942	−0.0425471	+	0.999094i	$0.513547\pi$
$32$	233.818	1.29167
$33$	15.8656	0.0836923
$34$	371.786	1.87532
$35$	0	0
$36$	101.504	0.469928
$37$	7.76174	0.0344871	0.0172435	−	0.999851i	$-0.494511\pi$
$37$	7.76174	0.0344871	0.0172435	+	0.999851i	$0.494511\pi$
$38$	−251.798	−1.07492
$39$	−17.9619	−0.0737490
$40$	0	0
$41$	−399.867	−1.52314	−0.761569	−	0.648084i	$-0.775571\pi$
$41$	−399.867	−1.52314	−0.761569	+	0.648084i	$0.775571\pi$
$42$	−415.364	−1.52600
$43$	17.9967	0.0638247	0.0319124	−	0.999491i	$-0.489840\pi$
$43$	17.9967	0.0638247	0.0319124	+	0.999491i	$0.489840\pi$
$44$	−59.6455	−0.204361
$45$	0	0
$46$	136.506	0.437538
$47$	262.798	0.815595	0.407798	−	0.913072i	$-0.366297\pi$
$47$	262.798	0.815595	0.407798	+	0.913072i	$0.366297\pi$
$48$	81.0800	0.243810
$49$	651.368	1.89903
$50$	0	0
$51$	254.027	0.697469
$52$	67.5265	0.180082
$53$	64.9925	0.168442	0.0842209	−	0.996447i	$-0.473160\pi$
$53$	64.9925	0.168442	0.0842209	+	0.996447i	$0.473160\pi$
$54$	118.549	0.298750
$55$	0	0
$56$	453.893	1.08311
$57$	−172.044	−0.399786
$58$	−127.330	−0.288264
$59$	122.788	0.270942	0.135471	−	0.990781i	$-0.456745\pi$
$59$	122.788	0.270942	0.135471	+	0.990781i	$0.456745\pi$
$60$	0	0
$61$	264.384	0.554932	0.277466	−	0.960735i	$-0.410505\pi$
$61$	264.384	0.554932	0.277466	+	0.960735i	$0.410505\pi$
$62$	64.4877	0.132096
$63$	−283.802	−0.567552
$64$	−810.411	−1.58283
$65$	0	0
$66$	−69.6611	−0.129920
$67$	−622.712	−1.13547	−0.567734	−	0.823212i	$-0.692180\pi$
$67$	−622.712	−1.13547	−0.567734	+	0.823212i	$0.692180\pi$
$68$	−954.997	−1.70309
$69$	93.2694	0.162729
$70$	0	0
$71$	327.717	0.547786	0.273893	−	0.961760i	$-0.411689\pi$
$71$	327.717	0.547786	0.273893	+	0.961760i	$0.411689\pi$
$72$	−129.545	−0.212043
$73$	833.518	1.33638	0.668191	−	0.743989i	$-0.267069\pi$
$73$	833.518	1.33638	0.668191	+	0.743989i	$0.267069\pi$
$74$	−34.0795	−0.0535360
$75$	0	0
$76$	646.787	0.976204
$77$	166.767	0.246816
$78$	78.8655	0.114484
$79$	−666.457	−0.949143	−0.474571	−	0.880217i	$-0.657397\pi$
$79$	−666.457	−0.949143	−0.474571	+	0.880217i	$0.657397\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	1755.70	2.36444
$83$	1320.72	1.74660	0.873301	−	0.487182i	$-0.161975\pi$
$83$	1320.72	1.74660	0.873301	+	0.487182i	$0.161975\pi$
$84$	1066.93	1.38586
$85$	0	0
$86$	−79.0180	−0.0990782
$87$	−87.0000	−0.107211
$88$	76.1229	0.0922128
$89$	−189.059	−0.225172	−0.112586	−	0.993642i	$-0.535913\pi$
$89$	−189.059	−0.225172	−0.112586	+	0.993642i	$0.535913\pi$
$90$	0	0
$91$	−188.801	−0.217492
$92$	−350.640	−0.397355
$93$	44.0620	0.0491292
$94$	−1153.87	−1.26609
$95$	0	0
$96$	−701.453	−0.745747
$97$	137.630	0.144064	0.0720321	−	0.997402i	$-0.477052\pi$
$97$	137.630	0.144064	0.0720321	+	0.997402i	$0.477052\pi$
$98$	−2859.97	−2.94796
$99$	−47.5968	−0.0483198
$100$	0	0
$101$	−1721.15	−1.69565	−0.847826	−	0.530275i	$-0.822089\pi$
$101$	−1721.15	−1.69565	−0.847826	+	0.530275i	$0.822089\pi$
$102$	−1115.36	−1.08272
$103$	−797.585	−0.762994	−0.381497	−	0.924370i	$-0.624591\pi$
$103$	−797.585	−0.762994	−0.381497	+	0.924370i	$0.624591\pi$
$104$	−86.1810	−0.0812571
$105$	0	0
$106$	−285.363	−0.261480
$107$	778.452	0.703325	0.351662	−	0.936127i	$-0.385616\pi$
$107$	778.452	0.703325	0.351662	+	0.936127i	$0.385616\pi$
$108$	−304.513	−0.271313
$109$	721.205	0.633752	0.316876	−	0.948467i	$-0.397366\pi$
$109$	721.205	0.633752	0.316876	+	0.948467i	$0.397366\pi$
$110$	0	0
$111$	−23.2852	−0.0199111
$112$	852.249	0.719017
$113$	559.024	0.465386	0.232693	−	0.972550i	$-0.425246\pi$
$113$	559.024	0.465386	0.232693	+	0.972550i	$0.425246\pi$
$114$	755.394	0.620607
$115$	0	0
$116$	327.070	0.261790
$117$	53.8858	0.0425790
$118$	−539.124	−0.420597
$119$	2670.13	2.05690
$120$	0	0
$121$	−1303.03	−0.978987
$122$	−1160.83	−0.861448
$123$	1199.60	0.879384
$124$	−165.648	−0.119965
$125$	0	0
$126$	1246.09	0.881038
$127$	626.324	0.437616	0.218808	−	0.975768i	$-0.429783\pi$
$127$	626.324	0.437616	0.218808	+	0.975768i	$0.429783\pi$
$128$	1687.73	1.16544
$129$	−53.9900	−0.0368492
$130$	0	0
$131$	1764.12	1.17658	0.588289	−	0.808651i	$-0.299802\pi$
$131$	1764.12	1.17658	0.588289	+	0.808651i	$0.299802\pi$
$132$	178.937	0.117988
$133$	−1808.39	−1.17900
$134$	2734.14	1.76264
$135$	0	0
$136$	1218.82	0.768476
$137$	433.723	0.270478	0.135239	−	0.990813i	$-0.456820\pi$
$137$	433.723	0.270478	0.135239	+	0.990813i	$0.456820\pi$
$138$	−409.518	−0.252613
$139$	−504.659	−0.307947	−0.153974	−	0.988075i	$-0.549207\pi$
$139$	−504.659	−0.307947	−0.153974	+	0.988075i	$0.549207\pi$
$140$	0	0
$141$	−788.393	−0.470884
$142$	−1438.91	−0.850355
$143$	−31.6641	−0.0185167
$144$	−243.240	−0.140764
$145$	0	0
$146$	−3659.73	−2.07453
$147$	−1954.11	−1.09641
$148$	87.5391	0.0486194
$149$	−354.655	−0.194997	−0.0974983	−	0.995236i	$-0.531084\pi$
$149$	−354.655	−0.194997	−0.0974983	+	0.995236i	$0.531084\pi$
$150$	0	0
