LMFDB - Embedded newform 2254.4.a.l.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:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 92x^{3} - 28x^{2} + 1593x - 1782 $ x^5 - 92x^3 - 28x^2 + 1593*x - 1782
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.39309$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} -8.39309 q^{3} +4.00000 q^{4} +1.25007 q^{5} -16.7862 q^{6} +8.00000 q^{8} +43.4440 q^{9} +2.50013 q^{10} +34.5209 q^{11} -33.5724 q^{12} +19.9510 q^{13} -10.4919 q^{15} +16.0000 q^{16} -12.2007 q^{17} +86.8880 q^{18} -149.358 q^{19} +5.00026 q^{20} +69.0418 q^{22} +23.0000 q^{23} -67.1447 q^{24} -123.437 q^{25} +39.9019 q^{26} -138.016 q^{27} +38.5535 q^{29} -20.9838 q^{30} -71.0694 q^{31} +32.0000 q^{32} -289.737 q^{33} -24.4014 q^{34} +173.776 q^{36} +82.1755 q^{37} -298.715 q^{38} -167.450 q^{39} +10.0005 q^{40} +217.469 q^{41} -393.106 q^{43} +138.084 q^{44} +54.3078 q^{45} +46.0000 q^{46} +504.758 q^{47} -134.289 q^{48} -246.875 q^{50} +102.402 q^{51} +79.8038 q^{52} -671.499 q^{53} -276.032 q^{54} +43.1534 q^{55} +1253.57 q^{57} +77.1070 q^{58} +16.0190 q^{59} -41.9677 q^{60} +507.596 q^{61} -142.139 q^{62} +64.0000 q^{64} +24.9400 q^{65} -579.474 q^{66} +398.674 q^{67} -48.8028 q^{68} -193.041 q^{69} +93.7266 q^{71} +347.552 q^{72} +15.9904 q^{73} +164.351 q^{74} +1036.02 q^{75} -597.431 q^{76} -334.900 q^{78} +592.216 q^{79} +20.0010 q^{80} -14.6075 q^{81} +434.937 q^{82} -760.409 q^{83} -15.2517 q^{85} -786.212 q^{86} -323.583 q^{87} +276.167 q^{88} +852.993 q^{89} +108.616 q^{90} +92.0000 q^{92} +596.492 q^{93} +1009.52 q^{94} -186.707 q^{95} -268.579 q^{96} +131.784 q^{97} +1499.73 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 10 q^{2} - 5 q^{3} + 20 q^{4} - 22 q^{5} - 10 q^{6} + 40 q^{8} + 54 q^{9} - 44 q^{10} + 42 q^{11} - 20 q^{12} - 107 q^{13} + 122 q^{15} + 80 q^{16} - 218 q^{17} + 108 q^{18} - 194 q^{19} - 88 q^{20}+ \cdots - 2616 q^{99}+O(q^{100}) $ 5 * q + 10 * q^2 - 5 * q^3 + 20 * q^4 - 22 * q^5 - 10 * q^6 + 40 * q^8 + 54 * q^9 - 44 * q^10 + 42 * q^11 - 20 * q^12 - 107 * q^13 + 122 * q^15 + 80 * q^16 - 218 * q^17 + 108 * q^18 - 194 * q^19 - 88 * q^20 + 84 * q^22 + 115 * q^23 - 40 * q^24 + 219 * q^25 - 214 * q^26 - 203 * q^27 + 33 * q^29 + 244 * q^30 - 19 * q^31 + 160 * q^32 - 240 * q^33 - 436 * q^34 + 216 * q^36 + 214 * q^37 - 388 * q^38 - 243 * q^39 - 176 * q^40 - 395 * q^41 - 284 * q^43 + 168 * q^44 - 470 * q^45 + 230 * q^46 + 493 * q^47 - 80 * q^48 + 438 * q^50 - 244 * q^51 - 428 * q^52 - 70 * q^53 - 406 * q^54 + 186 * q^55 + 22 * q^57 + 66 * q^58 - 226 * q^59 + 488 * q^60 - 1606 * q^61 - 38 * q^62 + 320 * q^64 + 362 * q^65 - 480 * q^66 + 298 * q^67 - 872 * q^68 - 115 * q^69 - 969 * q^71 + 432 * q^72 - 697 * q^73 + 428 * q^74 - 1065 * q^75 - 776 * q^76 - 486 * q^78 + 186 * q^79 - 352 * q^80 - 335 * q^81 - 790 * q^82 + 1104 * q^83 - 1344 * q^85 - 568 * q^86 + 1603 * q^87 + 336 * q^88 + 814 * q^89 - 940 * q^90 + 460 * q^92 - 3423 * q^93 + 986 * q^94 - 2812 * q^95 - 160 * q^96 + 256 * q^97 - 2616 * 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
$3$	−8.39309	−1.61525	−0.807626	−	0.589696i	$-0.799248\pi$
$3$	−8.39309	−1.61525	−0.807626	+	0.589696i	$0.799248\pi$
$4$	4.00000	0.500000
$5$	1.25007	0.111809	0.0559046	−	0.998436i	$-0.482196\pi$
$5$	1.25007	0.111809	0.0559046	+	0.998436i	$0.482196\pi$
$6$	−16.7862	−1.14216
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	43.4440	1.60904
$10$	2.50013	0.0790611
$11$	34.5209	0.946222	0.473111	−	0.881003i	$-0.343131\pi$
$11$	34.5209	0.946222	0.473111	+	0.881003i	$0.343131\pi$
$12$	−33.5724	−0.807626
$13$	19.9510	0.425646	0.212823	−	0.977091i	$-0.431734\pi$
$13$	19.9510	0.425646	0.212823	+	0.977091i	$0.431734\pi$
$14$	0	0
$15$	−10.4919	−0.180600
$16$	16.0000	0.250000
$17$	−12.2007	−0.174065	−0.0870324	−	0.996205i	$-0.527738\pi$
$17$	−12.2007	−0.174065	−0.0870324	+	0.996205i	$0.527738\pi$
$18$	86.8880	1.13776
$19$	−149.358	−1.80342	−0.901711	−	0.432340i	$-0.857688\pi$
$19$	−149.358	−1.80342	−0.901711	+	0.432340i	$0.857688\pi$
$20$	5.00026	0.0559046
$21$	0	0
$22$	69.0418	0.669080
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−67.1447	−0.571078
$25$	−123.437	−0.987499
$26$	39.9019	0.300977
$27$	−138.016	−0.983747
$28$	0	0
$29$	38.5535	0.246869	0.123435	−	0.992353i	$-0.460609\pi$
$29$	38.5535	0.246869	0.123435	+	0.992353i	$0.460609\pi$
$30$	−20.9838	−0.127704
$31$	−71.0694	−0.411756	−0.205878	−	0.978578i	$-0.566005\pi$
$31$	−71.0694	−0.411756	−0.205878	+	0.978578i	$0.566005\pi$
$32$	32.0000	0.176777
$33$	−289.737	−1.52839
$34$	−24.4014	−0.123082
$35$	0	0
$36$	173.776	0.804518
$37$	82.1755	0.365123	0.182562	−	0.983194i	$-0.441561\pi$
$37$	82.1755	0.365123	0.182562	+	0.983194i	$0.441561\pi$
$38$	−298.715	−1.27521
$39$	−167.450	−0.687526
$40$	10.0005	0.0395305
$41$	217.469	0.828363	0.414182	−	0.910194i	$-0.364068\pi$
$41$	217.469	0.828363	0.414182	+	0.910194i	$0.364068\pi$
$42$	0	0
$43$	−393.106	−1.39414	−0.697071	−	0.717002i	$-0.745514\pi$
$43$	−393.106	−1.39414	−0.697071	+	0.717002i	$0.745514\pi$
$44$	138.084	0.473111
$45$	54.3078	0.179905
$46$	46.0000	0.147442
$47$	504.758	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$47$	504.758	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$48$	−134.289	−0.403813
$49$	0	0
$50$	−246.875	−0.698267
$51$	102.402	0.281159
$52$	79.8038	0.212823
$53$	−671.499	−1.74033	−0.870165	−	0.492761i	$-0.835988\pi$
$53$	−671.499	−1.74033	−0.870165	+	0.492761i	$0.835988\pi$
$54$	−276.032	−0.695614
$55$	43.1534	0.105796
$56$	0	0
