LMFDB - Embedded newform 2254.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.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:	$132.990305153$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4 x^{10} - 212 x^{9} + 487 x^{8} + 16315 x^{7} - 9025 x^{6} - 516068 x^{5} - 504693 x^{4} + \cdots - 11394027 $ x^11 - 4x^10 - 212x^9 + 487x^8 + 16315x^7 - 9025x^6 - 516068x^5 - 504693x^4 + 5020566x^3 + 8901846x^2 - 4796415x - 11394027
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$9.96981$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -7.96981 q^{3} +4.00000 q^{4} +15.4032 q^{5} -15.9396 q^{6} +8.00000 q^{8} +36.5179 q^{9} +30.8063 q^{10} -59.6292 q^{11} -31.8793 q^{12} -17.4445 q^{13} -122.760 q^{15} +16.0000 q^{16} -100.674 q^{17} +73.0358 q^{18} -64.5559 q^{19} +61.6126 q^{20} -119.258 q^{22} +23.0000 q^{23} -63.7585 q^{24} +112.257 q^{25} -34.8889 q^{26} -75.8561 q^{27} -260.291 q^{29} -245.521 q^{30} +248.506 q^{31} +32.0000 q^{32} +475.233 q^{33} -201.348 q^{34} +146.072 q^{36} -358.250 q^{37} -129.112 q^{38} +139.029 q^{39} +123.225 q^{40} +432.148 q^{41} +470.883 q^{43} -238.517 q^{44} +562.491 q^{45} +46.0000 q^{46} -199.178 q^{47} -127.517 q^{48} +224.514 q^{50} +802.353 q^{51} -69.7778 q^{52} -120.925 q^{53} -151.712 q^{54} -918.477 q^{55} +514.498 q^{57} -520.583 q^{58} +881.429 q^{59} -491.041 q^{60} +320.352 q^{61} +497.013 q^{62} +64.0000 q^{64} -268.700 q^{65} +950.466 q^{66} +757.878 q^{67} -402.696 q^{68} -183.306 q^{69} -326.086 q^{71} +292.143 q^{72} -56.9293 q^{73} -716.500 q^{74} -894.668 q^{75} -258.224 q^{76} +278.058 q^{78} +562.924 q^{79} +246.450 q^{80} -381.425 q^{81} +864.296 q^{82} +342.227 q^{83} -1550.70 q^{85} +941.766 q^{86} +2074.47 q^{87} -477.033 q^{88} -96.5644 q^{89} +1124.98 q^{90} +92.0000 q^{92} -1980.55 q^{93} -398.356 q^{94} -994.364 q^{95} -255.034 q^{96} -102.191 q^{97} -2177.53 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 22 q^{2} + 18 q^{3} + 44 q^{4} + 33 q^{5} + 36 q^{6} + 88 q^{8} + 171 q^{9} + 66 q^{10} + 8 q^{11} + 72 q^{12} + 185 q^{13} - 186 q^{15} + 176 q^{16} + 107 q^{17} + 342 q^{18} + 114 q^{19} + 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) $ 11 * q + 22 * q^2 + 18 * q^3 + 44 * q^4 + 33 * q^5 + 36 * q^6 + 88 * q^8 + 171 * q^9 + 66 * q^10 + 8 * q^11 + 72 * q^12 + 185 * q^13 - 186 * q^15 + 176 * q^16 + 107 * q^17 + 342 * q^18 + 114 * q^19 + 132 * q^20 + 16 * q^22 + 253 * q^23 + 144 * q^24 + 494 * q^25 + 370 * q^26 + 561 * q^27 - 79 * q^29 - 372 * q^30 + 488 * q^31 + 352 * q^32 + 304 * q^33 + 214 * q^34 + 684 * q^36 - 41 * q^37 + 228 * q^38 + 425 * q^39 + 264 * q^40 + 812 * q^41 + 219 * q^43 + 32 * q^44 + 619 * q^45 + 506 * q^46 + 1141 * q^47 + 288 * q^48 + 988 * q^50 + 83 * q^51 + 740 * q^52 + 579 * q^53 + 1122 * q^54 + 951 * q^55 + 383 * q^57 - 158 * q^58 + 1695 * q^59 - 744 * q^60 + 1105 * q^61 + 976 * q^62 + 704 * q^64 - 347 * q^65 + 608 * q^66 + 695 * q^67 + 428 * q^68 + 414 * q^69 + 65 * q^71 + 1368 * q^72 + 1935 * q^73 - 82 * q^74 + 1712 * q^75 + 456 * q^76 + 850 * q^78 + 447 * q^79 + 528 * q^80 + 2203 * q^81 + 1624 * q^82 + 1389 * q^83 + 153 * q^85 + 438 * q^86 + 3354 * q^87 + 64 * q^88 + 2698 * q^89 + 1238 * q^90 + 1012 * q^92 - 686 * q^93 + 2282 * q^94 + 827 * q^95 + 576 * q^96 + 1589 * q^97 - 1729 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−7.96981	−1.53379	−0.766896	−	0.641772i	$-0.778200\pi$
$3$	−7.96981	−1.53379	−0.766896	+	0.641772i	$0.778200\pi$
$4$	4.00000	0.500000
$5$	15.4032	1.37770	0.688850	−	0.724904i	$-0.258116\pi$
$5$	15.4032	1.37770	0.688850	+	0.724904i	$0.258116\pi$
$6$	−15.9396	−1.08455
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	36.5179	1.35252
$10$	30.8063	0.974181
$11$	−59.6292	−1.63444	−0.817221	−	0.576324i	$-0.804487\pi$
$11$	−59.6292	−1.63444	−0.817221	+	0.576324i	$0.804487\pi$
$12$	−31.8793	−0.766896
$13$	−17.4445	−0.372171	−0.186085	−	0.982534i	$-0.559580\pi$
$13$	−17.4445	−0.372171	−0.186085	+	0.982534i	$0.559580\pi$
$14$	0	0
$15$	−122.760	−2.11310
$16$	16.0000	0.250000
$17$	−100.674	−1.43630	−0.718148	−	0.695891i	$-0.755010\pi$
$17$	−100.674	−1.43630	−0.718148	+	0.695891i	$0.755010\pi$
$18$	73.0358	0.956373
$19$	−64.5559	−0.779481	−0.389740	−	0.920925i	$-0.627435\pi$
$19$	−64.5559	−0.779481	−0.389740	+	0.920925i	$0.627435\pi$
$20$	61.6126	0.688850
$21$	0	0
$22$	−119.258	−1.15573
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−63.7585	−0.542277
$25$	112.257	0.898057
$26$	−34.8889	−0.263165
$27$	−75.8561	−0.540686
$28$	0	0
$29$	−260.291	−1.66672	−0.833361	−	0.552730i	$-0.813586\pi$
$29$	−260.291	−1.66672	−0.833361	+	0.552730i	$0.813586\pi$
$30$	−245.521	−1.49419
$31$	248.506	1.43978	0.719888	−	0.694090i	$-0.244193\pi$
$31$	248.506	1.43978	0.719888	+	0.694090i	$0.244193\pi$
$32$	32.0000	0.176777
$33$	475.233	2.50689
$34$	−201.348	−1.01561
$35$	0	0
$36$	146.072	0.676258
$37$	−358.250	−1.59178	−0.795891	−	0.605439i	$-0.792997\pi$
$37$	−358.250	−1.59178	−0.795891	+	0.605439i	$0.792997\pi$
$38$	−129.112	−0.551176
$39$	139.029	0.570833
$40$	123.225	0.487090
$41$	432.148	1.64610	0.823050	−	0.567968i	$-0.192270\pi$
$41$	432.148	1.64610	0.823050	+	0.567968i	$0.192270\pi$
$42$	0	0
$43$	470.883	1.66998	0.834988	−	0.550268i	$-0.185475\pi$
$43$	470.883	1.66998	0.834988	+	0.550268i	$0.185475\pi$
$44$	−238.517	−0.817221
$45$	562.491	1.86336
$46$	46.0000	0.147442
$47$	−199.178	−0.618152	−0.309076	−	0.951037i	$-0.600020\pi$
$47$	−199.178	−0.618152	−0.309076	+	0.951037i	$0.600020\pi$
$48$	−127.517	−0.383448
$49$	0	0
$50$	224.514	0.635022
$51$	802.353	2.20298