$151$	−325.611	−0.175483	−0.0877413	−	0.996143i	$-0.527965\pi$
$151$	−325.611	−0.175483	−0.0877413	+	0.996143i	$0.527965\pi$
$152$	−825.464	−0.440487
$153$	−762.082	−0.402684
$154$	−732.222	−0.383144
$155$	0	0
$156$	−202.580	−0.103970
$157$	2690.09	1.36747	0.683733	−	0.729732i	$-0.260355\pi$
$157$	2690.09	1.36747	0.683733	+	0.729732i	$0.260355\pi$
$158$	2926.22	1.47340
$159$	−194.978	−0.0972499
$160$	0	0
$161$	980.374	0.479902
$162$	−355.647	−0.172483
$163$	−1411.51	−0.678270	−0.339135	−	0.940738i	$-0.610134\pi$
$163$	−1411.51	−0.678270	−0.339135	+	0.940738i	$0.610134\pi$
$164$	−4509.81	−2.14730
$165$	0	0
$166$	−5798.89	−2.71133
$167$	2538.25	1.17614	0.588072	−	0.808809i	$-0.299887\pi$
$167$	2538.25	1.17614	0.588072	+	0.808809i	$0.299887\pi$
$168$	−1361.68	−0.625332
$169$	−2161.15	−0.983683
$170$	0	0
$171$	516.132	0.230816
$172$	202.971	0.0899791
$173$	305.030	0.134052	0.0670260	−	0.997751i	$-0.478649\pi$
$173$	305.030	0.134052	0.0670260	+	0.997751i	$0.478649\pi$
$174$	381.991	0.166429
$175$	0	0
$176$	142.932	0.0612152
$177$	−368.363	−0.156429
$178$	830.104	0.349545
$179$	3093.43	1.29170	0.645849	−	0.763465i	$-0.276504\pi$
$179$	3093.43	1.29170	0.645849	+	0.763465i	$0.276504\pi$
$180$	0	0
$181$	2800.38	1.15000	0.575001	−	0.818153i	$-0.305002\pi$
$181$	2800.38	1.15000	0.575001	+	0.818153i	$0.305002\pi$
$182$	828.971	0.337623
$183$	−793.151	−0.320390
$184$	447.505	0.179296
$185$	0	0
$186$	−193.463	−0.0762656
$187$	447.811	0.175118
$188$	2963.91	1.14981
$189$	851.407	0.327676
$190$	0	0
$191$	4208.78	1.59443	0.797217	−	0.603693i	$-0.206305\pi$
$191$	4208.78	1.59443	0.797217	+	0.603693i	$0.206305\pi$
$192$	2431.23	0.913849
$193$	247.563	0.0923315	0.0461658	−	0.998934i	$-0.485300\pi$
$193$	247.563	0.0923315	0.0461658	+	0.998934i	$0.485300\pi$
$194$	−604.293	−0.223638
$195$	0	0
$196$	7346.31	2.67723
$197$	4574.07	1.65426	0.827129	−	0.562013i	$-0.189973\pi$
$197$	4574.07	1.65426	0.827129	+	0.562013i	$0.189973\pi$
$198$	208.983	0.0750091
$199$	2601.05	0.926551	0.463276	−	0.886214i	$-0.346674\pi$
$199$	2601.05	0.926551	0.463276	+	0.886214i	$0.346674\pi$
$200$	0	0
$201$	1868.13	0.655563
$202$	7557.06	2.63224
$203$	−914.475	−0.316175
$204$	2864.99	0.983281
$205$	0	0
$206$	3501.96	1.18443
$207$	−279.808	−0.0939518
$208$	−161.817	−0.0539423
$209$	−303.287	−0.100377
$210$	0	0
$211$	244.148	0.0796582	0.0398291	−	0.999207i	$-0.487319\pi$
$211$	244.148	0.0796582	0.0398291	+	0.999207i	$0.487319\pi$
$212$	733.004	0.237467
$213$	−983.150	−0.316265
$214$	−3417.95	−1.09181
$215$	0	0
$216$	388.636	0.122423
$217$	463.144	0.144886
$218$	−3166.60	−0.983803
$219$	−2500.56	−0.771561
$220$	0	0
$221$	−506.980	−0.154313
$222$	102.239	0.0309090
$223$	−4179.26	−1.25500	−0.627498	−	0.778618i	$-0.715921\pi$
$223$	−4179.26	−1.25500	−0.627498	+	0.778618i	$0.715921\pi$
$224$	−7373.12	−2.19927
$225$	0	0
$226$	−2454.51	−0.722441
$227$	−4034.64	−1.17968	−0.589842	−	0.807518i	$-0.700810\pi$
$227$	−4034.64	−1.17968	−0.589842	+	0.807518i	$0.700810\pi$
$228$	−1940.36	−0.563612
$229$	4501.53	1.29899	0.649497	−	0.760364i	$-0.274980\pi$
$229$	4501.53	1.29899	0.649497	+	0.760364i	$0.274980\pi$
$230$	0	0
$231$	−500.300	−0.142499
$232$	−417.424	−0.118126
$233$	4710.72	1.32450	0.662252	−	0.749282i	$-0.269601\pi$
$233$	4710.72	1.32450	0.662252	+	0.749282i	$0.269601\pi$
$234$	−236.597	−0.0660974
$235$	0	0
$236$	1384.83	0.381970
$237$	1999.37	0.547988
$238$	−11723.8	−3.19302
$239$	−496.634	−0.134412	−0.0672062	−	0.997739i	$-0.521409\pi$
$239$	−496.634	−0.134412	−0.0672062	+	0.997739i	$0.521409\pi$
$240$	0	0
$241$	6414.62	1.71453	0.857266	−	0.514874i	$-0.172161\pi$
$241$	6414.62	1.71453	0.857266	+	0.514874i	$0.172161\pi$
$242$	5721.22	1.51973
$243$	−243.000	−0.0641500
$244$	2981.79	0.782335
$245$	0	0
$246$	−5267.09	−1.36511
$247$	343.360	0.0884514
$248$	211.409	0.0541309
$249$	−3962.16	−1.00840
$250$	0	0
$251$	4741.55	1.19237	0.596183	−	0.802848i	$-0.296683\pi$
$251$	4741.55	1.19237	0.596183	+	0.802848i	$0.296683\pi$
$252$	−3200.80	−0.800126
$253$	164.420	0.0408576
$254$	−2750.00	−0.679333
$255$	0	0
$256$	−927.044	−0.226329
$257$	6110.47	1.48312	0.741558	−	0.670889i	$-0.234087\pi$
$257$	6110.47	1.48312	0.741558	+	0.670889i	$0.234087\pi$
$258$	237.054	0.0572028
$259$	−244.756	−0.0587196
$260$	0	0
$261$	261.000	0.0618984
$262$	−7745.72	−1.82646
$263$	3905.40	0.915654	0.457827	−	0.889041i	$-0.348628\pi$
$263$	3905.40	0.915654	0.457827	+	0.889041i	$0.348628\pi$
$264$	−228.369	−0.0532391
$265$	0	0
$266$	7940.10	1.83022
$267$	567.178	0.130003
$268$	−7023.11	−1.60076
$269$	−6684.63	−1.51513	−0.757563	−	0.652762i	$-0.773610\pi$
$269$	−6684.63	−1.51513	−0.757563	+	0.652762i	$0.773610\pi$
$270$	0	0
$271$	−6784.76	−1.52083	−0.760415	−	0.649438i	$-0.775004\pi$
$271$	−6784.76	−1.52083	−0.760415	+	0.649438i	$0.775004\pi$
$272$	2288.50	0.510151
$273$	566.404	0.125569
$274$	−1904.35	−0.419875
$275$	0	0
$276$	1051.92	0.229413