$57$	1253.57	2.91298
$58$	77.1070	0.174563
$59$	16.0190	0.0353474	0.0176737	−	0.999844i	$-0.494374\pi$
$59$	16.0190	0.0353474	0.0176737	+	0.999844i	$0.494374\pi$
$60$	−41.9677	−0.0903000
$61$	507.596	1.06543	0.532713	−	0.846296i	$-0.321173\pi$
$61$	507.596	1.06543	0.532713	+	0.846296i	$0.321173\pi$
$62$	−142.139	−0.291156
$63$	0	0
$64$	64.0000	0.125000
$65$	24.9400	0.0475912
$66$	−579.474	−1.08073
$67$	398.674	0.726953	0.363476	−	0.931603i	$-0.381590\pi$
$67$	398.674	0.726953	0.363476	+	0.931603i	$0.381590\pi$
$68$	−48.8028	−0.0870324
$69$	−193.041	−0.336803
$70$	0	0
$71$	93.7266	0.156666	0.0783331	−	0.996927i	$-0.475040\pi$
$71$	93.7266	0.156666	0.0783331	+	0.996927i	$0.475040\pi$
$72$	347.552	0.568880
$73$	15.9904	0.0256374	0.0128187	−	0.999918i	$-0.495920\pi$
$73$	15.9904	0.0256374	0.0128187	+	0.999918i	$0.495920\pi$
$74$	164.351	0.258181
$75$	1036.02	1.59506
$76$	−597.431	−0.901711
$77$	0	0
$78$	−334.900	−0.486154
$79$	592.216	0.843411	0.421705	−	0.906733i	$-0.361432\pi$
$79$	592.216	0.843411	0.421705	+	0.906733i	$0.361432\pi$
$80$	20.0010	0.0279523
$81$	−14.6075	−0.0200377
$82$	434.937	0.585741
$83$	−760.409	−1.00561	−0.502806	−	0.864399i	$-0.667699\pi$
$83$	−760.409	−1.00561	−0.502806	+	0.864399i	$0.667699\pi$
$84$	0	0
$85$	−15.2517	−0.0194621
$86$	−786.212	−0.985808
$87$	−323.583	−0.398756
$88$	276.167	0.334540
$89$	852.993	1.01592	0.507961	−	0.861380i	$-0.330399\pi$
$89$	852.993	1.01592	0.507961	+	0.861380i	$0.330399\pi$
$90$	108.616	0.127212
$91$	0	0
$92$	92.0000	0.104257
$93$	596.492	0.665090
$94$	1009.52	1.10770
$95$	−186.707	−0.201639
$96$	−268.579	−0.285539
$97$	131.784	0.137945	0.0689726	−	0.997619i	$-0.478028\pi$
$97$	131.784	0.137945	0.0689726	+	0.997619i	$0.478028\pi$
$98$	0	0
$99$	1499.73	1.52251
$100$	−493.749	−0.493749
$101$	521.431	0.513706	0.256853	−	0.966451i	$-0.417314\pi$
$101$	521.431	0.513706	0.256853	+	0.966451i	$0.417314\pi$
$102$	204.803	0.198809
$103$	−1437.32	−1.37499	−0.687493	−	0.726191i	$-0.741289\pi$
$103$	−1437.32	−1.37499	−0.687493	+	0.726191i	$0.741289\pi$
$104$	159.608	0.150489
$105$	0	0
$106$	−1343.00	−1.23060
$107$	978.389	0.883967	0.441984	−	0.897023i	$-0.354275\pi$
$107$	978.389	0.883967	0.441984	+	0.897023i	$0.354275\pi$
$108$	−552.064	−0.491874
$109$	−1783.57	−1.56729	−0.783646	−	0.621208i	$-0.786642\pi$
$109$	−1783.57	−1.56729	−0.783646	+	0.621208i	$0.786642\pi$
$110$	86.3068	0.0748094
$111$	−689.706	−0.589766
$112$	0	0
$113$	1587.95	1.32196	0.660981	−	0.750403i	$-0.270140\pi$
$113$	1587.95	1.32196	0.660981	+	0.750403i	$0.270140\pi$
$114$	2507.15	2.05979
$115$	28.7515	0.0233138
$116$	154.214	0.123435
$117$	866.749	0.684880
$118$	32.0380	0.0249944
$119$	0	0
$120$	−83.9353	−0.0638518
$121$	−139.307	−0.104663
$122$	1015.19	0.753370
$123$	−1825.23	−1.33801
$124$	−284.278	−0.205878
$125$	−310.563	−0.222221
$126$	0	0
$127$	−2287.72	−1.59844	−0.799222	−	0.601036i	$-0.794755\pi$
$127$	−2287.72	−1.59844	−0.799222	+	0.601036i	$0.794755\pi$
$128$	128.000	0.0883883
$129$	3299.38	2.25189
$130$	49.8800	0.0336521
$131$	−898.206	−0.599058	−0.299529	−	0.954087i	$-0.596830\pi$
$131$	−898.206	−0.599058	−0.299529	+	0.954087i	$0.596830\pi$
$132$	−1158.95	−0.764193
$133$	0	0
$134$	797.349	0.514033
$135$	−172.529	−0.109992
$136$	−97.6056	−0.0615412
$137$	2301.27	1.43511	0.717557	−	0.696500i	$-0.245260\pi$
$137$	2301.27	1.43511	0.717557	+	0.696500i	$0.245260\pi$
$138$	−386.082	−0.238156
$139$	−778.242	−0.474889	−0.237445	−	0.971401i	$-0.576310\pi$
$139$	−778.242	−0.474889	−0.237445	+	0.971401i	$0.576310\pi$
$140$	0	0
$141$	−4236.48	−2.53032
$142$	187.453	0.110780
$143$	688.725	0.402756
$144$	695.104	0.402259
$145$	48.1944	0.0276023
$146$	31.9808	0.0181284
$147$	0	0
$148$	328.702	0.182562
$149$	−1694.76	−0.931811	−0.465906	−	0.884834i	$-0.654271\pi$
$149$	−1694.76	−0.931811	−0.465906	+	0.884834i	$0.654271\pi$
$150$	2072.04	1.12788
$151$	−2385.34	−1.28554	−0.642769	−	0.766060i	$-0.722214\pi$
$151$	−2385.34	−1.28554	−0.642769	+	0.766060i	$0.722214\pi$
$152$	−1194.86	−0.637606
$153$	−530.047	−0.280077
$154$	0	0
$155$	−88.8414	−0.0460382
$156$	−669.801	−0.343763
$157$	−1629.96	−0.828566	−0.414283	−	0.910148i	$-0.635968\pi$
$157$	−1629.96	−0.828566	−0.414283	+	0.910148i	$0.635968\pi$
$158$	1184.43	0.596381
$159$	5635.95	2.81107
$160$	40.0021	0.0197653
$161$	0	0
$162$	−29.2150	−0.0141688
$163$	1582.27	0.760326	0.380163	−	0.924920i	$-0.375868\pi$
$163$	1582.27	0.760326	0.380163	+	0.924920i	$0.375868\pi$
$164$	869.874	0.414182
$165$	−362.190	−0.170888
$166$	−1520.82	−0.711075
$167$	−126.908	−0.0588048	−0.0294024	−	0.999568i	$-0.509360\pi$
$167$	−126.908	−0.0588048	−0.0294024	+	0.999568i	$0.509360\pi$
$168$	0	0
$169$	−1798.96	−0.818825
$170$	−30.5033	−0.0137618
$171$	−6488.69	−2.90177
$172$	−1572.42	−0.697071
$173$	2012.99	0.884650	0.442325	−	0.896855i	$-0.354154\pi$
$173$	2012.99	0.884650	0.442325	+	0.896855i	$0.354154\pi$
$174$	−647.166	−0.281963
$175$	0	0
$176$	552.335	0.236556
$177$	−134.449	−0.0570950
$178$	1705.99	0.718365
$179$	−4013.83	−1.67602	−0.838010	−	0.545655i	$-0.816281\pi$
$179$	−4013.83	−1.67602	−0.838010	+	0.545655i	$0.816281\pi$
$180$	217.231	0.0899526
$181$	−151.474	−0.0622042	−0.0311021	−	0.999516i	$-0.509902\pi$
$181$	−151.474	−0.0622042	−0.0311021	+	0.999516i	$0.509902\pi$
$182$	0	0
$183$	−4260.30	−1.72093
$184$	184.000	0.0737210
$185$	102.725	0.0408242
$186$	1192.98	0.470290
$187$	−421.179	−0.164704
$188$	2019.03	0.783260
$189$	0	0
$190$	−373.414	−0.142580
$191$	378.019	0.143207	0.0716035	−	0.997433i	$-0.477188\pi$
$191$	378.019	0.143207	0.0716035	+	0.997433i	$0.477188\pi$