$52$	−69.7778	−0.186085
$53$	−120.925	−0.313402	−0.156701	−	0.987646i	$-0.550086\pi$
$53$	−120.925	−0.313402	−0.156701	+	0.987646i	$0.550086\pi$
$54$	−151.712	−0.382322
$55$	−918.477	−2.25177
$56$	0	0
$57$	514.498	1.19556
$58$	−520.583	−1.17855
$59$	881.429	1.94495	0.972477	−	0.233001i	$-0.0748545\pi$
$59$	881.429	1.94495	0.972477	+	0.233001i	$0.0748545\pi$
$60$	−491.041	−1.05655
$61$	320.352	0.672408	0.336204	−	0.941789i	$-0.390857\pi$
$61$	320.352	0.672408	0.336204	+	0.941789i	$0.390857\pi$
$62$	497.013	1.01808
$63$	0	0
$64$	64.0000	0.125000
$65$	−268.700	−0.512740
$66$	950.466	1.77264
$67$	757.878	1.38193	0.690967	−	0.722887i	$-0.257185\pi$
$67$	757.878	1.38193	0.690967	+	0.722887i	$0.257185\pi$
$68$	−402.696	−0.718148
$69$	−183.306	−0.319818
$70$	0	0
$71$	−326.086	−0.545061	−0.272530	−	0.962147i	$-0.587860\pi$
$71$	−326.086	−0.545061	−0.272530	+	0.962147i	$0.587860\pi$
$72$	292.143	0.478187
$73$	−56.9293	−0.0912750	−0.0456375	−	0.998958i	$-0.514532\pi$
$73$	−56.9293	−0.0912750	−0.0456375	+	0.998958i	$0.514532\pi$
$74$	−716.500	−1.12556
$75$	−894.668	−1.37743
$76$	−258.224	−0.389740
$77$	0	0
$78$	278.058	0.403640
$79$	562.924	0.801695	0.400848	−	0.916145i	$-0.368716\pi$
$79$	562.924	0.801695	0.400848	+	0.916145i	$0.368716\pi$
$80$	246.450	0.344425
$81$	−381.425	−0.523217
$82$	864.296	1.16397
$83$	342.227	0.452582	0.226291	−	0.974060i	$-0.427340\pi$
$83$	342.227	0.452582	0.226291	+	0.974060i	$0.427340\pi$
$84$	0	0
$85$	−1550.70	−1.97878
$86$	941.766	1.18085
$87$	2074.47	2.55640
$88$	−477.033	−0.577863
$89$	−96.5644	−0.115009	−0.0575045	−	0.998345i	$-0.518314\pi$
$89$	−96.5644	−0.115009	−0.0575045	+	0.998345i	$0.518314\pi$
$90$	1124.98	1.31760
$91$	0	0
$92$	92.0000	0.104257
$93$	−1980.55	−2.20832
$94$	−398.356	−0.437099
$95$	−994.364	−1.07389
$96$	−255.034	−0.271139
$97$	−102.191	−0.106968	−0.0534841	−	0.998569i	$-0.517033\pi$
$97$	−102.191	−0.106968	−0.0534841	+	0.998569i	$0.517033\pi$
$98$	0	0
$99$	−2177.53	−2.21061
$100$	449.028	0.449028
$101$	1327.01	1.30735	0.653675	−	0.756775i	$-0.273226\pi$
$101$	1327.01	1.30735	0.653675	+	0.756775i	$0.273226\pi$
$102$	1604.71	1.55774
$103$	179.661	0.171869	0.0859345	−	0.996301i	$-0.472612\pi$
$103$	179.661	0.171869	0.0859345	+	0.996301i	$0.472612\pi$
$104$	−139.556	−0.131582
$105$	0	0
$106$	−241.850	−0.221609
$107$	545.895	0.493212	0.246606	−	0.969116i	$-0.420685\pi$
$107$	545.895	0.493212	0.246606	+	0.969116i	$0.420685\pi$
$108$	−303.424	−0.270343
$109$	−1478.63	−1.29933	−0.649665	−	0.760221i	$-0.725091\pi$
$109$	−1478.63	−1.29933	−0.649665	+	0.760221i	$0.725091\pi$
$110$	−1836.95	−1.59224
$111$	2855.19	2.44146
$112$	0	0
$113$	456.443	0.379987	0.189993	−	0.981785i	$-0.439153\pi$
$113$	456.443	0.379987	0.189993	+	0.981785i	$0.439153\pi$
$114$	1029.00	0.845389
$115$	354.273	0.287270
$116$	−1041.17	−0.833361
$117$	−637.035	−0.503367
$118$	1762.86	1.37529
$119$	0	0
$120$	−982.082	−0.747095
$121$	2224.64	1.67140
$122$	640.705	0.475464
$123$	−3444.14	−2.52478
$124$	994.026	0.719888
$125$	−196.280	−0.140447
$126$	0	0
$127$	1176.09	0.821739	0.410870	−	0.911694i	$-0.365225\pi$
$127$	1176.09	0.821739	0.410870	+	0.911694i	$0.365225\pi$
$128$	128.000	0.0883883
$129$	−3752.85	−2.56139
$130$	−537.399	−0.362562
$131$	661.390	0.441114	0.220557	−	0.975374i	$-0.429212\pi$
$131$	661.390	0.441114	0.220557	+	0.975374i	$0.429212\pi$
$132$	1900.93	1.25345
$133$	0	0
$134$	1515.76	0.977174
$135$	−1168.42	−0.744902
$136$	−805.392	−0.507807
$137$	−1716.85	−1.07066	−0.535331	−	0.844642i	$-0.679813\pi$
$137$	−1716.85	−1.07066	−0.535331	+	0.844642i	$0.679813\pi$
$138$	−366.611	−0.226145
$139$	−888.405	−0.542112	−0.271056	−	0.962564i	$-0.587373\pi$
$139$	−888.405	−0.542112	−0.271056	+	0.962564i	$0.587373\pi$
$140$	0	0
$141$	1587.41	0.948116
$142$	−652.172	−0.385416
$143$	1040.20	0.608292
$144$	584.287	0.338129
$145$	−4009.31	−2.29624
$146$	−113.859	−0.0645412
$147$	0	0
$148$	−1433.00	−0.795891
$149$	−289.992	−0.159443	−0.0797216	−	0.996817i	$-0.525403\pi$
$149$	−289.992	−0.159443	−0.0797216	+	0.996817i	$0.525403\pi$
$150$	−1789.34	−0.973992
$151$	−2947.09	−1.58828	−0.794142	−	0.607732i	$-0.792080\pi$
$151$	−2947.09	−1.58828	−0.794142	+	0.607732i	$0.792080\pi$
$152$	−516.447	−0.275588
$153$	−3676.40	−1.94261
$154$	0	0
$155$	3827.78	1.98358
$156$	556.116	0.285416
$157$	608.838	0.309494	0.154747	−	0.987954i	$-0.450544\pi$
$157$	608.838	0.309494	0.154747	+	0.987954i	$0.450544\pi$
$158$	1125.85	0.566884
$159$	963.748	0.480693
$160$	492.901	0.243545
$161$	0	0
$162$	−762.850	−0.369970
$163$	−2741.67	−1.31745	−0.658725	−	0.752384i	$-0.728904\pi$
$163$	−2741.67	−1.31745	−0.658725	+	0.752384i	$0.728904\pi$
$164$	1728.59	0.823050
$165$	7320.09	3.45375
$166$	684.455	0.320024
$167$	1990.81	0.922475	0.461237	−	0.887277i	$-0.347406\pi$
$167$	1990.81	0.922475	0.461237	+	0.887277i	$0.347406\pi$
$168$	0	0
$169$	−1892.69	−0.861489
$170$	−3101.39	−1.39921
$171$	−2357.45	−1.05426
$172$	1883.53	0.834988
$173$	3786.37	1.66400	0.832001	−	0.554774i	$-0.187195\pi$
$173$	3786.37	1.66400	0.832001	+	0.554774i	$0.187195\pi$
$174$	4148.95	1.80765
$175$	0	0
$176$	−954.066	−0.408611
$177$	−7024.82	−2.98315
$178$	−193.129	−0.0813237
$179$	650.483	0.271617	0.135808	−	0.990735i	$-0.456637\pi$
$179$	650.483	0.271617	0.135808	+	0.990735i	$0.456637\pi$
$180$	2249.96	0.931680
$181$	1343.81	0.551847	0.275924	−	0.961180i	$-0.411016\pi$
$181$	1343.81	0.551847	0.275924	+	0.961180i	$0.411016\pi$
$182$	0	0
$183$	−2553.15	−1.03133
$184$	184.000	0.0737210
$185$	−5518.18	−2.19300
$186$	−3961.10	−1.56152
$187$	6003.10	2.34754