$277$	−3811.10	−0.826666	−0.413333	−	0.910580i	$-0.635636\pi$
$277$	−3811.10	−0.826666	−0.413333	+	0.910580i	$0.635636\pi$
$278$	2215.81	0.478041
$279$	−132.186	−0.0283647
$280$	0	0
$281$	1848.27	0.392380	0.196190	−	0.980566i	$-0.437143\pi$
$281$	1848.27	0.392380	0.196190	+	0.980566i	$0.437143\pi$
$282$	3461.60	0.730976
$283$	−1381.94	−0.290275	−0.145138	−	0.989411i	$-0.546362\pi$
$283$	−1381.94	−0.290275	−0.145138	+	0.989411i	$0.546362\pi$
$284$	3696.08	0.772261
$285$	0	0
$286$	139.028	0.0287443
$287$	12609.2	2.59338
$288$	2104.36	0.430557
$289$	2256.98	0.459390
$290$	0	0
$291$	−412.890	−0.0831755
$292$	9400.65	1.88401
$293$	−6221.44	−1.24048	−0.620239	−	0.784413i	$-0.712964\pi$
$293$	−6221.44	−1.24048	−0.620239	+	0.784413i	$0.712964\pi$
$294$	8579.90	1.70201
$295$	0	0
$296$	−111.722	−0.0219382
$297$	142.790	0.0278974
$298$	1557.19	0.302702
$299$	−186.144	−0.0360034
$300$	0	0
$301$	−567.499	−0.108671
$302$	1429.66	0.272410
$303$	5163.45	0.978985
$304$	−1549.93	−0.292416
$305$	0	0
$306$	3346.08	0.625106
$307$	−8135.50	−1.51244	−0.756218	−	0.654320i	$-0.772955\pi$
$307$	−8135.50	−1.51244	−0.756218	+	0.654320i	$0.772955\pi$
$308$	1880.84	0.347957
$309$	2392.75	0.440515
$310$	0	0
$311$	3117.65	0.568444	0.284222	−	0.958759i	$-0.408265\pi$
$311$	3117.65	0.568444	0.284222	+	0.958759i	$0.408265\pi$
$312$	258.543	0.0469138
$313$	−7536.36	−1.36096	−0.680480	−	0.732767i	$-0.738229\pi$
$313$	−7536.36	−1.36096	−0.680480	+	0.732767i	$0.738229\pi$
$314$	−11811.4	−2.12278
$315$	0	0
$316$	−7516.49	−1.33809
$317$	−3579.28	−0.634172	−0.317086	−	0.948397i	$-0.602704\pi$
$317$	−3579.28	−0.634172	−0.317086	+	0.948397i	$0.602704\pi$
$318$	856.089	0.150966
$319$	−153.367	−0.0269183
$320$	0	0
$321$	−2335.35	−0.406065
$322$	−4304.53	−0.744976
$323$	−4855.98	−0.836515
$324$	913.540	0.156643
$325$	0	0
$326$	6197.52	1.05291
$327$	−2163.62	−0.365897
$328$	5755.66	0.968911
$329$	−8286.96	−1.38868
$330$	0	0
$331$	−6257.20	−1.03905	−0.519527	−	0.854454i	$-0.673892\pi$
$331$	−6257.20	−1.03905	−0.519527	+	0.854454i	$0.673892\pi$
$332$	14895.5	2.46233
$333$	69.8557	0.0114957
$334$	−11144.7	−1.82578
$335$	0	0
$336$	−2556.75	−0.415125
$337$	4034.64	0.652168	0.326084	−	0.945341i	$-0.394271\pi$
$337$	4034.64	0.652168	0.326084	+	0.945341i	$0.394271\pi$
$338$	9488.98	1.52702
$339$	−1677.07	−0.268691
$340$	0	0
$341$	77.6744	0.0123352
$342$	−2266.18	−0.358307
$343$	−9723.97	−1.53074
$344$	−259.043	−0.0406007
$345$	0	0
$346$	−1339.30	−0.208095
$347$	10221.1	1.58126	0.790630	−	0.612294i	$-0.209753\pi$
$347$	10221.1	1.58126	0.790630	+	0.612294i	$0.209753\pi$
$348$	−981.210	−0.151145
$349$	−2543.26	−0.390079	−0.195040	−	0.980795i	$-0.562484\pi$
$349$	−2543.26	−0.390079	−0.195040	+	0.980795i	$0.562484\pi$
$350$	0	0
$351$	−161.657	−0.0245830
$352$	−1236.55	−0.187240
$353$	−11741.3	−1.77034	−0.885168	−	0.465272i	$-0.845956\pi$
$353$	−11741.3	−1.77034	−0.885168	+	0.465272i	$0.845956\pi$
$354$	1617.37	0.242832
$355$	0	0
$356$	−2132.27	−0.317443
$357$	−8010.40	−1.18755
$358$	−13582.3	−2.00517
$359$	5475.18	0.804927	0.402463	−	0.915436i	$-0.368154\pi$
$359$	5475.18	0.804927	0.402463	+	0.915436i	$0.368154\pi$
$360$	0	0
$361$	−3570.21	−0.520514
$362$	−12295.6	−1.78520
$363$	3909.09	0.565218
$364$	−2129.36	−0.306617
$365$	0	0
$366$	3482.49	0.497357
$367$	−11158.0	−1.58704	−0.793522	−	0.608542i	$-0.791755\pi$
$367$	−11158.0	−1.58704	−0.793522	+	0.608542i	$0.791755\pi$
$368$	840.254	0.119025
$369$	−3598.80	−0.507713
$370$	0	0
$371$	−2049.45	−0.286798
$372$	496.943	0.0692616
$373$	−5823.05	−0.808327	−0.404163	−	0.914687i	$-0.632437\pi$
$373$	−5823.05	−0.808327	−0.404163	+	0.914687i	$0.632437\pi$
$374$	−1966.20	−0.271845
$375$	0	0
$376$	−3782.70	−0.518823
$377$	173.632	0.0237202
$378$	−3738.28	−0.508667
$379$	2272.79	0.308036	0.154018	−	0.988068i	$-0.450779\pi$
$379$	2272.79	0.308036	0.154018	+	0.988068i	$0.450779\pi$
$380$	0	0
$381$	−1878.97	−0.252658
$382$	−18479.5	−2.47512
$383$	−8686.06	−1.15884	−0.579422	−	0.815028i	$-0.696722\pi$
$383$	−8686.06	−1.15884	−0.579422	+	0.815028i	$0.696722\pi$
$384$	−5063.19	−0.672865
$385$	0	0
$386$	−1086.98	−0.143331
$387$	161.970	0.0212749
$388$	1552.23	0.203099
$389$	5410.18	0.705159	0.352580	−	0.935782i	$-0.385305\pi$
$389$	5410.18	0.705159	0.352580	+	0.935782i	$0.385305\pi$
$390$	0	0
$391$	2632.55	0.340496
$392$	−9375.76	−1.20803
$393$	−5292.35	−0.679298
$394$	−20083.4	−2.56798
$395$	0	0
$396$	−536.810	−0.0681205
$397$	−13470.8	−1.70297	−0.851486	−	0.524377i	$-0.824298\pi$
$397$	−13470.8	−1.70297	−0.851486	+	0.524377i	$0.824298\pi$
$398$	−11420.4	−1.43833
$399$	5425.17	0.680697
$400$	0	0
$401$	1749.26	0.217840	0.108920	−	0.994050i	$-0.465261\pi$
$401$	1749.26	0.217840	0.108920	+	0.994050i	$0.465261\pi$
$402$	−8202.43	−1.01766
$403$	−87.9376	−0.0108697
$404$	−19411.6	−2.39050
$405$	0	0
$406$	4015.19	0.490814
$407$	−41.0482	−0.00499923