$192$	−537.158	−0.201906
$193$	−1548.82	−0.577649	−0.288825	−	0.957382i	$-0.593264\pi$
$193$	−1548.82	−0.577649	−0.288825	+	0.957382i	$0.593264\pi$
$194$	263.569	0.0975419
$195$	−209.324	−0.0768717
$196$	0	0
$197$	1915.98	0.692935	0.346467	−	0.938062i	$-0.387381\pi$
$197$	1915.98	0.692935	0.346467	+	0.938062i	$0.387381\pi$
$198$	2999.45	1.07657
$199$	−1337.74	−0.476532	−0.238266	−	0.971200i	$-0.576579\pi$
$199$	−1337.74	−0.476532	−0.238266	+	0.971200i	$0.576579\pi$
$200$	−987.499	−0.349134
$201$	−3346.11	−1.17421
$202$	1042.86	0.363245
$203$	0	0
$204$	409.606	0.140579
$205$	271.850	0.0926187
$206$	−2874.65	−0.972262
$207$	999.212	0.335507
$208$	319.215	0.106412
$209$	−5155.96	−1.70644
$210$	0	0
$211$	−1366.24	−0.445764	−0.222882	−	0.974845i	$-0.571546\pi$
$211$	−1366.24	−0.445764	−0.222882	+	0.974845i	$0.571546\pi$
$212$	−2686.00	−0.870165
$213$	−786.656	−0.253055
$214$	1956.78	0.625059
$215$	−491.409	−0.155878
$216$	−1104.13	−0.347807
$217$	0	0
$218$	−3567.14	−1.10824
$219$	−134.209	−0.0414109
$220$	172.614	0.0528982
$221$	−243.416	−0.0740901
$222$	−1379.41	−0.417028
$223$	−5804.62	−1.74308	−0.871538	−	0.490328i	$-0.836877\pi$
$223$	−5804.62	−1.74308	−0.871538	+	0.490328i	$0.836877\pi$
$224$	0	0
$225$	−5362.61	−1.58892
$226$	3175.90	0.934768
$227$	−5574.98	−1.63006	−0.815031	−	0.579417i	$-0.803280\pi$
$227$	−5574.98	−1.63006	−0.815031	+	0.579417i	$0.803280\pi$
$228$	5014.29	1.45649
$229$	1441.32	0.415919	0.207959	−	0.978137i	$-0.433318\pi$
$229$	1441.32	0.415919	0.207959	+	0.978137i	$0.433318\pi$
$230$	57.5030	0.0164854
$231$	0	0
$232$	308.428	0.0872815
$233$	−788.176	−0.221610	−0.110805	−	0.993842i	$-0.535343\pi$
$233$	−788.176	−0.221610	−0.110805	+	0.993842i	$0.535343\pi$
$234$	1733.50	0.484284
$235$	630.980	0.175152
$236$	64.0761	0.0176737
$237$	−4970.52	−1.36232
$238$	0	0
$239$	−703.348	−0.190359	−0.0951796	−	0.995460i	$-0.530343\pi$
$239$	−703.348	−0.190359	−0.0951796	+	0.995460i	$0.530343\pi$
$240$	−167.871	−0.0451500
$241$	−4957.65	−1.32510	−0.662552	−	0.749016i	$-0.730527\pi$
$241$	−4957.65	−1.32510	−0.662552	+	0.749016i	$0.730527\pi$
$242$	−278.613	−0.0740080
$243$	3849.03	1.01611
$244$	2030.38	0.532713
$245$	0	0
$246$	−3650.47	−0.946119
$247$	−2979.83	−0.767619
$248$	−568.555	−0.145578
$249$	6382.18	1.62432
$250$	−621.126	−0.157134
$251$	2110.00	0.530606	0.265303	−	0.964165i	$-0.414528\pi$
$251$	2110.00	0.530606	0.265303	+	0.964165i	$0.414528\pi$
$252$	0	0
$253$	793.981	0.197301
$254$	−4575.44	−1.13027
$255$	128.009	0.0314361
$256$	256.000	0.0625000
$257$	−3980.20	−0.966063	−0.483031	−	0.875603i	$-0.660464\pi$
$257$	−3980.20	−0.966063	−0.483031	+	0.875603i	$0.660464\pi$
$258$	6598.75	1.59233
$259$	0	0
$260$	99.7600	0.0237956
$261$	1674.92	0.397222
$262$	−1796.41	−0.423598
$263$	2994.29	0.702037	0.351018	−	0.936369i	$-0.385835\pi$
$263$	2994.29	0.702037	0.351018	+	0.936369i	$0.385835\pi$
$264$	−2317.90	−0.540366
$265$	−839.418	−0.194585
$266$	0	0
$267$	−7159.25	−1.64097
$268$	1594.70	0.363476
$269$	1578.96	0.357883	0.178942	−	0.983860i	$-0.442733\pi$
$269$	1578.96	0.357883	0.178942	+	0.983860i	$0.442733\pi$
$270$	−345.058	−0.0777761
$271$	2520.34	0.564944	0.282472	−	0.959276i	$-0.408846\pi$
$271$	2520.34	0.564944	0.282472	+	0.959276i	$0.408846\pi$
$272$	−195.211	−0.0435162
$273$	0	0
$274$	4602.54	1.01478
$275$	−4261.17	−0.934393
$276$	−772.164	−0.168402
$277$	3071.73	0.666290	0.333145	−	0.942876i	$-0.391890\pi$
$277$	3071.73	0.666290	0.333145	+	0.942876i	$0.391890\pi$
$278$	−1556.48	−0.335797
$279$	−3087.54	−0.662531
$280$	0	0
$281$	3477.45	0.738246	0.369123	−	0.929380i	$-0.379658\pi$
$281$	3477.45	0.738246	0.369123	+	0.929380i	$0.379658\pi$
$282$	−8472.96	−1.78921
$283$	−4496.56	−0.944497	−0.472248	−	0.881465i	$-0.656557\pi$
$283$	−4496.56	−0.944497	−0.472248	+	0.881465i	$0.656557\pi$
$284$	374.906	0.0783331
$285$	1567.05	0.325698
$286$	1377.45	0.284791
$287$	0	0
$288$	1390.21	0.284440
$289$	−4764.14	−0.969701
$290$	96.3889	0.0195178
$291$	−1106.08	−0.222816
$292$	63.9615	0.0128187
$293$	−991.604	−0.197714	−0.0988568	−	0.995102i	$-0.531519\pi$
$293$	−991.604	−0.197714	−0.0988568	+	0.995102i	$0.531519\pi$
$294$	0	0
$295$	20.0248	0.00395217
$296$	657.404	0.129091
$297$	−4764.44	−0.930844
$298$	−3389.51	−0.658890
$299$	458.872	0.0887534
$300$	4144.08	0.797529
$301$	0	0
$302$	−4770.68	−0.909012
$303$	−4376.42	−0.829764
$304$	−2389.72	−0.450855
$305$	634.528	0.119124
$306$	−1060.09	−0.198044
$307$	755.726	0.140494	0.0702468	−	0.997530i	$-0.477621\pi$
$307$	755.726	0.140494	0.0702468	+	0.997530i	$0.477621\pi$
$308$	0	0
$309$	12063.6	2.22095
$310$	−177.683	−0.0325539
$311$	−6839.74	−1.24709	−0.623547	−	0.781786i	$-0.714309\pi$
$311$	−6839.74	−1.24709	−0.623547	+	0.781786i	$0.714309\pi$
$312$	−1339.60	−0.243077
$313$	1051.93	0.189964	0.0949818	−	0.995479i	$-0.469721\pi$
$313$	1051.93	0.189964	0.0949818	+	0.995479i	$0.469721\pi$
$314$	−3259.92	−0.585885
$315$	0	0
$316$	2368.86	0.421705
$317$	8346.10	1.47875	0.739375	−	0.673294i	$-0.235121\pi$
$317$	8346.10	1.47875	0.739375	+	0.673294i	$0.235121\pi$
$318$	11271.9	1.98773
$319$	1330.90	0.233593
$320$	80.0042	0.0139762
$321$	−8211.71	−1.42783
$322$	0	0
$323$	1822.27	0.313912
$324$	−58.4300	−0.0100189
$325$	−2462.69	−0.420325
$326$	3164.54	0.537631
$327$	14969.6	2.53157
$328$	1739.75	0.292871
$329$	0	0
$330$	−724.381	−0.120836
$331$	−9846.13	−1.63502	−0.817511	−	0.575913i	$-0.804647\pi$
$331$	−9846.13	−1.63502	−0.817511	+	0.575913i	$0.804647\pi$
$332$	−3041.64	−0.502806
$333$	3570.03	0.587497