$188$	−796.713	−0.309076
$189$	0	0
$190$	−1988.73	−0.759355
$191$	1560.25	0.591077	0.295538	−	0.955331i	$-0.404501\pi$
$191$	1560.25	0.591077	0.295538	+	0.955331i	$0.404501\pi$
$192$	−510.068	−0.191724
$193$	844.757	0.315062	0.157531	−	0.987514i	$-0.449647\pi$
$193$	844.757	0.315062	0.157531	+	0.987514i	$0.449647\pi$
$194$	−204.382	−0.0756380
$195$	2141.49	0.786436
$196$	0	0
$197$	3215.32	1.16285	0.581427	−	0.813598i	$-0.302494\pi$
$197$	3215.32	1.16285	0.581427	+	0.813598i	$0.302494\pi$
$198$	−4355.07	−1.56314
$199$	2357.22	0.839694	0.419847	−	0.907595i	$-0.362084\pi$
$199$	2357.22	0.839694	0.419847	+	0.907595i	$0.362084\pi$
$200$	898.057	0.317511
$201$	−6040.14	−2.11960
$202$	2654.02	0.924436
$203$	0	0
$204$	3209.41	1.10149
$205$	6656.44	2.26783
$206$	359.322	0.121530
$207$	839.912	0.282019
$208$	−279.111	−0.0930427
$209$	3849.41	1.27402
$210$	0	0
$211$	3643.73	1.18884	0.594418	−	0.804156i	$-0.297382\pi$
$211$	3643.73	1.18884	0.594418	+	0.804156i	$0.297382\pi$
$212$	−483.699	−0.156701
$213$	2598.85	0.836009
$214$	1091.79	0.348753
$215$	7253.08	2.30073
$216$	−606.849	−0.191161
$217$	0	0
$218$	−2957.26	−0.918765
$219$	453.716	0.139997
$220$	−3673.91	−1.12589
$221$	1756.20	0.534548
$222$	5710.37	1.72637
$223$	−1785.62	−0.536205	−0.268102	−	0.963390i	$-0.586397\pi$
$223$	−1785.62	−0.536205	−0.268102	+	0.963390i	$0.586397\pi$
$224$	0	0
$225$	4099.40	1.21464
$226$	912.885	0.268691
$227$	940.953	0.275125	0.137562	−	0.990493i	$-0.456073\pi$
$227$	940.953	0.275125	0.137562	+	0.990493i	$0.456073\pi$
$228$	2057.99	0.597780
$229$	3171.87	0.915297	0.457649	−	0.889133i	$-0.348692\pi$
$229$	3171.87	0.915297	0.457649	+	0.889133i	$0.348692\pi$
$230$	708.545	0.203131
$231$	0	0
$232$	−2082.33	−0.589275
$233$	−712.021	−0.200198	−0.100099	−	0.994978i	$-0.531916\pi$
$233$	−712.021	−0.200198	−0.100099	+	0.994978i	$0.531916\pi$
$234$	−1274.07	−0.355934
$235$	−3067.97	−0.851628
$236$	3525.71	0.972477
$237$	−4486.40	−1.22963
$238$	0	0
$239$	4657.92	1.26065	0.630325	−	0.776331i	$-0.282921\pi$
$239$	4657.92	1.26065	0.630325	+	0.776331i	$0.282921\pi$
$240$	−1964.16	−0.528276
$241$	−511.972	−0.136842	−0.0684212	−	0.997657i	$-0.521796\pi$
$241$	−511.972	−0.136842	−0.0684212	+	0.997657i	$0.521796\pi$
$242$	4449.27	1.18186
$243$	5088.00	1.34319
$244$	1281.41	0.336204
$245$	0	0
$246$	−6888.27	−1.78529
$247$	1126.14	0.290100
$248$	1988.05	0.509038
$249$	−2727.49	−0.694167
$250$	−392.561	−0.0993109
$251$	3152.53	0.792774	0.396387	−	0.918084i	$-0.370264\pi$
$251$	3152.53	0.792774	0.396387	+	0.918084i	$0.370264\pi$
$252$	0	0
$253$	−1371.47	−0.340805
$254$	2352.18	0.581057
$255$	12358.8	3.03504
$256$	256.000	0.0625000
$257$	−1653.27	−0.401276	−0.200638	−	0.979665i	$-0.564302\pi$
$257$	−1653.27	−0.401276	−0.200638	+	0.979665i	$0.564302\pi$
$258$	−7505.70	−1.81118
$259$	0	0
$260$	−1074.80	−0.256370
$261$	−9505.30	−2.25427
$262$	1322.78	0.311915
$263$	2612.79	0.612592	0.306296	−	0.951936i	$-0.400910\pi$
$263$	2612.79	0.612592	0.306296	+	0.951936i	$0.400910\pi$
$264$	3801.87	0.886321
$265$	−1862.62	−0.431774
$266$	0	0
$267$	769.600	0.176400
$268$	3031.51	0.690967
$269$	6546.93	1.48392	0.741958	−	0.670447i	$-0.233898\pi$
$269$	6546.93	1.48392	0.741958	+	0.670447i	$0.233898\pi$
$270$	−2336.85	−0.526726
$271$	−2092.92	−0.469135	−0.234568	−	0.972100i	$-0.575367\pi$
$271$	−2092.92	−0.469135	−0.234568	+	0.972100i	$0.575367\pi$
$272$	−1610.78	−0.359074
$273$	0	0
$274$	−3433.71	−0.757073
$275$	−6693.80	−1.46782
$276$	−733.223	−0.159909
$277$	1711.78	0.371302	0.185651	−	0.982616i	$-0.440561\pi$
$277$	1711.78	0.371302	0.185651	+	0.982616i	$0.440561\pi$
$278$	−1776.81	−0.383331
$279$	9074.94	1.94732
$280$	0	0
$281$	−800.750	−0.169996	−0.0849978	−	0.996381i	$-0.527088\pi$
$281$	−800.750	−0.169996	−0.0849978	+	0.996381i	$0.527088\pi$
$282$	3174.83	0.670419
$283$	3460.14	0.726798	0.363399	−	0.931634i	$-0.381616\pi$
$283$	3460.14	0.726798	0.363399	+	0.931634i	$0.381616\pi$
$284$	−1304.34	−0.272530
$285$	7924.90	1.64712
$286$	2080.40	0.430127
$287$	0	0
$288$	1168.57	0.239093
$289$	5222.25	1.06295
$290$	−8018.62	−1.62369
$291$	814.443	0.164067
$292$	−227.717	−0.0456375
$293$	−7559.50	−1.50727	−0.753636	−	0.657292i	$-0.771702\pi$
$293$	−7559.50	−1.50727	−0.753636	+	0.657292i	$0.771702\pi$
$294$	0	0
$295$	13576.8	2.67956
$296$	−2866.00	−0.562780
$297$	4523.23	0.883719
$298$	−579.983	−0.112743
$299$	−401.223	−0.0776030
$300$	−3578.67	−0.688716
$301$	0	0
$302$	−5894.19	−1.12309
$303$	−10576.0	−2.00520
$304$	−1032.89	−0.194870
$305$	4934.44	0.926377
$306$	−7352.81	−1.37363
$307$	−2846.92	−0.529259	−0.264629	−	0.964350i	$-0.585250\pi$
$307$	−2846.92	−0.529259	−0.264629	+	0.964350i	$0.585250\pi$
$308$	0	0
$309$	−1431.86	−0.263611
$310$	7655.57	1.40260
$311$	−186.727	−0.0340461	−0.0170231	−	0.999855i	$-0.505419\pi$
$311$	−186.727	−0.0340461	−0.0170231	+	0.999855i	$0.505419\pi$
$312$	1112.23	0.201820
$313$	−6248.48	−1.12839	−0.564193	−	0.825643i	$-0.690813\pi$
$313$	−6248.48	−1.12839	−0.564193	+	0.825643i	$0.690813\pi$
$314$	1217.68	0.218845
$315$	0	0
$316$	2251.70	0.400848
$317$	8063.68	1.42871	0.714356	−	0.699783i	$-0.246720\pi$
$317$	8063.68	1.42871	0.714356	+	0.699783i	$0.246720\pi$
$318$	1927.50	0.339901
$319$	15521.0	2.72416
$320$	985.802	0.172212
$321$	−4350.68	−0.756484
$322$	0	0
$323$	6499.10	1.11957
$324$	−1525.70	−0.261608
$325$	−1958.26	−0.334231
$326$	−5483.34	−0.931578
$327$	11784.4	1.99290
$328$	3457.18	0.581985
$329$	0	0
$330$	14640.2	2.44217
$331$	8059.57	1.33835	0.669175	−	0.743105i	$-0.266648\pi$
$331$	8059.57	1.33835	0.669175	+	0.743105i	$0.266648\pi$