$408$	−3656.45	−0.443680
$409$	−4261.47	−0.515198	−0.257599	−	0.966252i	$-0.582931\pi$
$409$	−4261.47	−0.515198	−0.257599	+	0.966252i	$0.582931\pi$
$410$	0	0
$411$	−1301.17	−0.156160
$412$	−8995.38	−1.07566
$413$	−3871.94	−0.461321
$414$	1228.56	0.145846
$415$	0	0
$416$	1399.94	0.164994
$417$	1513.98	0.177793
$418$	1331.64	0.155820
$419$	−3355.37	−0.391218	−0.195609	−	0.980682i	$-0.562668\pi$
$419$	−3355.37	−0.391218	−0.195609	+	0.980682i	$0.562668\pi$
$420$	0	0
$421$	−7859.86	−0.909896	−0.454948	−	0.890518i	$-0.650342\pi$
$421$	−7859.86	−0.909896	−0.454948	+	0.890518i	$0.650342\pi$
$422$	−1071.98	−0.123657
$423$	2365.18	0.271865
$424$	−935.499	−0.107151
$425$	0	0
$426$	4316.72	0.490953
$427$	−8336.98	−0.944858
$428$	8779.59	0.991537
$429$	94.9923	0.0106906
$430$	0	0
$431$	−9083.57	−1.01517	−0.507587	−	0.861600i	$-0.669463\pi$
$431$	−9083.57	−1.01517	−0.507587	+	0.861600i	$0.669463\pi$
$432$	729.720	0.0812701
$433$	−10303.2	−1.14351	−0.571756	−	0.820424i	$-0.693738\pi$
$433$	−10303.2	−1.14351	−0.571756	+	0.820424i	$0.693738\pi$
$434$	−2033.53	−0.224914
$435$	0	0
$436$	8133.95	0.893453
$437$	−1782.94	−0.195171
$438$	10979.2	1.19773
$439$	10285.7	1.11824	0.559122	−	0.829086i	$-0.311139\pi$
$439$	10285.7	1.11824	0.559122	+	0.829086i	$0.311139\pi$
$440$	0	0
$441$	5862.32	0.633011
$442$	2226.00	0.239547
$443$	−11628.0	−1.24710	−0.623549	−	0.781784i	$-0.714310\pi$
$443$	−11628.0	−1.24710	−0.623549	+	0.781784i	$0.714310\pi$
$444$	−262.617	−0.0280704
$445$	0	0
$446$	18349.9	1.94819
$447$	1063.97	0.112581
$448$	25555.2	2.69502
$449$	−13543.0	−1.42346	−0.711730	−	0.702453i	$-0.752088\pi$
$449$	−13543.0	−1.42346	−0.711730	+	0.702453i	$0.752088\pi$
$450$	0	0
$451$	2114.71	0.220793
$452$	6304.83	0.656094
$453$	976.834	0.101315
$454$	17714.9	1.83128
$455$	0	0
$456$	2476.39	0.254315
$457$	−16383.2	−1.67696	−0.838481	−	0.544930i	$-0.816556\pi$
$457$	−16383.2	−1.67696	−0.838481	+	0.544930i	$0.816556\pi$
$458$	−19764.9	−2.01649
$459$	2286.25	0.232490
$460$	0	0
$461$	13577.7	1.37175	0.685877	−	0.727717i	$-0.259419\pi$
$461$	13577.7	1.37175	0.685877	+	0.727717i	$0.259419\pi$
$462$	2196.67	0.221208
$463$	17986.2	1.80538	0.902688	−	0.430297i	$-0.141591\pi$
$463$	17986.2	1.80538	0.902688	+	0.430297i	$0.141591\pi$
$464$	−783.774	−0.0784176
$465$	0	0
$466$	−20683.4	−2.05609
$467$	−792.472	−0.0785251	−0.0392625	−	0.999229i	$-0.512501\pi$
$467$	−792.472	−0.0785251	−0.0392625	+	0.999229i	$0.512501\pi$
$468$	607.739	0.0600272
$469$	19636.3	1.93331
$470$	0	0
$471$	−8070.26	−0.789507
$472$	−1767.40	−0.172354
$473$	−95.1759	−0.00925199
$474$	−8778.65	−0.850668
$475$	0	0
$476$	30114.5	2.89978
$477$	584.933	0.0561472
$478$	2180.57	0.208655
$479$	−6655.02	−0.634813	−0.317407	−	0.948290i	$-0.602812\pi$
$479$	−6655.02	−0.634813	−0.317407	+	0.948290i	$0.602812\pi$
$480$	0	0
$481$	46.4719	0.00440528
$482$	−28164.7	−2.66155
$483$	−2941.12	−0.277072
$484$	−14695.9	−1.38016
$485$	0	0
$486$	1066.94	0.0995832
$487$	13972.7	1.30013	0.650064	−	0.759880i	$-0.274742\pi$
$487$	13972.7	1.30013	0.650064	+	0.759880i	$0.274742\pi$
$488$	−3805.53	−0.353008
$489$	4234.53	0.391600
$490$	0	0
$491$	3638.63	0.334438	0.167219	−	0.985920i	$-0.446521\pi$
$491$	3638.63	0.334438	0.167219	+	0.985920i	$0.446521\pi$
$492$	13529.4	1.23974
$493$	−2455.60	−0.224330
$494$	−1507.59	−0.137307
$495$	0	0
$496$	396.950	0.0359346
$497$	−10334.1	−0.932691
$498$	17396.7	1.56539
$499$	11313.5	1.01496	0.507478	−	0.861665i	$-0.330578\pi$
$499$	11313.5	1.01496	0.507478	+	0.861665i	$0.330578\pi$
$500$	0	0
$501$	−7614.76	−0.679047
$502$	−20818.7	−1.85097
$503$	4598.92	0.407665	0.203833	−	0.979006i	$-0.434660\pi$
$503$	4598.92	0.407665	0.203833	+	0.979006i	$0.434660\pi$
$504$	4085.04	0.361036
$505$	0	0
$506$	−721.917	−0.0634252
$507$	6483.46	0.567930
$508$	7063.86	0.616945
$509$	6967.41	0.606729	0.303364	−	0.952875i	$-0.401890\pi$
$509$	6967.41	0.606729	0.303364	+	0.952875i	$0.401890\pi$
$510$	0	0
$511$	−26283.8	−2.27540
$512$	−9431.48	−0.814095
$513$	−1548.40	−0.133262
$514$	−26829.3	−2.30231
$515$	0	0
$516$	−608.914	−0.0519495
$517$	−1389.81	−0.118228
$518$	1074.65	0.0911533
$519$	−915.090	−0.0773949
$520$	0	0
$521$	−11038.5	−0.928228	−0.464114	−	0.885776i	$-0.653627\pi$
$521$	−11038.5	−0.928228	−0.464114	+	0.885776i	$0.653627\pi$
$522$	−1145.97	−0.0960879
$523$	15368.6	1.28493	0.642467	−	0.766314i	$-0.277911\pi$
$523$	15368.6	1.28493	0.642467	+	0.766314i	$0.277911\pi$
$524$	19896.2	1.65872
$525$	0	0
$526$	−17147.4	−1.42141
$527$	1243.66	0.102798
$528$	−428.795	−0.0353426
$529$	−11200.4	−0.920558
$530$	0	0
$531$	1105.09	0.0903141
$532$	−20395.5	−1.66214
$533$	−2394.12	−0.194561
$534$	−2490.31	−0.201810
$535$	0	0
$536$	8963.28	0.722303
$537$	−9280.30	−0.745762
$538$	29350.2	2.35200
$539$	−3444.78	−0.275283
$540$	0	0
$541$	−10945.5	−0.869839	−0.434920	−	0.900469i	$-0.643223\pi$