$334$	−253.815	−0.0415813
$335$	498.369	0.0812800
$336$	0	0
$337$	5907.09	0.954836	0.477418	−	0.878676i	$-0.341573\pi$
$337$	5907.09	0.954836	0.477418	+	0.878676i	$0.341573\pi$
$338$	−3597.92	−0.578997
$339$	−13327.8	−2.13530
$340$	−61.0067	−0.00973103
$341$	−2453.38	−0.389613
$342$	−12977.4	−2.05186
$343$	0	0
$344$	−3144.85	−0.492904
$345$	−241.314	−0.0376577
$346$	4025.97	0.625542
$347$	10316.5	1.59602	0.798011	−	0.602643i	$-0.205886\pi$
$347$	10316.5	1.59602	0.798011	+	0.602643i	$0.205886\pi$
$348$	−1294.33	−0.199378
$349$	−3529.49	−0.541344	−0.270672	−	0.962672i	$-0.587246\pi$
$349$	−3529.49	−0.541344	−0.270672	+	0.962672i	$0.587246\pi$
$350$	0	0
$351$	−2753.55	−0.418728
$352$	1104.67	0.167270
$353$	−9742.65	−1.46898	−0.734489	−	0.678621i	$-0.762578\pi$
$353$	−9742.65	−1.46898	−0.734489	+	0.678621i	$0.762578\pi$
$354$	−268.898	−0.0403723
$355$	117.164	0.0175167
$356$	3411.97	0.507961
$357$	0	0
$358$	−8027.66	−1.18513
$359$	901.912	0.132594	0.0662968	−	0.997800i	$-0.478882\pi$
$359$	901.912	0.132594	0.0662968	+	0.997800i	$0.478882\pi$
$360$	434.463	0.0636061
$361$	15448.7	2.25233
$362$	−302.948	−0.0439850
$363$	1169.21	0.169057
$364$	0	0
$365$	19.9890	0.00286650
$366$	−8520.60	−1.21688
$367$	4791.96	0.681575	0.340788	−	0.940140i	$-0.389306\pi$
$367$	4791.96	0.681575	0.340788	+	0.940140i	$0.389306\pi$
$368$	368.000	0.0521286
$369$	9447.70	1.33287
$370$	205.449	0.0288671
$371$	0	0
$372$	2385.97	0.332545
$373$	−9960.67	−1.38269	−0.691346	−	0.722524i	$-0.742982\pi$
$373$	−9960.67	−1.38269	−0.691346	+	0.722524i	$0.742982\pi$
$374$	−842.358	−0.116463
$375$	2606.58	0.358942
$376$	4038.06	0.553849
$377$	769.180	0.105079
$378$	0	0
$379$	−13945.2	−1.89002	−0.945008	−	0.327047i	$-0.893946\pi$
$379$	−13945.2	−1.89002	−0.945008	+	0.327047i	$0.893946\pi$
$380$	−746.828	−0.100820
$381$	19201.0	2.58189
$382$	756.039	0.101263
$383$	3973.16	0.530076	0.265038	−	0.964238i	$-0.414616\pi$
$383$	3973.16	0.530076	0.265038	+	0.964238i	$0.414616\pi$
$384$	−1074.32	−0.142769
$385$	0	0
$386$	−3097.63	−0.408460
$387$	−17078.1	−2.24323
$388$	527.137	0.0689726
$389$	−10529.6	−1.37242	−0.686210	−	0.727404i	$-0.740727\pi$
$389$	−10529.6	−1.37242	−0.686210	+	0.727404i	$0.740727\pi$
$390$	−418.648	−0.0543565
$391$	−280.616	−0.0362950
$392$	0	0
$393$	7538.73	0.967630
$394$	3831.97	0.489979
$395$	740.308	0.0943011
$396$	5998.90	0.761253
$397$	−1596.09	−0.201777	−0.100888	−	0.994898i	$-0.532169\pi$
$397$	−1596.09	−0.201777	−0.100888	+	0.994898i	$0.532169\pi$
$398$	−2675.48	−0.336959
$399$	0	0
$400$	−1975.00	−0.246875
$401$	5269.53	0.656229	0.328114	−	0.944638i	$-0.393587\pi$
$401$	5269.53	0.656229	0.328114	+	0.944638i	$0.393587\pi$
$402$	−6692.22	−0.830293
$403$	−1417.90	−0.175262
$404$	2085.72	0.256853
$405$	−18.2603	−0.00224040
$406$	0	0
$407$	2836.77	0.345488
$408$	819.212	0.0994045
$409$	109.202	0.0132022	0.00660109	−	0.999978i	$-0.497899\pi$
$409$	109.202	0.0132022	0.00660109	+	0.999978i	$0.497899\pi$
$410$	543.700	0.0654913
$411$	−19314.7	−2.31807
$412$	−5749.29	−0.687493
$413$	0	0
$414$	1998.42	0.237240
$415$	−950.561	−0.112437
$416$	638.431	0.0752443
$417$	6531.85	0.767065
$418$	−10311.9	−1.20663
$419$	11287.2	1.31602	0.658012	−	0.753007i	$-0.271398\pi$
$419$	11287.2	1.31602	0.658012	+	0.753007i	$0.271398\pi$
$420$	0	0
$421$	9146.41	1.05883	0.529416	−	0.848362i	$-0.322411\pi$
$421$	9146.41	1.05883	0.529416	+	0.848362i	$0.322411\pi$
$422$	−2732.49	−0.315203
$423$	21928.7	2.52059
$424$	−5371.99	−0.615299
$425$	1506.02	0.171889
$426$	−1573.31	−0.178937
$427$	0	0
$428$	3913.56	0.441984
$429$	−5780.54	−0.650552
$430$	−982.817	−0.110222
$431$	6053.67	0.676554	0.338277	−	0.941047i	$-0.390156\pi$
$431$	6053.67	0.676554	0.338277	+	0.941047i	$0.390156\pi$
$432$	−2208.25	−0.245937
$433$	−7890.31	−0.875714	−0.437857	−	0.899045i	$-0.644262\pi$
$433$	−7890.31	−0.875714	−0.437857	+	0.899045i	$0.644262\pi$
$434$	0	0
$435$	−404.500	−0.0445846
$436$	−7134.27	−0.783646
$437$	−3435.23	−0.376039
$438$	−268.417	−0.0292819
$439$	−16943.5	−1.84207	−0.921033	−	0.389484i	$-0.872653\pi$
$439$	−16943.5	−1.84207	−0.921033	+	0.389484i	$0.872653\pi$
$440$	345.227	0.0374047
$441$	0	0
$442$	−486.831	−0.0523896
$443$	−7145.24	−0.766322	−0.383161	−	0.923682i	$-0.625165\pi$
$443$	−7145.24	−0.766322	−0.383161	+	0.923682i	$0.625165\pi$
$444$	−2758.83	−0.294883
$445$	1066.30	0.113589
$446$	−11609.2	−1.23254
$447$	14224.3	1.50511
$448$	0	0
$449$	−7130.35	−0.749448	−0.374724	−	0.927136i	$-0.622263\pi$
$449$	−7130.35	−0.749448	−0.374724	+	0.927136i	$0.622263\pi$
$450$	−10725.2	−1.12354
$451$	7507.21	0.783816
$452$	6351.79	0.660981
$453$	20020.4	2.07647
$454$	−11150.0	−1.15263
$455$	0	0
$456$	10028.6	1.02989
$457$	−10782.0	−1.10363	−0.551817	−	0.833965i	$-0.686065\pi$
$457$	−10782.0	−1.10363	−0.551817	+	0.833965i	$0.686065\pi$
$458$	2882.65	0.294099
$459$	1683.89	0.171236
$460$	115.006	0.0116569
$461$	−13766.4	−1.39081	−0.695405	−	0.718618i	$-0.744775\pi$
$461$	−13766.4	−1.39081	−0.695405	+	0.718618i	$0.744775\pi$
$462$	0	0
$463$	−7453.28	−0.748128	−0.374064	−	0.927403i	$-0.622036\pi$
$463$	−7453.28	−0.748128	−0.374064	+	0.927403i	$0.622036\pi$
$464$	616.856	0.0617173
$465$	745.654	0.0743632
$466$	−1576.35	−0.156702
$467$	−4411.82	−0.437162	−0.218581	−	0.975819i	$-0.570143\pi$
$467$	−4411.82	−0.437162	−0.218581	+	0.975819i	$0.570143\pi$
$468$	3467.00	0.342440
$469$	0	0
$470$	1261.96	0.123851
$471$	13680.4	1.33834
$472$	128.152	0.0124972
$473$	−13570.4	−1.31917
$474$	−9941.04	−0.963306
$475$	18436.3	1.78088
$476$	0	0
$477$	−29172.6	−2.80025