$332$	1368.91	0.226291
$333$	−13082.6	−2.15291
$334$	3981.61	0.652288
$335$	11673.7	1.90389
$336$	0	0
$337$	−6296.85	−1.01784	−0.508919	−	0.860814i	$-0.669955\pi$
$337$	−6296.85	−1.01784	−0.508919	+	0.860814i	$0.669955\pi$
$338$	−3785.38	−0.609165
$339$	−3637.76	−0.582821
$340$	−6202.79	−0.989392
$341$	−14818.2	−2.35323
$342$	−4714.89	−0.745474
$343$	0	0
$344$	3767.06	0.590426
$345$	−2823.49	−0.440613
$346$	7572.74	1.17663
$347$	−43.9881	−0.00680521	−0.00340260	−	0.999994i	$-0.501083\pi$
$347$	−43.9881	−0.00680521	−0.00340260	+	0.999994i	$0.501083\pi$
$348$	8297.90	1.27820
$349$	−6164.14	−0.945441	−0.472720	−	0.881212i	$-0.656728\pi$
$349$	−6164.14	−0.945441	−0.472720	+	0.881212i	$0.656728\pi$
$350$	0	0
$351$	1323.27	0.201227
$352$	−1908.13	−0.288931
$353$	5049.15	0.761301	0.380650	−	0.924719i	$-0.375700\pi$
$353$	5049.15	0.761301	0.380650	+	0.924719i	$0.375700\pi$
$354$	−14049.6	−2.10941
$355$	−5022.75	−0.750930
$356$	−386.257	−0.0575045
$357$	0	0
$358$	1300.97	0.192062
$359$	−8019.73	−1.17901	−0.589506	−	0.807764i	$-0.700677\pi$
$359$	−8019.73	−1.17901	−0.589506	+	0.807764i	$0.700677\pi$
$360$	4499.93	0.658798
$361$	−2691.54	−0.392410
$362$	2687.61	0.390215
$363$	−17729.9	−2.56358
$364$	0	0
$365$	−876.891	−0.125750
$366$	−5106.30	−0.729263
$367$	−13785.4	−1.96074	−0.980372	−	0.197156i	$-0.936830\pi$
$367$	−13785.4	−1.96074	−0.980372	+	0.197156i	$0.936830\pi$
$368$	368.000	0.0521286
$369$	15781.1	2.22638
$370$	−11036.4	−1.55068
$371$	0	0
$372$	−7922.20	−1.10416
$373$	−4957.51	−0.688177	−0.344089	−	0.938937i	$-0.611812\pi$
$373$	−4957.51	−0.688177	−0.344089	+	0.938937i	$0.611812\pi$
$374$	12006.2	1.65996
$375$	1564.32	0.215416
$376$	−1593.43	−0.218550
$377$	4540.64	0.620305
$378$	0	0
$379$	6053.52	0.820444	0.410222	−	0.911986i	$-0.365451\pi$
$379$	6053.52	0.820444	0.410222	+	0.911986i	$0.365451\pi$
$380$	−3977.46	−0.536945
$381$	−9373.20	−1.26038
$382$	3120.50	0.417954
$383$	−6923.15	−0.923646	−0.461823	−	0.886972i	$-0.652805\pi$
$383$	−6923.15	−0.923646	−0.461823	+	0.886972i	$0.652805\pi$
$384$	−1020.14	−0.135569
$385$	0	0
$386$	1689.51	0.222782
$387$	17195.7	2.25867
$388$	−408.764	−0.0534841
$389$	7909.68	1.03094	0.515471	−	0.856907i	$-0.327617\pi$
$389$	7909.68	1.03094	0.515471	+	0.856907i	$0.327617\pi$
$390$	4282.97	0.556094
$391$	−2315.50	−0.299488
$392$	0	0
$393$	−5271.16	−0.676577
$394$	6430.65	0.822262
$395$	8670.81	1.10450
$396$	−8710.13	−1.10530
$397$	11152.7	1.40992	0.704962	−	0.709245i	$-0.250964\pi$
$397$	11152.7	1.40992	0.704962	+	0.709245i	$0.250964\pi$
$398$	4714.45	0.593753
$399$	0	0
$400$	1796.11	0.224514
$401$	−3710.67	−0.462100	−0.231050	−	0.972942i	$-0.574216\pi$
$401$	−3710.67	−0.462100	−0.231050	+	0.972942i	$0.574216\pi$
$402$	−12080.3	−1.49878
$403$	−4335.06	−0.535843
$404$	5308.04	0.653675
$405$	−5875.15	−0.720836
$406$	0	0
$407$	21362.2	2.60168
$408$	6418.82	0.778870
$409$	3842.05	0.464492	0.232246	−	0.972657i	$-0.425393\pi$
$409$	3842.05	0.464492	0.232246	+	0.972657i	$0.425393\pi$
$410$	13312.9	1.60360
$411$	13683.0	1.64217
$412$	718.643	0.0859345
$413$	0	0
$414$	1679.82	0.199418
$415$	5271.38	0.623523
$416$	−558.223	−0.0657911
$417$	7080.42	0.831486
$418$	7698.83	0.900866
$419$	−10311.4	−1.20226	−0.601129	−	0.799152i	$-0.705282\pi$
$419$	−10311.4	−1.20226	−0.601129	+	0.799152i	$0.705282\pi$
$420$	0	0
$421$	6068.12	0.702475	0.351238	−	0.936286i	$-0.385761\pi$
$421$	6068.12	0.702475	0.351238	+	0.936286i	$0.385761\pi$
$422$	7287.45	0.840634
$423$	−7273.58	−0.836060
$424$	−967.399	−0.110804
$425$	−11301.4	−1.28988
$426$	5197.69	0.591148
$427$	0	0
$428$	2183.58	0.246606
$429$	−8290.19	−0.932993
$430$	14506.2	1.62686
$431$	4463.21	0.498806	0.249403	−	0.968400i	$-0.419766\pi$
$431$	4463.21	0.498806	0.249403	+	0.968400i	$0.419766\pi$
$432$	−1213.70	−0.135171
$433$	−2649.47	−0.294055	−0.147027	−	0.989132i	$-0.546971\pi$
$433$	−2649.47	−0.294055	−0.147027	+	0.989132i	$0.546971\pi$
$434$	0	0
$435$	31953.4	3.52196
$436$	−5914.51	−0.649665
$437$	−1484.79	−0.162533
$438$	907.433	0.0989927
$439$	−7598.28	−0.826073	−0.413037	−	0.910714i	$-0.635532\pi$
$439$	−7598.28	−0.826073	−0.413037	+	0.910714i	$0.635532\pi$
$440$	−7347.82	−0.796121
$441$	0	0
$442$	3512.41	0.377982
$443$	9065.68	0.972287	0.486144	−	0.873879i	$-0.338403\pi$
$443$	9065.68	0.972287	0.486144	+	0.873879i	$0.338403\pi$
$444$	11420.7	1.22073
$445$	−1487.40	−0.158448
$446$	−3571.23	−0.379154
$447$	2311.18	0.244553
$448$	0	0
$449$	13830.2	1.45365	0.726823	−	0.686825i	$-0.240996\pi$
$449$	13830.2	1.45365	0.726823	+	0.686825i	$0.240996\pi$
$450$	8198.79	0.858877
$451$	−25768.6	−2.69046
$452$	1825.77	0.189993
$453$	23487.8	2.43610
$454$	1881.91	0.194542
$455$	0	0
$456$	4115.99	0.422695
$457$	674.277	0.0690183	0.0345092	−	0.999404i	$-0.489013\pi$
$457$	674.277	0.0690183	0.0345092	+	0.999404i	$0.489013\pi$
$458$	6343.74	0.647213
$459$	7636.73	0.776584
$460$	1417.09	0.143635
$461$	899.140	0.0908398	0.0454199	−	0.998968i	$-0.485537\pi$
$461$	899.140	0.0908398	0.0454199	+	0.998968i	$0.485537\pi$
$462$	0	0
$463$	−11739.4	−1.17835	−0.589173	−	0.808007i	$-0.700546\pi$
$463$	−11739.4	−1.17835	−0.589173	+	0.808007i	$0.700546\pi$
$464$	−4164.66	−0.416680
$465$	−30506.7	−3.04240
$466$	−1424.04	−0.141561
$467$	−132.823	−0.0131613	−0.00658065	−	0.999978i	$-0.502095\pi$
$467$	−132.823	−0.0131613	−0.00658065	+	0.999978i	$0.502095\pi$
$468$	−2548.14	−0.251684
$469$	0	0
$470$	−6135.95	−0.602192
$471$	−4852.32	−0.474699
$472$	7051.43	0.687645
$473$	−28078.3	−2.72948
$474$	−8972.80	−0.869482
$475$	−7246.86	−0.700018
$476$	0	0
$477$	−4415.92	−0.423881