$541$	−10945.5	−0.869839	−0.434920	+	0.900469i	$0.643223\pi$
$542$	29789.9	2.36086
$543$	−8401.13	−0.663954
$544$	−19798.7	−1.56041
$545$	0	0
$546$	−2486.91	−0.194927
$547$	−10140.1	−0.792616	−0.396308	−	0.918118i	$-0.629709\pi$
$547$	−10140.1	−0.792616	−0.396308	+	0.918118i	$0.629709\pi$
$548$	4891.64	0.381315
$549$	2379.45	0.184977
$550$	0	0
$551$	1663.09	0.128585
$552$	−1342.52	−0.103517
$553$	21015.8	1.61606
$554$	16733.4	1.28327
$555$	0	0
$556$	−5691.69	−0.434139
$557$	−18329.0	−1.39430	−0.697151	−	0.716925i	$-0.745549\pi$
$557$	−18329.0	−1.39430	−0.697151	+	0.716925i	$0.745549\pi$
$558$	580.389	0.0440320
$559$	107.752	0.00815278
$560$	0	0
$561$	−1343.43	−0.101105
$562$	−8115.21	−0.609110
$563$	−2868.99	−0.214766	−0.107383	−	0.994218i	$-0.534247\pi$
$563$	−2868.99	−0.214766	−0.107383	+	0.994218i	$0.534247\pi$
$564$	−8891.72	−0.663845
$565$	0	0
$566$	6067.69	0.450608
$567$	−2554.22	−0.189184
$568$	−4717.14	−0.348462
$569$	18562.0	1.36759	0.683795	−	0.729674i	$-0.260328\pi$
$569$	18562.0	1.36759	0.683795	+	0.729674i	$0.260328\pi$
$570$	0	0
$571$	−10162.6	−0.744815	−0.372408	−	0.928069i	$-0.621468\pi$
$571$	−10162.6	−0.744815	−0.372408	+	0.928069i	$0.621468\pi$
$572$	−357.116	−0.0261045
$573$	−12626.4	−0.920547
$574$	−55363.4	−4.02583
$575$	0	0
$576$	−7293.70	−0.527611
$577$	−2525.24	−0.182196	−0.0910981	−	0.995842i	$-0.529038\pi$
$577$	−2525.24	−0.182196	−0.0910981	+	0.995842i	$0.529038\pi$
$578$	−9909.74	−0.713133
$579$	−742.689	−0.0533076
$580$	0	0
$581$	−41647.1	−2.97386
$582$	1812.88	0.129117
$583$	−343.715	−0.0244172
$584$	−11997.6	−0.850111
$585$	0	0
$586$	27316.5	1.92565
$587$	−3327.35	−0.233960	−0.116980	−	0.993134i	$-0.537321\pi$
$587$	−3327.35	−0.233960	−0.116980	+	0.993134i	$0.537321\pi$
$588$	−22038.9	−1.54570
$589$	−842.289	−0.0589234
$590$	0	0
$591$	−13722.2	−0.955086
$592$	−209.774	−0.0145636
$593$	16731.4	1.15864	0.579321	−	0.815099i	$-0.303318\pi$
$593$	16731.4	1.15864	0.579321	+	0.815099i	$0.303318\pi$
$594$	−626.950	−0.0433065
$595$	0	0
$596$	−3999.90	−0.274903
$597$	−7803.16	−0.534945
$598$	817.305	0.0558898
$599$	−12851.1	−0.876600	−0.438300	−	0.898829i	$-0.644419\pi$
$599$	−12851.1	−0.876600	−0.438300	+	0.898829i	$0.644419\pi$
$600$	0	0
$601$	−8773.05	−0.595441	−0.297720	−	0.954653i	$-0.596226\pi$
$601$	−8773.05	−0.595441	−0.297720	+	0.954653i	$0.596226\pi$
$602$	2491.72	0.168696
$603$	−5604.40	−0.378489
$604$	−3672.33	−0.247393
$605$	0	0
$606$	−22671.2	−1.51973
$607$	19083.1	1.27605	0.638023	−	0.770017i	$-0.279753\pi$
$607$	19083.1	1.27605	0.638023	+	0.770017i	$0.279753\pi$
$608$	13409.0	0.894417
$609$	2743.42	0.182544
$610$	0	0
$611$	1573.45	0.104182
$612$	−8594.97	−0.567698
$613$	−20976.8	−1.38213	−0.691063	−	0.722794i	$-0.742858\pi$
$613$	−20976.8	−1.38213	−0.691063	+	0.722794i	$0.742858\pi$
$614$	35720.6	2.34783
$615$	0	0
$616$	−2400.43	−0.157007
$617$	15290.1	0.997658	0.498829	−	0.866700i	$-0.333764\pi$
$617$	15290.1	0.997658	0.498829	+	0.866700i	$0.333764\pi$
$618$	−10505.9	−0.683832
$619$	18786.8	1.21988	0.609938	−	0.792449i	$-0.291194\pi$
$619$	18786.8	1.21988	0.609938	+	0.792449i	$0.291194\pi$
$620$	0	0
$621$	839.425	0.0542431
$622$	−13688.7	−0.882422
$623$	5961.73	0.383389
$624$	485.451	0.0311436
$625$	0	0
$626$	33089.9	2.11268
$627$	909.860	0.0579527
$628$	30339.5	1.92783
$629$	−657.231	−0.0416622
$630$	0	0
$631$	14273.7	0.900516	0.450258	−	0.892898i	$-0.351332\pi$
$631$	14273.7	0.900516	0.450258	+	0.892898i	$0.351332\pi$
$632$	9592.94	0.603777
$633$	−732.445	−0.0459907
$634$	15715.6	0.984456
$635$	0	0
$636$	−2199.01	−0.137101
$637$	3899.95	0.242577
$638$	673.391	0.0417865
$639$	2949.45	0.182595
$640$	0	0
$641$	10936.9	0.673919	0.336959	−	0.941519i	$-0.390602\pi$
$641$	10936.9	0.673919	0.336959	+	0.941519i	$0.390602\pi$
$642$	10253.9	0.630354
$643$	−5329.32	−0.326855	−0.163427	−	0.986555i	$-0.552255\pi$
$643$	−5329.32	−0.326855	−0.163427	+	0.986555i	$0.552255\pi$
$644$	11056.9	0.676559
$645$	0	0
$646$	21321.2	1.29856
$647$	15807.3	0.960509	0.480255	−	0.877129i	$-0.340544\pi$
$647$	15807.3	0.960509	0.480255	+	0.877129i	$0.340544\pi$
$648$	−1165.91	−0.0706809
$649$	−649.367	−0.0392756
$650$	0	0
$651$	−1389.43	−0.0836500
$652$	−15919.4	−0.956215
$653$	11047.2	0.662037	0.331019	−	0.943624i	$-0.392608\pi$
$653$	11047.2	0.662037	0.331019	+	0.943624i	$0.392608\pi$
$654$	9499.79	0.567999
$655$	0	0
$656$	10807.1	0.643209
$657$	7501.67	0.445461
$658$	36385.6	2.15571
$659$	−20355.2	−1.20322	−0.601612	−	0.798789i	$-0.705475\pi$
$659$	−20355.2	−1.20322	−0.601612	+	0.798789i	$0.705475\pi$
$660$	0	0
$661$	−27873.5	−1.64017	−0.820086	−	0.572241i	$-0.806074\pi$
$661$	−27873.5	−1.64017	−0.820086	+	0.572241i	$0.806074\pi$
$662$	27473.5	1.61297
$663$	1520.94	0.0890926
$664$	−19010.4	−1.11106
$665$	0	0
$666$	−306.716	−0.0178453
$667$	−901.605	−0.0523392
$668$	28627.1	1.65811
$669$	12537.8	0.724572
$670$	0	0