$478$	−1406.70	−0.134604
$479$	−9812.08	−0.935961	−0.467981	−	0.883739i	$-0.655018\pi$
$479$	−9812.08	−0.935961	−0.467981	+	0.883739i	$0.655018\pi$
$480$	−335.741	−0.0319259
$481$	1639.48	0.155413
$482$	−9915.30	−0.936990
$483$	0	0
$484$	−557.226	−0.0523316
$485$	164.739	0.0154235
$486$	7698.06	0.718501
$487$	−8594.81	−0.799729	−0.399865	−	0.916574i	$-0.630943\pi$
$487$	−8594.81	−0.799729	−0.399865	+	0.916574i	$0.630943\pi$
$488$	4060.77	0.376685
$489$	−13280.2	−1.22812
$490$	0	0
$491$	−6523.36	−0.599583	−0.299792	−	0.954005i	$-0.596917\pi$
$491$	−6523.36	−0.599583	−0.299792	+	0.954005i	$0.596917\pi$
$492$	−7300.94	−0.669007
$493$	−470.380	−0.0429713
$494$	−5959.66	−0.542789
$495$	1874.76	0.170230
$496$	−1137.11	−0.102939
$497$	0	0
$498$	12764.4	1.14856
$499$	−9190.98	−0.824539	−0.412270	−	0.911062i	$-0.635264\pi$
$499$	−9190.98	−0.824539	−0.412270	+	0.911062i	$0.635264\pi$
$500$	−1242.25	−0.111110
$501$	1065.15	0.0949846
$502$	4220.01	0.375195
$503$	−6508.34	−0.576924	−0.288462	−	0.957491i	$-0.593144\pi$
$503$	−6508.34	−0.576924	−0.288462	+	0.957491i	$0.593144\pi$
$504$	0	0
$505$	651.822	0.0574371
$506$	1587.96	0.139513
$507$	15098.8	1.32261
$508$	−9150.88	−0.799222
$509$	−6925.71	−0.603098	−0.301549	−	0.953451i	$-0.597504\pi$
$509$	−6925.71	−0.603098	−0.301549	+	0.953451i	$0.597504\pi$
$510$	256.017	0.0222287
$511$	0	0
$512$	512.000	0.0441942
$513$	20613.7	1.77411
$514$	−7960.40	−0.683110
$515$	−1796.75	−0.153736
$516$	13197.5	1.12595
$517$	17424.7	1.48228
$518$	0	0
$519$	−16895.2	−1.42893
$520$	199.520	0.0168260
$521$	7987.54	0.671671	0.335835	−	0.941921i	$-0.390981\pi$
$521$	7987.54	0.671671	0.335835	+	0.941921i	$0.390981\pi$
$522$	3349.84	0.280878
$523$	15046.6	1.25801	0.629006	−	0.777400i	$-0.283462\pi$
$523$	15046.6	1.25801	0.629006	+	0.777400i	$0.283462\pi$
$524$	−3592.83	−0.299529
$525$	0	0
$526$	5988.58	0.496415
$527$	867.096	0.0716723
$528$	−4635.79	−0.382097
$529$	529.000	0.0434783
$530$	−1678.84	−0.137592
$531$	695.930	0.0568753
$532$	0	0
$533$	4338.71	0.352590
$534$	−14318.5	−1.16034
$535$	1223.05	0.0988357
$536$	3189.39	0.257017
$537$	33688.4	2.70719
$538$	3157.91	0.253062
$539$	0	0
$540$	−690.116	−0.0549960
$541$	17694.0	1.40615	0.703074	−	0.711117i	$-0.251810\pi$
$541$	17694.0	1.40615	0.703074	+	0.711117i	$0.251810\pi$
$542$	5040.68	0.399476
$543$	1271.33	0.100475
$544$	−390.422	−0.0307706
$545$	−2229.58	−0.175238
$546$	0	0
$547$	−14053.9	−1.09854	−0.549270	−	0.835645i	$-0.685094\pi$
$547$	−14053.9	−1.09854	−0.549270	+	0.835645i	$0.685094\pi$
$548$	9205.07	0.717557
$549$	22052.0	1.71431
$550$	−8522.34	−0.660716
$551$	−5758.27	−0.445209
$552$	−1544.33	−0.119078
$553$	0	0
$554$	6143.46	0.471138
$555$	−862.178	−0.0659413
$556$	−3112.97	−0.237445
$557$	−9700.89	−0.737953	−0.368977	−	0.929439i	$-0.620292\pi$
$557$	−9700.89	−0.737953	−0.368977	+	0.929439i	$0.620292\pi$
$558$	−6175.08	−0.468480
$559$	−7842.85	−0.593412
$560$	0	0
$561$	3534.99	0.266038
$562$	6954.90	0.522019
$563$	−16195.9	−1.21239	−0.606194	−	0.795317i	$-0.707305\pi$
$563$	−16195.9	−1.21239	−0.606194	+	0.795317i	$0.707305\pi$
$564$	−16945.9	−1.26516
$565$	1985.04	0.147808
$566$	−8993.12	−0.667860
$567$	0	0
$568$	749.813	0.0553899
$569$	10736.0	0.790993	0.395496	−	0.918468i	$-0.370573\pi$
$569$	10736.0	0.790993	0.395496	+	0.918468i	$0.370573\pi$
$570$	3134.10	0.230303
$571$	−2527.76	−0.185260	−0.0926302	−	0.995701i	$-0.529527\pi$
$571$	−2527.76	−0.185260	−0.0926302	+	0.995701i	$0.529527\pi$
$572$	2754.90	0.201378
$573$	−3172.75	−0.231315
$574$	0	0
$575$	−2839.06	−0.205908
$576$	2780.42	0.201130
$577$	−17209.1	−1.24164	−0.620819	−	0.783954i	$-0.713200\pi$
$577$	−17209.1	−1.24164	−0.620819	+	0.783954i	$0.713200\pi$
$578$	−9528.29	−0.685682
$579$	12999.4	0.933048
$580$	192.778	0.0138011
$581$	0	0
$582$	−2212.16	−0.157555
$583$	−23180.8	−1.64674
$584$	127.923	0.00906420
$585$	1083.49	0.0765760
$586$	−1983.21	−0.139805
$587$	−23203.8	−1.63156	−0.815778	−	0.578365i	$-0.803691\pi$
$587$	−23203.8	−1.63156	−0.815778	+	0.578365i	$0.803691\pi$
$588$	0	0
$589$	10614.8	0.742570
$590$	40.0497	0.00279461
$591$	−16081.0	−1.11926
$592$	1314.81	0.0912809
$593$	−24829.9	−1.71946	−0.859731	−	0.510748i	$-0.829369\pi$
$593$	−24829.9	−1.71946	−0.859731	+	0.510748i	$0.829369\pi$
$594$	−9528.87	−0.658206
$595$	0	0
$596$	−6779.03	−0.465906
$597$	11227.8	0.769719
$598$	917.744	0.0627581
$599$	10861.6	0.740888	0.370444	−	0.928855i	$-0.379206\pi$
$599$	10861.6	0.740888	0.370444	+	0.928855i	$0.379206\pi$
$600$	8288.17	0.563938
$601$	−8744.86	−0.593528	−0.296764	−	0.954951i	$-0.595907\pi$
$601$	−8744.86	−0.593528	−0.296764	+	0.954951i	$0.595907\pi$
$602$	0	0
$603$	17320.0	1.16969
$604$	−9541.36	−0.642769
$605$	−174.142	−0.0117023
$606$	−8752.83	−0.586732
$607$	18776.5	1.25555	0.627773	−	0.778397i	$-0.283967\pi$
$607$	18776.5	1.25555	0.627773	+	0.778397i	$0.283967\pi$
$608$	−4779.45	−0.318803
$609$	0	0
$610$	1269.06	0.0842337
$611$	10070.4	0.666784
$612$	−2120.19	−0.140038
$613$	−13936.3	−0.918243	−0.459121	−	0.888374i	$-0.651836\pi$
$613$	−13936.3	−0.918243	−0.459121	+	0.888374i	$0.651836\pi$
$614$	1511.45	0.0993440
$615$	−2281.66	−0.149602
$616$	0	0
$617$	−20181.9	−1.31685	−0.658423	−	0.752648i	$-0.728776\pi$
$617$	−20181.9	−1.31685	−0.658423	+	0.752648i	$0.728776\pi$
$618$	24127.2	1.57045
$619$	2085.40	0.135411	0.0677056	−	0.997705i	$-0.478432\pi$
$619$	2085.40	0.135411	0.0677056	+	0.997705i	$0.478432\pi$
$620$	−355.366	−0.0230191
$621$	−3174.37	−0.205125
$622$	−13679.5	−0.881828
$623$	0	0
$624$	−2679.20	−0.171881
$625$	15041.4	0.962652