$478$	9315.83	0.891415
$479$	−4716.34	−0.449885	−0.224943	−	0.974372i	$-0.572220\pi$
$479$	−4716.34	−0.449885	−0.224943	+	0.974372i	$0.572220\pi$
$480$	−3928.33	−0.373548
$481$	6249.48	0.592415
$482$	−1023.94	−0.0967622
$483$	0	0
$484$	8898.54	0.835701
$485$	−1574.06	−0.147370
$486$	10176.0	0.949780
$487$	18058.7	1.68032	0.840160	−	0.542339i	$-0.182461\pi$
$487$	18058.7	1.68032	0.840160	+	0.542339i	$0.182461\pi$
$488$	2562.82	0.237732
$489$	21850.6	2.02069
$490$	0	0
$491$	3733.72	0.343178	0.171589	−	0.985169i	$-0.445110\pi$
$491$	3733.72	0.343178	0.171589	+	0.985169i	$0.445110\pi$
$492$	−13776.5	−1.26239
$493$	26204.6	2.39390
$494$	2252.28	0.205132
$495$	−33540.9	−3.04556
$496$	3976.10	0.359944
$497$	0	0
$498$	−5454.98	−0.490850
$499$	−13918.6	−1.24866	−0.624332	−	0.781159i	$-0.714629\pi$
$499$	−13918.6	−1.24866	−0.624332	+	0.781159i	$0.714629\pi$
$500$	−785.122	−0.0702234
$501$	−15866.4	−1.41488
$502$	6305.07	0.560576
$503$	15996.1	1.41795	0.708977	−	0.705232i	$-0.249157\pi$
$503$	15996.1	1.41795	0.708977	+	0.705232i	$0.249157\pi$
$504$	0	0
$505$	20440.1	1.80114
$506$	−2742.94	−0.240985
$507$	15084.4	1.32134
$508$	4704.35	0.410870
$509$	−9209.23	−0.801949	−0.400975	−	0.916089i	$-0.631328\pi$
$509$	−9209.23	−0.801949	−0.400975	+	0.916089i	$0.631328\pi$
$510$	24717.5	2.14610
$511$	0	0
$512$	512.000	0.0441942
$513$	4896.96	0.421454
$514$	−3306.54	−0.283745
$515$	2767.34	0.236784
$516$	−15011.4	−1.28070
$517$	11876.8	1.01033
$518$	0	0
$519$	−30176.7	−2.55223
$520$	−2149.60	−0.181281
$521$	−9400.35	−0.790474	−0.395237	−	0.918579i	$-0.629338\pi$
$521$	−9400.35	−0.790474	−0.395237	+	0.918579i	$0.629338\pi$
$522$	−19010.6	−1.59401
$523$	−2027.12	−0.169483	−0.0847415	−	0.996403i	$-0.527006\pi$
$523$	−2027.12	−0.169483	−0.0847415	+	0.996403i	$0.527006\pi$
$524$	2645.56	0.220557
$525$	0	0
$526$	5225.59	0.433168
$527$	−25018.1	−2.06794
$528$	7603.73	0.626723
$529$	529.000	0.0434783
$530$	−3725.25	−0.305310
$531$	32187.9	2.63058
$532$	0	0
$533$	−7538.58	−0.612631
$534$	1539.20	0.124734
$535$	8408.50	0.679498
$536$	6063.02	0.488587
$537$	−5184.23	−0.416603
$538$	13093.9	1.04929
$539$	0	0
$540$	−4673.69	−0.372451
$541$	14892.7	1.18353	0.591764	−	0.806111i	$-0.298432\pi$
$541$	14892.7	1.18353	0.591764	+	0.806111i	$0.298432\pi$
$542$	−4185.83	−0.331729
$543$	−10709.9	−0.846419
$544$	−3221.57	−0.253904
$545$	−22775.5	−1.79009
$546$	0	0
$547$	5957.23	0.465654	0.232827	−	0.972518i	$-0.425202\pi$
$547$	5957.23	0.465654	0.232827	+	0.972518i	$0.425202\pi$
$548$	−6867.42	−0.535331
$549$	11698.6	0.909443
$550$	−13387.6	−1.03791
$551$	16803.3	1.29918
$552$	−1466.45	−0.113073
$553$	0	0
$554$	3423.56	0.262550
$555$	43978.9	3.36360
$556$	−3553.62	−0.271056
$557$	−4821.93	−0.366807	−0.183404	−	0.983038i	$-0.558712\pi$
$557$	−4821.93	−0.366807	−0.183404	+	0.983038i	$0.558712\pi$
$558$	18149.9	1.37696
$559$	−8214.30	−0.621516
$560$	0	0
$561$	−47843.6	−3.60064
$562$	−1601.50	−0.120205
$563$	15077.7	1.12868	0.564341	−	0.825542i	$-0.309130\pi$
$563$	15077.7	1.12868	0.564341	+	0.825542i	$0.309130\pi$
$564$	6349.65	0.474058
$565$	7030.66	0.523508
$566$	6920.27	0.513924
$567$	0	0
$568$	−2608.69	−0.192708
$569$	14270.2	1.05139	0.525693	−	0.850675i	$-0.323806\pi$
$569$	14270.2	1.05139	0.525693	+	0.850675i	$0.323806\pi$
$570$	15849.8	1.16469
$571$	13692.0	1.00349	0.501744	−	0.865016i	$-0.332692\pi$
$571$	13692.0	1.00349	0.501744	+	0.865016i	$0.332692\pi$
$572$	4160.79	0.304146
$573$	−12434.9	−0.906588
$574$	0	0
$575$	2581.91	0.187258
$576$	2337.15	0.169064
$577$	−22213.0	−1.60267	−0.801333	−	0.598218i	$-0.795876\pi$
$577$	−22213.0	−1.60267	−0.801333	+	0.598218i	$0.795876\pi$
$578$	10444.5	0.751616
$579$	−6732.55	−0.483239
$580$	−16037.2	−1.14812
$581$	0	0
$582$	1628.89	0.116013
$583$	7210.64	0.512237
$584$	−455.435	−0.0322706
$585$	−9812.35	−0.693489
$586$	−15119.0	−1.06580
$587$	−388.512	−0.0273179	−0.0136590	−	0.999907i	$-0.504348\pi$
$587$	−388.512	−0.0273179	−0.0136590	+	0.999907i	$0.504348\pi$
$588$	0	0
$589$	−16042.6	−1.12228
$590$	27153.6	1.89474
$591$	−25625.5	−1.78358
$592$	−5732.00	−0.397946
$593$	12154.3	0.841680	0.420840	−	0.907135i	$-0.361735\pi$
$593$	12154.3	0.841680	0.420840	+	0.907135i	$0.361735\pi$
$594$	9046.47	0.624884
$595$	0	0
$596$	−1159.97	−0.0797216
$597$	−18786.6	−1.28792
$598$	−802.445	−0.0548736
$599$	−21860.7	−1.49116	−0.745580	−	0.666417i	$-0.767827\pi$
$599$	−21860.7	−1.49116	−0.745580	+	0.666417i	$0.767827\pi$
$600$	−7157.35	−0.486996
$601$	−2299.43	−0.156066	−0.0780330	−	0.996951i	$-0.524864\pi$
$601$	−2299.43	−0.156066	−0.0780330	+	0.996951i	$0.524864\pi$
$602$	0	0
$603$	27676.1	1.86909
$604$	−11788.4	−0.794142
$605$	34266.4	2.30269
$606$	−21152.0	−1.41789
$607$	−401.714	−0.0268617	−0.0134309	−	0.999910i	$-0.504275\pi$
$607$	−401.714	−0.0268617	−0.0134309	+	0.999910i	$0.504275\pi$
$608$	−2065.79	−0.137794
$609$	0	0
$610$	9868.87	0.655047
$611$	3474.56	0.230058
$612$	−14705.6	−0.971306
$613$	4192.17	0.276215	0.138108	−	0.990417i	$-0.455898\pi$
$613$	4192.17	0.276215	0.138108	+	0.990417i	$0.455898\pi$
$614$	−5693.84	−0.374242
$615$	−53050.6	−3.47838
$616$	0	0
$617$	−4018.32	−0.262190	−0.131095	−	0.991370i	$-0.541849\pi$
$617$	−4018.32	−0.262190	−0.131095	+	0.991370i	$0.541849\pi$
$618$	−2863.73	−0.186401
$619$	813.547	0.0528259	0.0264129	−	0.999651i	$-0.491592\pi$
$619$	813.547	0.0528259	0.0264129	+	0.999651i	$0.491592\pi$
$620$	15311.1	0.991790
$621$	−1744.69	−0.112741
$622$	−373.455	−0.0240742
$623$	0	0
$624$	2224.47	0.142708
$625$	−17055.5	−1.09155
$626$	−12497.0	−0.797890
$627$	−30679.1	−1.95408