$671$	−1398.20	−0.0804427
$672$	22119.3	1.26975
$673$	9555.18	0.547288	0.273644	−	0.961831i	$-0.411771\pi$
$673$	9555.18	0.547288	0.273644	+	0.961831i	$0.411771\pi$
$674$	−17714.9	−1.01239
$675$	0	0
$676$	−24374.1	−1.38678
$677$	−17556.1	−0.996658	−0.498329	−	0.866988i	$-0.666053\pi$
$677$	−17556.1	−0.996658	−0.498329	+	0.866988i	$0.666053\pi$
$678$	7363.53	0.417101
$679$	−4339.97	−0.245292
$680$	0	0
$681$	12103.9	0.681091
$682$	−341.045	−0.0191485
$683$	−11046.7	−0.618875	−0.309437	−	0.950920i	$-0.600141\pi$
$683$	−11046.7	−0.618875	−0.309437	+	0.950920i	$0.600141\pi$
$684$	5821.08	0.325401
$685$	0	0
$686$	42695.1	2.37625
$687$	−13504.6	−0.749974
$688$	−486.390	−0.0269527
$689$	389.130	0.0215162
$690$	0	0
$691$	−7999.45	−0.440396	−0.220198	−	0.975455i	$-0.570670\pi$
$691$	−7999.45	−0.440396	−0.220198	+	0.975455i	$0.570670\pi$
$692$	3440.21	0.188984
$693$	1500.90	0.0822719
$694$	−44877.8	−2.45467
$695$	0	0
$696$	1252.27	0.0682001
$697$	33859.0	1.84003
$698$	11166.7	0.605538
$699$	−14132.1	−0.764702
$700$	0	0
$701$	−25401.3	−1.36861	−0.684303	−	0.729198i	$-0.739893\pi$
$701$	−25401.3	−1.36861	−0.684303	+	0.729198i	$0.739893\pi$
$702$	709.790	0.0381614
$703$	445.120	0.0238806
$704$	4285.88	0.229447
$705$	0	0
$706$	51552.7	2.74818
$707$	54274.1	2.88711
$708$	−4154.50	−0.220531
$709$	−9479.34	−0.502122	−0.251061	−	0.967971i	$-0.580779\pi$
$709$	−9479.34	−0.502122	−0.251061	+	0.967971i	$0.580779\pi$
$710$	0	0
$711$	−5998.11	−0.316381
$712$	2721.31	0.143238
$713$	456.626	0.0239843
$714$	35171.3	1.84349
$715$	0	0
$716$	34888.6	1.82102
$717$	1489.90	0.0776030
$718$	−24039.9	−1.24953
$719$	−32420.3	−1.68161	−0.840803	−	0.541341i	$-0.817917\pi$
$719$	−32420.3	−1.68161	−0.840803	+	0.541341i	$0.817917\pi$
$720$	0	0
$721$	25150.7	1.29912
$722$	15675.7	0.808019
$723$	−19243.9	−0.989885
$724$	31583.4	1.62126
$725$	0	0
$726$	−17163.7	−0.877416
$727$	6808.51	0.347337	0.173668	−	0.984804i	$-0.444438\pi$
$727$	6808.51	0.347337	0.173668	+	0.984804i	$0.444438\pi$
$728$	2717.60	0.138353
$729$	729.000	0.0370370
$730$	0	0
$731$	−1523.88	−0.0771036
$732$	−8945.38	−0.451681
$733$	−20190.4	−1.01740	−0.508698	−	0.860945i	$-0.669873\pi$
$733$	−20190.4	−1.01740	−0.508698	+	0.860945i	$0.669873\pi$
$734$	48991.6	2.46364
$735$	0	0
$736$	−7269.35	−0.364065
$737$	3293.23	0.164597
$738$	15801.3	0.788147
$739$	15130.0	0.753134	0.376567	−	0.926389i	$-0.377105\pi$
$739$	15130.0	0.753134	0.376567	+	0.926389i	$0.377105\pi$
$740$	0	0
$741$	−1030.08	−0.0510674
$742$	8998.52	0.445210
$743$	13315.6	0.657470	0.328735	−	0.944422i	$-0.393378\pi$
$743$	13315.6	0.657470	0.328735	+	0.944422i	$0.393378\pi$
$744$	−634.226	−0.0312525
$745$	0	0
$746$	25567.3	1.25480
$747$	11886.5	0.582200
$748$	5050.53	0.246879
$749$	−24547.4	−1.19752
$750$	0	0
$751$	−18846.4	−0.915734	−0.457867	−	0.889021i	$-0.651386\pi$
$751$	−18846.4	−0.915734	−0.457867	+	0.889021i	$0.651386\pi$
$752$	−7102.55	−0.344419
$753$	−14224.7	−0.688413
$754$	−762.367	−0.0368220
$755$	0	0
$756$	9602.41	0.461953
$757$	−699.522	−0.0335859	−0.0167930	−	0.999859i	$-0.505346\pi$
$757$	−699.522	−0.0335859	−0.0167930	+	0.999859i	$0.505346\pi$
$758$	−9979.16	−0.478178
$759$	−493.259	−0.0235891
$760$	0	0
$761$	−37430.3	−1.78298	−0.891489	−	0.453041i	$-0.850339\pi$
$761$	−37430.3	−1.78298	−0.891489	+	0.453041i	$0.850339\pi$
$762$	8250.01	0.392213
$763$	−22742.2	−1.07906
$764$	47467.8	2.24781
$765$	0	0
$766$	38137.9	1.79893
$767$	735.168	0.0346093
$768$	2781.13	0.130671
$769$	30299.7	1.42085	0.710426	−	0.703772i	$-0.248502\pi$
$769$	30299.7	1.42085	0.710426	+	0.703772i	$0.248502\pi$
$770$	0	0
$771$	−18331.4	−0.856277
$772$	2792.09	0.130168
$773$	−16926.9	−0.787603	−0.393802	−	0.919195i	$-0.628840\pi$
$773$	−16926.9	−0.787603	−0.393802	+	0.919195i	$0.628840\pi$
$774$	−711.162	−0.0330261
$775$	0	0
$776$	−1981.04	−0.0916433
$777$	734.267	0.0339018
$778$	−23754.5	−1.09465
$779$	−22931.5	−1.05470
$780$	0	0
$781$	−1733.14	−0.0794068
$782$	−11558.8	−0.528568
$783$	−783.000	−0.0357371
$784$	−17604.3	−0.801946
$785$	0	0
$786$	23237.2	1.05451
$787$	20726.9	0.938799	0.469400	−	0.882986i	$-0.344470\pi$
$787$	20726.9	0.938799	0.469400	+	0.882986i	$0.344470\pi$
$788$	51587.6	2.33215
$789$	−11716.2	−0.528653
$790$	0	0
$791$	−17628.1	−0.792391
$792$	685.106	0.0307376
$793$	1582.95	0.0708854
$794$	59146.3	2.64361
$795$	0	0
$796$	29335.4	1.30624
$797$	25560.6	1.13601	0.568006	−	0.823025i	$-0.307715\pi$
$797$	25560.6	1.13601	0.568006	+	0.823025i	$0.307715\pi$
$798$	−23820.3	−1.05668
$799$	−22252.6	−0.985282
$800$	0	0
$801$	−1701.54	−0.0750572
$802$	−7680.49	−0.338164
$803$	−4408.09	−0.193721
$804$	21069.3	0.924202
$805$	0	0
$806$	386.108	0.0168735
$807$	20053.9	0.874758
$808$	24774.1	1.07865
$809$	−45525.7	−1.97849	−0.989244	−	0.146273i	$-0.953272\pi$
$809$	−45525.7	−1.97849	−0.989244	+	0.146273i	$0.953272\pi$
$810$	0	0