$626$	2103.86	0.134325
$627$	43274.5	2.75633
$628$	−6519.84	−0.414283
$629$	−1002.60	−0.0635552
$630$	0	0
$631$	−8948.77	−0.564572	−0.282286	−	0.959330i	$-0.591093\pi$
$631$	−8948.77	−0.564572	−0.282286	+	0.959330i	$0.591093\pi$
$632$	4737.72	0.298191
$633$	11467.0	0.720021
$634$	16692.2	1.04563
$635$	−2859.80	−0.178721
$636$	22543.8	1.40553
$637$	0	0
$638$	2661.81	0.165175
$639$	4071.86	0.252082
$640$	160.008	0.00988264
$641$	30471.3	1.87760	0.938802	−	0.344457i	$-0.111937\pi$
$641$	30471.3	1.87760	0.938802	+	0.344457i	$0.111937\pi$
$642$	−16423.4	−1.00963
$643$	−13214.1	−0.810441	−0.405220	−	0.914219i	$-0.632805\pi$
$643$	−13214.1	−0.810441	−0.405220	+	0.914219i	$0.632805\pi$
$644$	0	0
$645$	4124.44	0.251782
$646$	3644.54	0.221970
$647$	4347.80	0.264188	0.132094	−	0.991237i	$-0.457830\pi$
$647$	4347.80	0.264188	0.132094	+	0.991237i	$0.457830\pi$
$648$	−116.860	−0.00708441
$649$	552.991	0.0334465
$650$	−4925.39	−0.297215
$651$	0	0
$652$	6329.09	0.380163
$653$	20836.5	1.24869	0.624344	−	0.781149i	$-0.285366\pi$
$653$	20836.5	1.24869	0.624344	+	0.781149i	$0.285366\pi$
$654$	29939.3	1.79009
$655$	−1122.82	−0.0669803
$656$	3479.50	0.207091
$657$	694.686	0.0412516
$658$	0	0
$659$	17281.8	1.02155	0.510775	−	0.859714i	$-0.329358\pi$
$659$	17281.8	1.02155	0.510775	+	0.859714i	$0.329358\pi$
$660$	−1448.76	−0.0854439
$661$	30113.8	1.77200	0.885998	−	0.463689i	$-0.153474\pi$
$661$	30113.8	1.77200	0.885998	+	0.463689i	$0.153474\pi$
$662$	−19692.3	−1.15613
$663$	2043.01	0.119674
$664$	−6083.27	−0.355537
$665$	0	0
$666$	7140.06	0.415423
$667$	886.731	0.0514758
$668$	−507.630	−0.0294024
$669$	48718.7	2.81551
$670$	996.738	0.0574737
$671$	17522.7	1.00813
$672$	0	0
$673$	15778.7	0.903748	0.451874	−	0.892082i	$-0.350756\pi$
$673$	15778.7	0.903748	0.451874	+	0.892082i	$0.350756\pi$
$674$	11814.2	0.675171
$675$	17036.3	0.971449
$676$	−7195.84	−0.409413
$677$	−13864.5	−0.787084	−0.393542	−	0.919307i	$-0.628750\pi$
$677$	−13864.5	−0.787084	−0.393542	+	0.919307i	$0.628750\pi$
$678$	−26655.6	−1.50989
$679$	0	0
$680$	−122.013	−0.00688088
$681$	46791.3	2.63296
$682$	−4906.76	−0.275498
$683$	−9949.71	−0.557416	−0.278708	−	0.960376i	$-0.589906\pi$
$683$	−9949.71	−0.557416	−0.278708	+	0.960376i	$0.589906\pi$
$684$	−25954.8	−1.45089
$685$	2876.74	0.160459
$686$	0	0
$687$	−12097.2	−0.671813
$688$	−6289.70	−0.348536
$689$	−13397.0	−0.740765
$690$	−482.628	−0.0266280
$691$	23364.5	1.28629	0.643146	−	0.765744i	$-0.277629\pi$
$691$	23364.5	1.28629	0.643146	+	0.765744i	$0.277629\pi$
$692$	8051.94	0.442325
$693$	0	0
$694$	20633.0	1.12856
$695$	−972.853	−0.0530970
$696$	−2588.67	−0.140982
$697$	−2653.27	−0.144189
$698$	−7058.98	−0.382788
$699$	6615.23	0.357956
$700$	0	0
$701$	24204.5	1.30413	0.652063	−	0.758165i	$-0.273904\pi$
$701$	24204.5	1.30413	0.652063	+	0.758165i	$0.273904\pi$
$702$	−5507.10	−0.296086
$703$	−12273.5	−0.658471
$704$	2209.34	0.118278
$705$	−5295.87	−0.282914
$706$	−19485.3	−1.03872
$707$	0	0
$708$	−537.796	−0.0285475
$709$	−24636.4	−1.30499	−0.652497	−	0.757791i	$-0.726279\pi$
$709$	−24636.4	−1.30499	−0.652497	+	0.757791i	$0.726279\pi$
$710$	234.329	0.0123862
$711$	25728.2	1.35708
$712$	6823.94	0.359183
$713$	−1634.60	−0.0858571
$714$	0	0
$715$	860.952	0.0450319
$716$	−16055.3	−0.838010
$717$	5903.27	0.307478
$718$	1803.82	0.0937579
$719$	23300.2	1.20856	0.604278	−	0.796774i	$-0.293462\pi$
$719$	23300.2	1.20856	0.604278	+	0.796774i	$0.293462\pi$
$720$	868.925	0.0449763
$721$	0	0
$722$	30897.4	1.59264
$723$	41610.0	2.14038
$724$	−605.896	−0.0311021
$725$	−4758.94	−0.243783
$726$	2338.43	0.119542
$727$	7216.97	0.368174	0.184087	−	0.982910i	$-0.441067\pi$
$727$	7216.97	0.368174	0.184087	+	0.982910i	$0.441067\pi$
$728$	0	0
$729$	−31910.9	−1.62124
$730$	39.9780	0.00202692
$731$	4796.17	0.242671
$732$	−17041.2	−0.860465
$733$	33652.3	1.69574	0.847868	−	0.530207i	$-0.177886\pi$
$733$	33652.3	1.69574	0.847868	+	0.530207i	$0.177886\pi$
$734$	9583.91	0.481946
$735$	0	0
$736$	736.000	0.0368605
$737$	13762.6	0.687859
$738$	18895.4	0.942479
$739$	4892.42	0.243533	0.121766	−	0.992559i	$-0.461144\pi$
$739$	4892.42	0.243533	0.121766	+	0.992559i	$0.461144\pi$
$740$	410.899	0.0204121
$741$	25010.0	1.23990
$742$	0	0
$743$	37793.4	1.86609	0.933046	−	0.359757i	$-0.117141\pi$
$743$	37793.4	1.86609	0.933046	+	0.359757i	$0.117141\pi$
$744$	4771.94	0.235145
$745$	−2118.56	−0.104185
$746$	−19921.3	−0.977710
$747$	−33035.2	−1.61807
$748$	−1684.72	−0.0823520
$749$	0	0
$750$	5213.17	0.253811
$751$	−16410.3	−0.797363	−0.398681	−	0.917090i	$-0.630532\pi$
$751$	−16410.3	−0.797363	−0.398681	+	0.917090i	$0.630532\pi$
$752$	8076.12	0.391630
$753$	−17709.4	−0.857063
$754$	1538.36	0.0743021
$755$	−2981.83	−0.143735
$756$	0	0
$757$	556.810	0.0267340	0.0133670	−	0.999911i	$-0.495745\pi$
$757$	556.810	0.0267340	0.0133670	+	0.999911i	$0.495745\pi$
$758$	−27890.4	−1.33644
$759$	−6663.96	−0.318691
$760$	−1493.66	−0.0712902
$761$	28693.1	1.36678	0.683392	−	0.730051i	$-0.260504\pi$
$761$	28693.1	1.36678	0.683392	+	0.730051i	$0.260504\pi$
$762$	38402.1	1.82567
$763$	0	0
$764$	1512.08	0.0716035
$765$	−662.593	−0.0313152
$766$	7946.32	0.374820
$767$	319.595	0.0150455
$768$	−2148.63	−0.100953
$769$	7724.86	0.362244	0.181122	−	0.983461i	$-0.442027\pi$
$769$	7724.86	0.362244	0.181122	+	0.983461i	$0.442027\pi$
$770$	0	0
$771$	33406.2	1.56043
$772$	−6195.27	−0.288825
$773$	13661.3	0.635657	0.317829	−	0.948148i	$-0.397046\pi$
$773$	13661.3	0.635657	0.317829	+	0.948148i	$0.397046\pi$
$774$	−34156.2	−1.58620
$775$	8772.62	0.406609