$628$	2435.35	0.154747
$629$	36066.5	2.28627
$630$	0	0
$631$	−7571.14	−0.477658	−0.238829	−	0.971062i	$-0.576764\pi$
$631$	−7571.14	−0.477658	−0.238829	+	0.971062i	$0.576764\pi$
$632$	4503.39	0.283442
$633$	−29039.8	−1.82343
$634$	16127.4	1.01025
$635$	18115.5	1.13211
$636$	3854.99	0.240347
$637$	0	0
$638$	31041.9	1.92627
$639$	−11908.0	−0.737203
$640$	1971.60	0.121773
$641$	7871.02	0.485002	0.242501	−	0.970151i	$-0.422032\pi$
$641$	7871.02	0.485002	0.242501	+	0.970151i	$0.422032\pi$
$642$	−8701.36	−0.534915
$643$	12067.2	0.740100	0.370050	−	0.929012i	$-0.379340\pi$
$643$	12067.2	0.740100	0.370050	+	0.929012i	$0.379340\pi$
$644$	0	0
$645$	−57805.7	−3.52883
$646$	12998.2	0.791652
$647$	−12351.5	−0.750520	−0.375260	−	0.926919i	$-0.622447\pi$
$647$	−12351.5	−0.750520	−0.375260	+	0.926919i	$0.622447\pi$
$648$	−3051.40	−0.184985
$649$	−52558.8	−3.17891
$650$	−3916.53	−0.236337
$651$	0	0
$652$	−10966.7	−0.658725
$653$	1910.93	0.114518	0.0572592	−	0.998359i	$-0.481764\pi$
$653$	1910.93	0.114518	0.0572592	+	0.998359i	$0.481764\pi$
$654$	23568.8	1.40919
$655$	10187.5	0.607723
$656$	6914.36	0.411525
$657$	−2078.94	−0.123451
$658$	0	0
$659$	−23086.9	−1.36470	−0.682351	−	0.731025i	$-0.739042\pi$
$659$	−23086.9	−1.36470	−0.682351	+	0.731025i	$0.739042\pi$
$660$	29280.4	1.72687
$661$	24129.5	1.41986	0.709931	−	0.704271i	$-0.248726\pi$
$661$	24129.5	1.41986	0.709931	+	0.704271i	$0.248726\pi$
$662$	16119.1	0.946356
$663$	−13996.6	−0.819884
$664$	2737.82	0.160012
$665$	0	0
$666$	−26165.1	−1.52234
$667$	−5986.70	−0.347535
$668$	7963.23	0.461237
$669$	14231.0	0.822426
$670$	23347.4	1.34625
$671$	−19102.3	−1.09901
$672$	0	0
$673$	29226.2	1.67398	0.836990	−	0.547219i	$-0.184313\pi$
$673$	29226.2	1.67398	0.836990	+	0.547219i	$0.184313\pi$
$674$	−12593.7	−0.719720
$675$	−8515.39	−0.485566
$676$	−7570.76	−0.430744
$677$	−8812.55	−0.500286	−0.250143	−	0.968209i	$-0.580478\pi$
$677$	−8812.55	−0.500286	−0.250143	+	0.968209i	$0.580478\pi$
$678$	−7275.53	−0.412116
$679$	0	0
$680$	−12405.6	−0.699606
$681$	−7499.22	−0.421984
$682$	−29636.5	−1.66399
$683$	−5070.92	−0.284090	−0.142045	−	0.989860i	$-0.545368\pi$
$683$	−5070.92	−0.284090	−0.142045	+	0.989860i	$0.545368\pi$
$684$	−9429.79	−0.527130
$685$	−26445.0	−1.47505
$686$	0	0
$687$	−25279.2	−1.40387
$688$	7534.13	0.417494
$689$	2109.47	0.116639
$690$	−5646.97	−0.311560
$691$	16394.6	0.902577	0.451289	−	0.892378i	$-0.350964\pi$
$691$	16394.6	0.902577	0.451289	+	0.892378i	$0.350964\pi$
$692$	15145.5	0.832001
$693$	0	0
$694$	−87.9763	−0.00481201
$695$	−13684.2	−0.746867
$696$	16595.8	0.903825
$697$	−43506.0	−2.36429
$698$	−12328.3	−0.668528
$699$	5674.67	0.307061
$700$	0	0
$701$	−21162.5	−1.14022	−0.570111	−	0.821567i	$-0.693100\pi$
$701$	−21162.5	−1.14022	−0.570111	+	0.821567i	$0.693100\pi$
$702$	2646.54	0.142289
$703$	23127.2	1.24076
$704$	−3816.27	−0.204305
$705$	24451.2	1.30622
$706$	10098.3	0.538321
$707$	0	0
$708$	−28099.3	−1.49158
$709$	−12610.6	−0.667986	−0.333993	−	0.942576i	$-0.608396\pi$
$709$	−12610.6	−0.667986	−0.333993	+	0.942576i	$0.608396\pi$
$710$	−10045.5	−0.530988
$711$	20556.8	1.08431
$712$	−772.515	−0.0406618
$713$	5715.65	0.300214
$714$	0	0
$715$	16022.3	0.838044
$716$	2601.93	0.135808
$717$	−37122.7	−1.93358
$718$	−16039.5	−0.833687
$719$	2986.22	0.154892	0.0774458	−	0.996997i	$-0.475324\pi$
$719$	2986.22	0.154892	0.0774458	+	0.996997i	$0.475324\pi$
$720$	8999.86	0.465840
$721$	0	0
$722$	−5383.07	−0.277475
$723$	4080.32	0.209888
$724$	5375.23	0.275924
$725$	−29219.6	−1.49681
$726$	−35459.9	−1.81273
$727$	−9678.12	−0.493730	−0.246865	−	0.969050i	$-0.579400\pi$
$727$	−9678.12	−0.493730	−0.246865	+	0.969050i	$0.579400\pi$
$728$	0	0
$729$	−30251.9	−1.53696
$730$	−1753.78	−0.0889184
$731$	−47405.6	−2.39858
$732$	−10212.6	−0.515667
$733$	−10650.7	−0.536689	−0.268344	−	0.963323i	$-0.586477\pi$
$733$	−10650.7	−0.536689	−0.268344	+	0.963323i	$0.586477\pi$
$734$	−27570.8	−1.38646
$735$	0	0
$736$	736.000	0.0368605
$737$	−45191.6	−2.25869
$738$	31562.3	1.57429
$739$	765.563	0.0381079	0.0190539	−	0.999818i	$-0.493935\pi$
$739$	765.563	0.0381079	0.0190539	+	0.999818i	$0.493935\pi$
$740$	−22072.7	−1.09650
$741$	−8975.15	−0.444953
$742$	0	0
$743$	−255.490	−0.0126151	−0.00630754	−	0.999980i	$-0.502008\pi$
$743$	−255.490	−0.0126151	−0.00630754	+	0.999980i	$0.502008\pi$
$744$	−15844.4	−0.780758
$745$	−4466.79	−0.219665
$746$	−9915.02	−0.486615
$747$	12497.4	0.612125
$748$	24012.4	1.17377
$749$	0	0
$750$	3128.64	0.152322
$751$	15374.7	0.747046	0.373523	−	0.927621i	$-0.378150\pi$
$751$	15374.7	0.747046	0.373523	+	0.927621i	$0.378150\pi$
$752$	−3186.85	−0.154538
$753$	−25125.1	−1.21595
$754$	9081.29	0.438622
$755$	−45394.5	−2.18818
$756$	0	0
$757$	8167.12	0.392125	0.196063	−	0.980591i	$-0.437184\pi$
$757$	8167.12	0.392125	0.196063	+	0.980591i	$0.437184\pi$
$758$	12107.0	0.580141
$759$	10930.4	0.522723
$760$	−7954.91	−0.379678
$761$	8199.89	0.390599	0.195299	−	0.980744i	$-0.437432\pi$
$761$	8199.89	0.390599	0.195299	+	0.980744i	$0.437432\pi$
$762$	−18746.4	−0.891221
$763$	0	0
$764$	6241.00	0.295538
$765$	−56628.2	−2.67634
$766$	−13846.3	−0.653117
$767$	−15376.0	−0.723855
$768$	−2040.27	−0.0958620
$769$	26199.4	1.22858	0.614289	−	0.789081i	$-0.289443\pi$
$769$	26199.4	1.22858	0.614289	+	0.789081i	$0.289443\pi$
$770$	0	0
$771$	13176.2	0.615474
$772$	3379.03	0.157531
$773$	38258.8	1.78017	0.890087	−	0.455790i	$-0.150643\pi$
$773$	38258.8	1.78017	0.890087	+	0.455790i	$0.150643\pi$
$774$	34391.3	1.59712
$775$	27896.6	1.29300
$776$	−817.528	−0.0378190
$777$	0	0
$778$	15819.4	0.728986