$811$	8515.77	0.368717	0.184358	−	0.982859i	$-0.440979\pi$
$811$	8515.77	0.368717	0.184358	+	0.982859i	$0.440979\pi$
$812$	−10313.7	−0.445739
$813$	20354.3	0.878051
$814$	180.231	0.00776054
$815$	0	0
$816$	−6865.51	−0.294536
$817$	1032.07	0.0441954
$818$	18710.8	0.799767
$819$	−1699.21	−0.0724973
$820$	0	0
$821$	18922.1	0.804369	0.402185	−	0.915559i	$-0.368251\pi$
$821$	18922.1	0.804369	0.402185	+	0.915559i	$0.368251\pi$
$822$	5713.04	0.242415
$823$	13297.5	0.563210	0.281605	−	0.959530i	$-0.409133\pi$
$823$	13297.5	0.563210	0.281605	+	0.959530i	$0.409133\pi$
$824$	11480.4	0.485362
$825$	0	0
$826$	17000.5	0.716131
$827$	26020.8	1.09411	0.547056	−	0.837096i	$-0.315748\pi$
$827$	26020.8	1.09411	0.547056	+	0.837096i	$0.315748\pi$
$828$	−3155.76	−0.132452
$829$	8756.88	0.366874	0.183437	−	0.983031i	$-0.441278\pi$
$829$	8756.88	0.366874	0.183437	+	0.983031i	$0.441278\pi$
$830$	0	0
$831$	11433.3	0.477276
$832$	−4852.18	−0.202186
$833$	−55155.1	−2.29413
$834$	−6647.43	−0.275997
$835$	0	0
$836$	−3420.55	−0.141510
$837$	396.558	0.0163764
$838$	14732.4	0.607307
$839$	−1131.02	−0.0465400	−0.0232700	−	0.999729i	$-0.507408\pi$
$839$	−1131.02	−0.0465400	−0.0232700	+	0.999729i	$0.507408\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	34510.3	1.41248
$843$	−5544.82	−0.226540
$844$	2753.57	0.112301
$845$	0	0
$846$	−10384.8	−0.422029
$847$	41089.3	1.66688
$848$	−1756.53	−0.0711316
$849$	4145.82	0.167590
$850$	0	0
$851$	−241.311	−0.00972037
$852$	−11088.2	−0.445865
$853$	12199.3	0.489679	0.244839	−	0.969564i	$-0.421265\pi$
$853$	12199.3	0.489679	0.244839	+	0.969564i	$0.421265\pi$
$854$	36605.2	1.46675
$855$	0	0
$856$	−11205.0	−0.447405
$857$	−37800.4	−1.50670	−0.753348	−	0.657622i	$-0.771562\pi$
$857$	−37800.4	−1.50670	−0.753348	+	0.657622i	$0.771562\pi$
$858$	−417.083	−0.0165955
$859$	30888.1	1.22688	0.613438	−	0.789743i	$-0.289786\pi$
$859$	30888.1	1.22688	0.613438	+	0.789743i	$0.289786\pi$
$860$	0	0
$861$	−37827.7	−1.49729
$862$	39883.3	1.57590
$863$	−38008.4	−1.49921	−0.749607	−	0.661883i	$-0.769758\pi$
$863$	−38008.4	−1.49921	−0.749607	+	0.661883i	$0.769758\pi$
$864$	−6313.08	−0.248582
$865$	0	0
$866$	45238.3	1.77513
$867$	−6770.95	−0.265229
$868$	5223.47	0.204258
$869$	3524.58	0.137587
$870$	0	0
$871$	−3728.37	−0.145041
$872$	−10381.0	−0.403147
$873$	1238.67	0.0480214
$874$	7828.35	0.302973
$875$	0	0
$876$	−28202.0	−1.08773
$877$	−45609.0	−1.75611	−0.878055	−	0.478561i	$-0.841159\pi$
$877$	−45609.0	−1.75611	−0.878055	+	0.478561i	$0.841159\pi$
$878$	−45161.4	−1.73590
$879$	18664.3	0.716191
$880$	0	0
$881$	15306.9	0.585360	0.292680	−	0.956210i	$-0.405453\pi$
$881$	15306.9	0.585360	0.292680	+	0.956210i	$0.405453\pi$
$882$	−25739.7	−0.982654
$883$	4293.37	0.163628	0.0818140	−	0.996648i	$-0.473929\pi$
$883$	4293.37	0.163628	0.0818140	+	0.996648i	$0.473929\pi$
$884$	−5717.86	−0.217548
$885$	0	0
$886$	51055.3	1.93593
$887$	−15161.7	−0.573935	−0.286968	−	0.957940i	$-0.592647\pi$
$887$	−15161.7	−0.573935	−0.286968	+	0.957940i	$0.592647\pi$
$888$	335.166	0.0126660
$889$	−19750.3	−0.745110
$890$	0	0
$891$	−428.371	−0.0161066
$892$	−47134.9	−1.76927
$893$	15070.9	0.564758
$894$	−4671.56	−0.174765
$895$	0	0
$896$	−53220.3	−1.98434
$897$	558.433	0.0207866
$898$	59463.3	2.20971
$899$	−425.932	−0.0158016
$900$	0	0
$901$	−5503.29	−0.203486
$902$	−9285.05	−0.342748
$903$	1702.50	0.0627415
$904$	−8046.57	−0.296045
$905$	0	0
$906$	−4288.99	−0.157276
$907$	−45095.6	−1.65091	−0.825454	−	0.564469i	$-0.809081\pi$
$907$	−45095.6	−1.65091	−0.825454	+	0.564469i	$0.809081\pi$
$908$	−45503.8	−1.66310
$909$	−15490.3	−0.565217
$910$	0	0
$911$	10578.3	0.384714	0.192357	−	0.981325i	$-0.438387\pi$
$911$	10578.3	0.384714	0.192357	+	0.981325i	$0.438387\pi$
$912$	4649.78	0.168826
$913$	−6984.67	−0.253186
$914$	71933.6	2.60323
$915$	0	0
$916$	50769.5	1.83130
$917$	−55629.0	−2.00331
$918$	−10038.2	−0.360905
$919$	50326.0	1.80642	0.903210	−	0.429198i	$-0.141204\pi$
$919$	50326.0	1.80642	0.903210	+	0.429198i	$0.141204\pi$
$920$	0	0
$921$	24406.5	0.873205
$922$	−59615.9	−2.12944
$923$	1962.14	0.0699726
$924$	−5642.52	−0.200893
$925$	0	0
$926$	−78972.0	−2.80257
$927$	−7178.26	−0.254331
$928$	6780.71	0.239858
$929$	8373.10	0.295708	0.147854	−	0.989009i	$-0.452763\pi$
$929$	8373.10	0.295708	0.147854	+	0.989009i	$0.452763\pi$
$930$	0	0
$931$	37354.7	1.31498
$932$	53128.8	1.86726
$933$	−9352.96	−0.328191
$934$	3479.51	0.121898
$935$	0	0
$936$	−775.629	−0.0270857
$937$	−53048.3	−1.84953	−0.924766	−	0.380537i	$-0.875739\pi$
$937$	−53048.3	−1.84953	−0.924766	+	0.380537i	$0.875739\pi$
$938$	−86217.4	−3.00117
$939$	22609.1	0.785750
$940$	0	0
$941$	−1015.79	−0.0351899	−0.0175949	−	0.999845i	$-0.505601\pi$
$941$	−1015.79	−0.0351899	−0.0175949	+	0.999845i	$0.505601\pi$
$942$	35434.1	1.22559
$943$	12431.8	0.429305
$944$	−3318.54	−0.114417
$945$	0	0
$946$	417.889	0.0143623
$947$	26038.4	0.893490	0.446745	−	0.894661i	$-0.352583\pi$