$776$	1054.27	0.0487710
$777$	0	0
$778$	−21059.2	−0.970447
$779$	−32480.6	−1.49389
$780$	−837.295	−0.0384359
$781$	3235.53	0.148241
$782$	−561.232	−0.0256645
$783$	−5321.00	−0.242857
$784$	0	0
$785$	−2037.56	−0.0926414
$786$	15077.5	0.684218
$787$	15229.2	0.689789	0.344895	−	0.938641i	$-0.387915\pi$
$787$	15229.2	0.689789	0.344895	+	0.938641i	$0.387915\pi$
$788$	7663.93	0.346467
$789$	−25131.3	−1.13397
$790$	1480.62	0.0666810
$791$	0	0
$792$	11997.8	0.538287
$793$	10127.0	0.453495
$794$	−3192.18	−0.142678
$795$	7045.31	0.314304
$796$	−5350.96	−0.238266
$797$	39301.5	1.74671	0.873357	−	0.487081i	$-0.161938\pi$
$797$	39301.5	1.74671	0.873357	+	0.487081i	$0.161938\pi$
$798$	0	0
$799$	−6158.39	−0.272676
$800$	−3949.99	−0.174567
$801$	37057.4	1.63466
$802$	10539.1	0.464024
$803$	552.002	0.0242587
$804$	−13384.4	−0.587105
$805$	0	0
$806$	−2835.81	−0.123929
$807$	−13252.3	−0.578072
$808$	4171.45	0.181622
$809$	−17700.5	−0.769242	−0.384621	−	0.923075i	$-0.625668\pi$
$809$	−17700.5	−0.769242	−0.384621	+	0.923075i	$0.625668\pi$
$810$	−36.5207	−0.00158420
$811$	−40258.2	−1.74310	−0.871552	−	0.490302i	$-0.836886\pi$
$811$	−40258.2	−1.74310	−0.871552	+	0.490302i	$0.836886\pi$
$812$	0	0
$813$	−21153.5	−0.912527
$814$	5673.55	0.244297
$815$	1977.94	0.0850114
$816$	1638.42	0.0702896
$817$	58713.4	2.51423
$818$	218.404	0.00933534
$819$	0	0
$820$	1087.40	0.0463093
$821$	−8530.30	−0.362618	−0.181309	−	0.983426i	$-0.558033\pi$
$821$	−8530.30	−0.362618	−0.181309	+	0.983426i	$0.558033\pi$
$822$	−38629.5	−1.63912
$823$	2167.02	0.0917831	0.0458915	−	0.998946i	$-0.485387\pi$
$823$	2167.02	0.0917831	0.0458915	+	0.998946i	$0.485387\pi$
$824$	−11498.6	−0.486131
$825$	35764.4	1.50928
$826$	0	0
$827$	963.326	0.0405056	0.0202528	−	0.999795i	$-0.493553\pi$
$827$	963.326	0.0405056	0.0202528	+	0.999795i	$0.493553\pi$
$828$	3996.85	0.167754
$829$	24689.1	1.03436	0.517182	−	0.855876i	$-0.326981\pi$
$829$	24689.1	1.03436	0.517182	+	0.855876i	$0.326981\pi$
$830$	−1901.12	−0.0795047
$831$	−25781.3	−1.07623
$832$	1276.86	0.0532058
$833$	0	0
$834$	13063.7	0.542397
$835$	−158.643	−0.00657492
$836$	−20623.9	−0.853219
$837$	9808.71	0.405064
$838$	22574.3	0.930569
$839$	19791.5	0.814398	0.407199	−	0.913340i	$-0.366506\pi$
$839$	19791.5	0.814398	0.407199	+	0.913340i	$0.366506\pi$
$840$	0	0
$841$	−22902.6	−0.939056
$842$	18292.8	0.748708
$843$	−29186.5	−1.19245
$844$	−5464.98	−0.222882
$845$	−2248.82	−0.0915523
$846$	43857.4	1.78233
$847$	0	0
$848$	−10744.0	−0.435082
$849$	37740.0	1.52560
$850$	3012.04	0.121544
$851$	1890.04	0.0761335
$852$	−3146.62	−0.126528
$853$	40827.7	1.63882	0.819410	−	0.573208i	$-0.194301\pi$
$853$	40827.7	1.63882	0.819410	+	0.573208i	$0.194301\pi$
$854$	0	0
$855$	−8111.29	−0.324445
$856$	7827.12	0.312530
$857$	−18564.8	−0.739978	−0.369989	−	0.929036i	$-0.620638\pi$
$857$	−18564.8	−0.739978	−0.369989	+	0.929036i	$0.620638\pi$
$858$	−11561.1	−0.460010
$859$	−36746.1	−1.45956	−0.729779	−	0.683683i	$-0.760377\pi$
$859$	−36746.1	−1.45956	−0.729779	+	0.683683i	$0.760377\pi$
$860$	−1965.63	−0.0779390
$861$	0	0
$862$	12107.3	0.478396
$863$	19545.9	0.770974	0.385487	−	0.922713i	$-0.374034\pi$
$863$	19545.9	0.770974	0.385487	+	0.922713i	$0.374034\pi$
$864$	−4416.51	−0.173904
$865$	2516.36	0.0989121
$866$	−15780.6	−0.619223
$867$	39985.9	1.56631
$868$	0	0
$869$	20443.8	0.798054
$870$	−809.001	−0.0315261
$871$	7953.94	0.309425
$872$	−14268.5	−0.554121
$873$	5725.24	0.221959
$874$	−6870.45	−0.265900
$875$	0	0
$876$	−536.835	−0.0207054
$877$	42195.4	1.62467	0.812336	−	0.583190i	$-0.198196\pi$
$877$	42195.4	1.62467	0.812336	+	0.583190i	$0.198196\pi$
$878$	−33886.9	−1.30254
$879$	8322.62	0.319357
$880$	690.454	0.0264491
$881$	−17683.6	−0.676248	−0.338124	−	0.941101i	$-0.609792\pi$
$881$	−17683.6	−0.676248	−0.338124	+	0.941101i	$0.609792\pi$
$882$	0	0
$883$	8463.26	0.322550	0.161275	−	0.986910i	$-0.448439\pi$
$883$	8463.26	0.322550	0.161275	+	0.986910i	$0.448439\pi$
$884$	−973.662	−0.0370450
$885$	−168.070	−0.00638375
$886$	−14290.5	−0.541871
$887$	14341.0	0.542867	0.271434	−	0.962457i	$-0.412502\pi$
$887$	14341.0	0.542867	0.271434	+	0.962457i	$0.412502\pi$
$888$	−5517.65	−0.208514
$889$	0	0
$890$	2132.59	0.0803199
$891$	−504.264	−0.0189601
$892$	−23218.5	−0.871538
$893$	−75389.4	−2.82510
$894$	28448.5	1.06427
$895$	−5017.55	−0.187395
$896$	0	0
$897$	−3851.36	−0.143359
$898$	−14260.7	−0.529940
$899$	−2739.98	−0.101650
$900$	−21450.4	−0.794461
$901$	8192.75	0.302930
$902$	15014.4	0.554241
$903$	0	0
$904$	12703.6	0.467384
$905$	−189.352	−0.00695501
$906$	40040.7	1.46828
$907$	−24548.4	−0.898696	−0.449348	−	0.893357i	$-0.648344\pi$
$907$	−24548.4	−0.898696	−0.449348	+	0.893357i	$0.648344\pi$
$908$	−22299.9	−0.815031
$909$	22653.0	0.826571
$910$	0	0
$911$	−1013.37	−0.0368543	−0.0184272	−	0.999830i	$-0.505866\pi$
$911$	−1013.37	−0.0368543	−0.0184272	+	0.999830i	$0.505866\pi$
$912$	20057.2	0.728245
$913$	−26250.0	−0.951532
$914$	−21564.0	−0.780387
$915$	−5325.65	−0.192416
$916$	5765.30	0.207959
$917$	0	0
$918$	3367.78	0.121082
$919$	−35469.5	−1.27316	−0.636578	−	0.771212i	$-0.719651\pi$
$919$	−35469.5	−1.27316	−0.636578	+	0.771212i	$0.719651\pi$
$920$	230.012	0.00824269
$921$	−6342.87	−0.226933
$922$	−27532.7	−0.983451
$923$	1869.94	0.0666844
$924$	0	0
$925$	−10143.5	−0.360559
$926$	−14906.6	−0.529006
$927$	−62443.0	−2.21240
$928$	1233.71	0.0436407
$929$	44668.7	1.57754	0.788770	−	0.614689i	$-0.210718\pi$
$929$	44668.7	1.57754	0.788770	+	0.614689i	$0.210718\pi$
$930$	1491.31	0.0525827
$931$	0	0
$932$	−3152.70	−0.110805