$779$	−27897.7	−1.28310
$780$	8565.94	0.393218
$781$	19444.2	0.890870
$782$	−4631.00	−0.211770
$783$	19744.7	0.901172
$784$	0	0
$785$	9378.02	0.426390
$786$	−10542.3	−0.478412
$787$	−10063.3	−0.455805	−0.227903	−	0.973684i	$-0.573187\pi$
$787$	−10063.3	−0.455805	−0.227903	+	0.973684i	$0.573187\pi$
$788$	12861.3	0.581427
$789$	−20823.5	−0.939589
$790$	17341.6	0.780996
$791$	0	0
$792$	−17420.3	−0.781568
$793$	−5588.37	−0.250251
$794$	22305.5	0.996966
$795$	14844.8	0.662251
$796$	9428.89	0.419847
$797$	−25523.9	−1.13438	−0.567191	−	0.823586i	$-0.691970\pi$
$797$	−25523.9	−1.13438	−0.567191	+	0.823586i	$0.691970\pi$
$798$	0	0
$799$	20052.1	0.887849
$800$	3592.23	0.158756
$801$	−3526.33	−0.155552
$802$	−7421.34	−0.326754
$803$	3394.65	0.149184
$804$	−24160.6	−1.05980
$805$	0	0
$806$	−8670.12	−0.378898
$807$	−52177.8	−2.27602
$808$	10616.1	0.462218
$809$	7745.70	0.336618	0.168309	−	0.985734i	$-0.446169\pi$
$809$	7745.70	0.336618	0.168309	+	0.985734i	$0.446169\pi$
$810$	−11750.3	−0.509708
$811$	−29179.0	−1.26340	−0.631698	−	0.775214i	$-0.717642\pi$
$811$	−29179.0	−1.26340	−0.631698	+	0.775214i	$0.717642\pi$
$812$	0	0
$813$	16680.2	0.719556
$814$	42724.3	1.83966
$815$	−42230.4	−1.81505
$816$	12837.6	0.550744
$817$	−30398.3	−1.30171
$818$	7684.10	0.328445
$819$	0	0
$820$	26625.8	1.13392
$821$	−15828.5	−0.672861	−0.336431	−	0.941708i	$-0.609220\pi$
$821$	−15828.5	−0.672861	−0.336431	+	0.941708i	$0.609220\pi$
$822$	27366.0	1.16119
$823$	−34843.9	−1.47580	−0.737900	−	0.674910i	$-0.764182\pi$
$823$	−34843.9	−1.47580	−0.737900	+	0.674910i	$0.764182\pi$
$824$	1437.29	0.0607648
$825$	53348.3	2.25133
$826$	0	0
$827$	−11330.2	−0.476408	−0.238204	−	0.971215i	$-0.576559\pi$
$827$	−11330.2	−0.476408	−0.238204	+	0.971215i	$0.576559\pi$
$828$	3359.65	0.141010
$829$	6364.05	0.266626	0.133313	−	0.991074i	$-0.457439\pi$
$829$	6364.05	0.266626	0.133313	+	0.991074i	$0.457439\pi$
$830$	10542.8	0.440897
$831$	−13642.6	−0.569500
$832$	−1116.45	−0.0465214
$833$	0	0
$834$	14160.8	0.587949
$835$	30664.7	1.27089
$836$	15397.7	0.637008
$837$	−18850.7	−0.778466
$838$	−20622.8	−0.850124
$839$	−11672.0	−0.480288	−0.240144	−	0.970737i	$-0.577195\pi$
$839$	−11672.0	−0.480288	−0.240144	+	0.970737i	$0.577195\pi$
$840$	0	0
$841$	43362.6	1.77796
$842$	12136.2	0.496725
$843$	6381.83	0.260738
$844$	14574.9	0.594418
$845$	−29153.4	−1.18687
$846$	−14547.2	−0.591184
$847$	0	0
$848$	−1934.80	−0.0783505
$849$	−27576.6	−1.11476
$850$	−22602.7	−0.912080
$851$	−8239.75	−0.331910
$852$	10395.4	0.418005
$853$	42128.8	1.69105	0.845523	−	0.533939i	$-0.179289\pi$
$853$	42128.8	1.69105	0.845523	+	0.533939i	$0.179289\pi$
$854$	0	0
$855$	−36312.1	−1.45245
$856$	4367.16	0.174377
$857$	40576.1	1.61733	0.808667	−	0.588267i	$-0.200190\pi$
$857$	40576.1	1.61733	0.808667	+	0.588267i	$0.200190\pi$
$858$	−16580.4	−0.659726
$859$	14917.0	0.592506	0.296253	−	0.955109i	$-0.404263\pi$
$859$	14917.0	0.592506	0.296253	+	0.955109i	$0.404263\pi$
$860$	29012.3	1.15036
$861$	0	0
$862$	8926.42	0.352709
$863$	−14364.7	−0.566604	−0.283302	−	0.959031i	$-0.591430\pi$
$863$	−14364.7	−0.566604	−0.283302	+	0.959031i	$0.591430\pi$
$864$	−2427.39	−0.0955806
$865$	58322.0	2.29250
$866$	−5298.95	−0.207928
$867$	−41620.4	−1.63034
$868$	0	0
$869$	−33566.7	−1.31032
$870$	63906.9	2.49040
$871$	−13220.8	−0.514315
$872$	−11829.0	−0.459382
$873$	−3731.80	−0.144676
$874$	−2969.57	−0.114928
$875$	0	0
$876$	1814.87	0.0699984
$877$	−4038.50	−0.155497	−0.0777483	−	0.996973i	$-0.524773\pi$
$877$	−4038.50	−0.155497	−0.0777483	+	0.996973i	$0.524773\pi$
$878$	−15196.6	−0.584122
$879$	60247.8	2.31184
$880$	−14695.6	−0.562943
$881$	−2959.92	−0.113192	−0.0565960	−	0.998397i	$-0.518025\pi$
$881$	−2959.92	−0.113192	−0.0565960	+	0.998397i	$0.518025\pi$
$882$	0	0
$883$	−15568.0	−0.593322	−0.296661	−	0.954983i	$-0.595873\pi$
$883$	−15568.0	−0.593322	−0.296661	+	0.954983i	$0.595873\pi$
$884$	7024.81	0.267274
$885$	−108204.	−4.10989
$886$	18131.4	0.687511
$887$	1825.62	0.0691074	0.0345537	−	0.999403i	$-0.488999\pi$
$887$	1825.62	0.0691074	0.0345537	+	0.999403i	$0.488999\pi$
$888$	22841.5	0.863187
$889$	0	0
$890$	−2974.79	−0.112040
$891$	22744.1	0.855168
$892$	−7142.46	−0.268102
$893$	12858.1	0.481837
$894$	4622.36	0.172925
$895$	10019.5	0.374206
$896$	0	0
$897$	3197.67	0.119027
$898$	27660.4	1.02788
$899$	−64684.1	−2.39971
$900$	16397.6	0.607318
$901$	12174.0	0.450138
$902$	−51537.2	−1.90244
$903$	0	0
$904$	3651.54	0.134346
$905$	20698.9	0.760280
$906$	46975.6	1.72258
$907$	31949.6	1.16965	0.584823	−	0.811161i	$-0.301164\pi$
$907$	31949.6	1.16965	0.584823	+	0.811161i	$0.301164\pi$
$908$	3763.81	0.137562
$909$	48459.6	1.76821
$910$	0	0
$911$	44549.8	1.62020	0.810099	−	0.586294i	$-0.199413\pi$
$911$	44549.8	1.62020	0.810099	+	0.586294i	$0.199413\pi$
$912$	8231.97	0.298890
$913$	−20406.7	−0.739720
$914$	1348.55	0.0488033
$915$	−39326.5	−1.42087
$916$	12687.5	0.457649
$917$	0	0
$918$	15273.5	0.549128
$919$	16796.0	0.602884	0.301442	−	0.953484i	$-0.402532\pi$
$919$	16796.0	0.602884	0.301442	+	0.953484i	$0.402532\pi$
$920$	2834.18	0.101565
$921$	22689.4	0.811772
$922$	1798.28	0.0642334
$923$	5688.39	0.202856
$924$	0	0
$925$	−40216.1	−1.42951
$926$	−23478.7	−0.833217
$927$	6560.84	0.232455
$928$	−8329.33	−0.294637
$929$	11688.3	0.412787	0.206394	−	0.978469i	$-0.433827\pi$
$929$	11688.3	0.412787	0.206394	+	0.978469i	$0.433827\pi$
$930$	−61013.4	−2.15130
$931$	0	0
$932$	−2848.08	−0.100099
$933$	1488.18	0.0522196
$934$	−265.647	−0.00930645
$935$	92466.7	3.23421
$936$	−5096.28	−0.177967
$937$	12287.6	0.428408	0.214204	−	0.976789i	$-0.431284\pi$