$947$	26038.4	0.893490	0.446745	+	0.894661i	$0.352583\pi$
$948$	22549.5	0.772545
$949$	4990.53	0.170706
$950$	0	0
$951$	10737.9	0.366140
$952$	−38433.7	−1.30845
$953$	22396.1	0.761260	0.380630	−	0.924727i	$-0.375707\pi$
$953$	22396.1	0.761260	0.380630	+	0.924727i	$0.375707\pi$
$954$	−2568.27	−0.0871601
$955$	0	0
$956$	−5601.17	−0.189492
$957$	460.102	0.0155413
$958$	29220.2	0.985451
$959$	−13676.8	−0.460530
$960$	0	0
$961$	−29575.3	−0.992759
$962$	−204.045	−0.00683852
$963$	7006.06	0.234442
$964$	72345.9	2.41712
$965$	0	0
$966$	12913.6	0.430112
$967$	33614.9	1.11787	0.558936	−	0.829211i	$-0.311210\pi$
$967$	33614.9	1.11787	0.558936	+	0.829211i	$0.311210\pi$
$968$	18755.8	0.622761
$969$	14568.0	0.482962
$970$	0	0
$971$	−30440.6	−1.00606	−0.503030	−	0.864269i	$-0.667781\pi$
$971$	−30440.6	−1.00606	−0.503030	+	0.864269i	$0.667781\pi$
$972$	−2740.62	−0.0904377
$973$	15913.7	0.524328
$974$	−61349.8	−2.01825
$975$	0	0
$976$	−7145.42	−0.234343
$977$	−18698.6	−0.612305	−0.306153	−	0.951982i	$-0.599042\pi$
$977$	−18698.6	−0.612305	−0.306153	+	0.951982i	$0.599042\pi$
$978$	−18592.6	−0.607899
$979$	999.847	0.0326407
$980$	0	0
$981$	6490.85	0.211251
$982$	−15976.1	−0.519164
$983$	10438.4	0.338692	0.169346	−	0.985557i	$-0.445834\pi$
$983$	10438.4	0.338692	0.169346	+	0.985557i	$0.445834\pi$
$984$	−17267.0	−0.559401
$985$	0	0
$986$	10781.8	0.348238
$987$	24860.9	0.801753
$988$	3872.51	0.124697
$989$	−559.513	−0.0179893
$990$	0	0
$991$	20547.5	0.658641	0.329320	−	0.944218i	$-0.393180\pi$
$991$	20547.5	0.658641	0.329320	+	0.944218i	$0.393180\pi$
$992$	−3434.16	−0.109914
$993$	18771.6	0.599898
$994$	45373.9	1.44786
$995$	0	0
$996$	−44686.4	−1.42163
$997$	−6486.05	−0.206033	−0.103017	−	0.994680i	$-0.532850\pi$
$997$	−6486.05	−0.206033	−0.103017	+	0.994680i	$0.532850\pi$
$998$	−49674.3	−1.57556
$999$	−209.567	−0.00663704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.4.a.k.1.2		6
5.4	even	2		435.4.a.h.1.5	✓	6
15.14	odd	2		1305.4.a.h.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.h.1.5	✓	6	5.4	even	2
1305.4.a.h.1.2		6	15.14	odd	2
2175.4.a.k.1.2		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q-4.39070 q^{2} -3.00000 q^{3} +11.2783 q^{4} +13.1721 q^{6} -31.5336 q^{7} -14.3939 q^{8} +9.00000 q^{9} -5.28853 q^{11} -33.8348 q^{12} +5.98731 q^{13} +138.455 q^{14} -27.0267 q^{16} -84.6758 q^{17} -39.5163 q^{18} +57.3480 q^{19} +94.6008 q^{21} +23.2204 q^{22} -31.0898 q^{23} +43.1818 q^{24} -26.2885 q^{26} -27.0000 q^{27} -355.645 q^{28} +29.0000 q^{29} -14.6873 q^{31} +233.818 q^{32} +15.8656 q^{33} +371.786 q^{34} +101.504 q^{36} +7.76174 q^{37} -251.798 q^{38} -17.9619 q^{39} -399.867 q^{41} -415.364 q^{42} +17.9967 q^{43} -59.6455 q^{44} +136.506 q^{46} +262.798 q^{47} +81.0800 q^{48} +651.368 q^{49} +254.027 q^{51} +67.5265 q^{52} +64.9925 q^{53} +118.549 q^{54} +453.893 q^{56} -172.044 q^{57} -127.330 q^{58} +122.788 q^{59} +264.384 q^{61} +64.4877 q^{62} -283.802 q^{63} -810.411 q^{64} -69.6611 q^{66} -622.712 q^{67} -954.997 q^{68} +93.2694 q^{69} +327.717 q^{71} -129.545 q^{72} +833.518 q^{73} -34.0795 q^{74} +646.787 q^{76} +166.767 q^{77} +78.8655 q^{78} -666.457 q^{79} +81.0000 q^{81} +1755.70 q^{82} +1320.72 q^{83} +1066.93 q^{84} -79.0180 q^{86} -87.0000 q^{87} +76.1229 q^{88} -189.059 q^{89} -188.801 q^{91} -350.640 q^{92} +44.0620 q^{93} -1153.87 q^{94} -701.453 q^{96} +137.630 q^{97} -2859.97 q^{98} -47.5968 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 18 q^{3} + 51 q^{4} + 3 q^{6} - 47 q^{7} - 51 q^{8} + 54 q^{9} + 81 q^{11} - 153 q^{12} - 169 q^{13} - 30 q^{14} + 131 q^{16} + q^{17} - 9 q^{18} + 116 q^{19} + 141 q^{21} - 90 q^{22} + 52 q^{23}+ \cdots + 729 q^{99}+O(q^{100}) \) 6 * q - q^2 - 18 * q^3 + 51 * q^4 + 3 * q^6 - 47 * q^7 - 51 * q^8 + 54 * q^9 + 81 * q^11 - 153 * q^12 - 169 * q^13 - 30 * q^14 + 131 * q^16 + q^17 - 9 * q^18 + 116 * q^19 + 141 * q^21 - 90 * q^22 + 52 * q^23 + 153 * q^24 + 294 * q^26 - 162 * q^27 - 344 * q^28 + 174 * q^29 + 340 * q^31 - 499 * q^32 - 243 * q^33 + 920 * q^34 + 459 * q^36 - 332 * q^37 + 378 * q^38 + 507 * q^39 - 616 * q^41 + 90 * q^42 - 334 * q^43 - 52 * q^44 - 158 * q^46 + 85 * q^47 - 393 * q^48 + 879 * q^49 - 3 * q^51 - 2220 * q^52 + 850 * q^53 + 27 * q^54 - 624 * q^56 - 348 * q^57 - 29 * q^58 - 758 * q^59 - 36 * q^61 + 152 * q^62 - 423 * q^63 + 1795 * q^64 + 270 * q^66 - 939 * q^67 + 186 * q^68 - 156 * q^69 - 1388 * q^71 - 459 * q^72 - 1708 * q^73 - 814 * q^74 + 566 * q^76 - 2585 * q^77 - 882 * q^78 + 1250 * q^79 + 486 * q^81 - 1372 * q^82 + 748 * q^83 + 1032 * q^84 - 800 * q^86 - 522 * q^87 - 536 * q^88 + 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 1020 * q^93 - 4542 * q^94 + 1497 * q^96 + 22 * q^97 + 1433 * q^98 + 729 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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