$933$	57406.6	2.01437
$934$	−8823.63	−0.309120
$935$	−526.501	−0.0184154
$936$	6933.99	0.242142
$937$	−29296.9	−1.02144	−0.510719	−	0.859748i	$-0.670621\pi$
$937$	−29296.9	−1.02144	−0.510719	+	0.859748i	$0.670621\pi$
$938$	0	0
$939$	−8828.95	−0.306839
$940$	2523.92	0.0875758
$941$	−32731.8	−1.13393	−0.566965	−	0.823742i	$-0.691882\pi$
$941$	−32731.8	−1.13393	−0.566965	+	0.823742i	$0.691882\pi$
$942$	27360.8	0.946351
$943$	5001.78	0.172726
$944$	256.304	0.00883686
$945$	0	0
$946$	−27140.8	−0.932793
$947$	−43440.7	−1.49064	−0.745319	−	0.666709i	$-0.767703\pi$
$947$	−43440.7	−1.49064	−0.745319	+	0.666709i	$0.767703\pi$
$948$	−19882.1	−0.681160
$949$	319.023	0.0109125
$950$	36872.6	1.25927
$951$	−70049.6	−2.38855
$952$	0	0
$953$	32198.3	1.09445	0.547223	−	0.836987i	$-0.315685\pi$
$953$	32198.3	1.09445	0.547223	+	0.836987i	$0.315685\pi$
$954$	−58345.2	−1.98008
$955$	472.549	0.0160119
$956$	−2813.39	−0.0951796
$957$	−11170.4	−0.377312
$958$	−19624.2	−0.661825
$959$	0	0
$960$	−671.483	−0.0225750
$961$	−24740.1	−0.830457
$962$	3278.96	0.109894
$963$	42505.1	1.42234
$964$	−19830.6	−0.662552
$965$	−1936.12	−0.0645865
$966$	0	0
$967$	9822.55	0.326651	0.163326	−	0.986572i	$-0.447778\pi$
$967$	9822.55	0.326651	0.163326	+	0.986572i	$0.447778\pi$
$968$	−1114.45	−0.0370040
$969$	−15294.5	−0.507047
$970$	329.478	0.0109061
$971$	−39113.0	−1.29268	−0.646342	−	0.763048i	$-0.723702\pi$
$971$	−39113.0	−1.29268	−0.646342	+	0.763048i	$0.723702\pi$
$972$	15396.1	0.508057
$973$	0	0
$974$	−17189.6	−0.565494
$975$	20669.6	0.678931
$976$	8121.53	0.266356
$977$	12293.0	0.402546	0.201273	−	0.979535i	$-0.435492\pi$
$977$	12293.0	0.402546	0.201273	+	0.979535i	$0.435492\pi$
$978$	−26560.3	−0.868410
$979$	29446.1	0.961288
$980$	0	0
$981$	−77485.3	−2.52183
$982$	−13046.7	−0.423969
$983$	−11332.8	−0.367711	−0.183855	−	0.982953i	$-0.558858\pi$
$983$	−11332.8	−0.367711	−0.183855	+	0.982953i	$0.558858\pi$
$984$	−14601.9	−0.473060
$985$	2395.10	0.0774765
$986$	−940.759	−0.0303853
$987$	0	0
$988$	−11919.3	−0.383810
$989$	−9041.44	−0.290699
$990$	3749.51	0.120371
$991$	−3870.67	−0.124072	−0.0620362	−	0.998074i	$-0.519759\pi$
$991$	−3870.67	−0.124072	−0.0620362	+	0.998074i	$0.519759\pi$
$992$	−2274.22	−0.0727889
$993$	82639.5	2.64097
$994$	0	0
$995$	−1672.26	−0.0532807
$996$	25528.7	0.812158
$997$	−50890.5	−1.61657	−0.808284	−	0.588792i	$-0.799604\pi$
$997$	−50890.5	−1.61657	−0.808284	+	0.588792i	$0.799604\pi$
$998$	−18382.0	−0.583037
$999$	−11341.5	−0.359189

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.l.1.1		5
7.6	odd	2		322.4.a.h.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.a.h.1.5	✓	5	7.6	odd	2
2254.4.a.l.1.1		5	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -8.39309 q^{3} +4.00000 q^{4} +1.25007 q^{5} -16.7862 q^{6} +8.00000 q^{8} +43.4440 q^{9} +2.50013 q^{10} +34.5209 q^{11} -33.5724 q^{12} +19.9510 q^{13} -10.4919 q^{15} +16.0000 q^{16} -12.2007 q^{17} +86.8880 q^{18} -149.358 q^{19} +5.00026 q^{20} +69.0418 q^{22} +23.0000 q^{23} -67.1447 q^{24} -123.437 q^{25} +39.9019 q^{26} -138.016 q^{27} +38.5535 q^{29} -20.9838 q^{30} -71.0694 q^{31} +32.0000 q^{32} -289.737 q^{33} -24.4014 q^{34} +173.776 q^{36} +82.1755 q^{37} -298.715 q^{38} -167.450 q^{39} +10.0005 q^{40} +217.469 q^{41} -393.106 q^{43} +138.084 q^{44} +54.3078 q^{45} +46.0000 q^{46} +504.758 q^{47} -134.289 q^{48} -246.875 q^{50} +102.402 q^{51} +79.8038 q^{52} -671.499 q^{53} -276.032 q^{54} +43.1534 q^{55} +1253.57 q^{57} +77.1070 q^{58} +16.0190 q^{59} -41.9677 q^{60} +507.596 q^{61} -142.139 q^{62} +64.0000 q^{64} +24.9400 q^{65} -579.474 q^{66} +398.674 q^{67} -48.8028 q^{68} -193.041 q^{69} +93.7266 q^{71} +347.552 q^{72} +15.9904 q^{73} +164.351 q^{74} +1036.02 q^{75} -597.431 q^{76} -334.900 q^{78} +592.216 q^{79} +20.0010 q^{80} -14.6075 q^{81} +434.937 q^{82} -760.409 q^{83} -15.2517 q^{85} -786.212 q^{86} -323.583 q^{87} +276.167 q^{88} +852.993 q^{89} +108.616 q^{90} +92.0000 q^{92} +596.492 q^{93} +1009.52 q^{94} -186.707 q^{95} -268.579 q^{96} +131.784 q^{97} +1499.73 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 10 q^{2} - 5 q^{3} + 20 q^{4} - 22 q^{5} - 10 q^{6} + 40 q^{8} + 54 q^{9} - 44 q^{10} + 42 q^{11} - 20 q^{12} - 107 q^{13} + 122 q^{15} + 80 q^{16} - 218 q^{17} + 108 q^{18} - 194 q^{19} - 88 q^{20}+ \cdots - 2616 q^{99}+O(q^{100}) \) 5 * q + 10 * q^2 - 5 * q^3 + 20 * q^4 - 22 * q^5 - 10 * q^6 + 40 * q^8 + 54 * q^9 - 44 * q^10 + 42 * q^11 - 20 * q^12 - 107 * q^13 + 122 * q^15 + 80 * q^16 - 218 * q^17 + 108 * q^18 - 194 * q^19 - 88 * q^20 + 84 * q^22 + 115 * q^23 - 40 * q^24 + 219 * q^25 - 214 * q^26 - 203 * q^27 + 33 * q^29 + 244 * q^30 - 19 * q^31 + 160 * q^32 - 240 * q^33 - 436 * q^34 + 216 * q^36 + 214 * q^37 - 388 * q^38 - 243 * q^39 - 176 * q^40 - 395 * q^41 - 284 * q^43 + 168 * q^44 - 470 * q^45 + 230 * q^46 + 493 * q^47 - 80 * q^48 + 438 * q^50 - 244 * q^51 - 428 * q^52 - 70 * q^53 - 406 * q^54 + 186 * q^55 + 22 * q^57 + 66 * q^58 - 226 * q^59 + 488 * q^60 - 1606 * q^61 - 38 * q^62 + 320 * q^64 + 362 * q^65 - 480 * q^66 + 298 * q^67 - 872 * q^68 - 115 * q^69 - 969 * q^71 + 432 * q^72 - 697 * q^73 + 428 * q^74 - 1065 * q^75 - 776 * q^76 - 486 * q^78 + 186 * q^79 - 352 * q^80 - 335 * q^81 - 790 * q^82 + 1104 * q^83 - 1344 * q^85 - 568 * q^86 + 1603 * q^87 + 336 * q^88 + 814 * q^89 - 940 * q^90 + 460 * q^92 - 3423 * q^93 + 986 * q^94 - 2812 * q^95 - 160 * q^96 + 256 * q^97 - 2616 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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