$937$	12287.6	0.428408	0.214204	+	0.976789i	$0.431284\pi$
$938$	0	0
$939$	49799.2	1.73071
$940$	−12271.9	−0.425814
$941$	−15234.5	−0.527769	−0.263885	−	0.964554i	$-0.585004\pi$
$941$	−15234.5	−0.527769	−0.263885	+	0.964554i	$0.585004\pi$
$942$	−9704.65	−0.335663
$943$	9939.40	0.343236
$944$	14102.9	0.486238
$945$	0	0
$946$	−56156.7	−1.93003
$947$	31525.3	1.08177	0.540884	−	0.841097i	$-0.318090\pi$
$947$	31525.3	1.08177	0.540884	+	0.841097i	$0.318090\pi$
$948$	−17945.6	−0.614817
$949$	993.102	0.0339699
$950$	−14493.7	−0.494988
$951$	−64266.1	−2.19135
$952$	0	0
$953$	−33318.0	−1.13250	−0.566251	−	0.824233i	$-0.691607\pi$
$953$	−33318.0	−1.13250	−0.566251	+	0.824233i	$0.691607\pi$
$954$	−8831.85	−0.299729
$955$	24032.8	0.814326
$956$	18631.7	0.630325
$957$	−123699.	−4.17829
$958$	−9432.68	−0.318117
$959$	0	0
$960$	−7856.66	−0.264138
$961$	31964.5	1.07296
$962$	12499.0	0.418901
$963$	19934.9	0.667076
$964$	−2047.89	−0.0684212
$965$	13011.9	0.434061
$966$	0	0
$967$	−20850.2	−0.693379	−0.346689	−	0.937980i	$-0.612694\pi$
$967$	−20850.2	−0.693379	−0.346689	+	0.937980i	$0.612694\pi$
$968$	17797.1	0.590930
$969$	−51796.6	−1.71718
$970$	−3148.13	−0.104206
$971$	36281.5	1.19910	0.599551	−	0.800336i	$-0.295346\pi$
$971$	36281.5	1.19910	0.599551	+	0.800336i	$0.295346\pi$
$972$	20352.0	0.671596
$973$	0	0
$974$	36117.3	1.18817
$975$	15607.0	0.512640
$976$	5125.64	0.168102
$977$	31848.9	1.04292	0.521462	−	0.853274i	$-0.325387\pi$
$977$	31848.9	1.04292	0.521462	+	0.853274i	$0.325387\pi$
$978$	43701.2	1.42885
$979$	5758.05	0.187976
$980$	0	0
$981$	−53996.4	−1.75736
$982$	7467.44	0.242663
$983$	16776.4	0.544337	0.272169	−	0.962250i	$-0.412259\pi$
$983$	16776.4	0.544337	0.272169	+	0.962250i	$0.412259\pi$
$984$	−27553.1	−0.892643
$985$	49526.1	1.60206
$986$	52409.2	1.69275
$987$	0	0
$988$	4504.57	0.145050
$989$	10830.3	0.348214
$990$	−67081.7	−2.15353
$991$	21653.8	0.694103	0.347052	−	0.937846i	$-0.387183\pi$
$991$	21653.8	0.694103	0.347052	+	0.937846i	$0.387183\pi$
$992$	7952.21	0.254519
$993$	−64233.2	−2.05275
$994$	0	0
$995$	36308.7	1.15685
$996$	−10910.0	−0.347084
$997$	19249.6	0.611475	0.305737	−	0.952116i	$-0.401097\pi$
$997$	19249.6	0.611475	0.305737	+	0.952116i	$0.401097\pi$
$998$	−27837.3	−0.882939
$999$	27175.5	0.860654

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.y.1.1		11
7.2	even	3		322.4.e.a.277.11	yes	22
7.4	even	3		322.4.e.a.93.11	✓	22
7.6	odd	2		2254.4.a.v.1.11		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.a.93.11	✓	22	7.4	even	3
322.4.e.a.277.11	yes	22	7.2	even	3
2254.4.a.v.1.11		11	7.6	odd	2
2254.4.a.y.1.1		11	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:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -7.96981 q^{3} +4.00000 q^{4} +15.4032 q^{5} -15.9396 q^{6} +8.00000 q^{8} +36.5179 q^{9} +30.8063 q^{10} -59.6292 q^{11} -31.8793 q^{12} -17.4445 q^{13} -122.760 q^{15} +16.0000 q^{16} -100.674 q^{17} +73.0358 q^{18} -64.5559 q^{19} +61.6126 q^{20} -119.258 q^{22} +23.0000 q^{23} -63.7585 q^{24} +112.257 q^{25} -34.8889 q^{26} -75.8561 q^{27} -260.291 q^{29} -245.521 q^{30} +248.506 q^{31} +32.0000 q^{32} +475.233 q^{33} -201.348 q^{34} +146.072 q^{36} -358.250 q^{37} -129.112 q^{38} +139.029 q^{39} +123.225 q^{40} +432.148 q^{41} +470.883 q^{43} -238.517 q^{44} +562.491 q^{45} +46.0000 q^{46} -199.178 q^{47} -127.517 q^{48} +224.514 q^{50} +802.353 q^{51} -69.7778 q^{52} -120.925 q^{53} -151.712 q^{54} -918.477 q^{55} +514.498 q^{57} -520.583 q^{58} +881.429 q^{59} -491.041 q^{60} +320.352 q^{61} +497.013 q^{62} +64.0000 q^{64} -268.700 q^{65} +950.466 q^{66} +757.878 q^{67} -402.696 q^{68} -183.306 q^{69} -326.086 q^{71} +292.143 q^{72} -56.9293 q^{73} -716.500 q^{74} -894.668 q^{75} -258.224 q^{76} +278.058 q^{78} +562.924 q^{79} +246.450 q^{80} -381.425 q^{81} +864.296 q^{82} +342.227 q^{83} -1550.70 q^{85} +941.766 q^{86} +2074.47 q^{87} -477.033 q^{88} -96.5644 q^{89} +1124.98 q^{90} +92.0000 q^{92} -1980.55 q^{93} -398.356 q^{94} -994.364 q^{95} -255.034 q^{96} -102.191 q^{97} -2177.53 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 18 q^{3} + 44 q^{4} + 33 q^{5} + 36 q^{6} + 88 q^{8} + 171 q^{9} + 66 q^{10} + 8 q^{11} + 72 q^{12} + 185 q^{13} - 186 q^{15} + 176 q^{16} + 107 q^{17} + 342 q^{18} + 114 q^{19} + 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) \) 11 * q + 22 * q^2 + 18 * q^3 + 44 * q^4 + 33 * q^5 + 36 * q^6 + 88 * q^8 + 171 * q^9 + 66 * q^10 + 8 * q^11 + 72 * q^12 + 185 * q^13 - 186 * q^15 + 176 * q^16 + 107 * q^17 + 342 * q^18 + 114 * q^19 + 132 * q^20 + 16 * q^22 + 253 * q^23 + 144 * q^24 + 494 * q^25 + 370 * q^26 + 561 * q^27 - 79 * q^29 - 372 * q^30 + 488 * q^31 + 352 * q^32 + 304 * q^33 + 214 * q^34 + 684 * q^36 - 41 * q^37 + 228 * q^38 + 425 * q^39 + 264 * q^40 + 812 * q^41 + 219 * q^43 + 32 * q^44 + 619 * q^45 + 506 * q^46 + 1141 * q^47 + 288 * q^48 + 988 * q^50 + 83 * q^51 + 740 * q^52 + 579 * q^53 + 1122 * q^54 + 951 * q^55 + 383 * q^57 - 158 * q^58 + 1695 * q^59 - 744 * q^60 + 1105 * q^61 + 976 * q^62 + 704 * q^64 - 347 * q^65 + 608 * q^66 + 695 * q^67 + 428 * q^68 + 414 * q^69 + 65 * q^71 + 1368 * q^72 + 1935 * q^73 - 82 * q^74 + 1712 * q^75 + 456 * q^76 + 850 * q^78 + 447 * q^79 + 528 * q^80 + 2203 * q^81 + 1624 * q^82 + 1389 * q^83 + 153 * q^85 + 438 * q^86 + 3354 * q^87 + 64 * q^88 + 2698 * q^89 + 1238 * q^90 + 1012 * q^92 - 686 * q^93 + 2282 * q^94 + 827 * q^95 + 576 * q^96 + 1589 * q^97 - 1729 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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