LMFDB - Embedded newform 650.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,4,Mod(1,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, 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("650.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	650.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:	$38.3512415037$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{105}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 26 $ x^2 - x - 26
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.62348$ of defining polynomial
Character	$\chi$	$=$	650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -1.62348 q^{3} +4.00000 q^{4} +3.24695 q^{6} -2.62348 q^{7} -8.00000 q^{8} -24.3643 q^{9} -51.1174 q^{11} -6.49390 q^{12} -13.0000 q^{13} +5.24695 q^{14} +16.0000 q^{16} -4.36433 q^{17} +48.7287 q^{18} -47.4817 q^{19} +4.25915 q^{21} +102.235 q^{22} +91.5991 q^{23} +12.9878 q^{24} +26.0000 q^{26} +83.3887 q^{27} -10.4939 q^{28} -139.988 q^{29} +31.1418 q^{31} -32.0000 q^{32} +82.9878 q^{33} +8.72866 q^{34} -97.4573 q^{36} +377.482 q^{37} +94.9634 q^{38} +21.1052 q^{39} -7.71646 q^{41} -8.51830 q^{42} -75.5991 q^{43} -204.470 q^{44} -183.198 q^{46} -186.785 q^{47} -25.9756 q^{48} -336.117 q^{49} +7.08538 q^{51} -52.0000 q^{52} -236.457 q^{53} -166.777 q^{54} +20.9878 q^{56} +77.0854 q^{57} +279.976 q^{58} -4.36433 q^{59} +380.360 q^{61} -62.2835 q^{62} +63.9192 q^{63} +64.0000 q^{64} -165.976 q^{66} -98.2424 q^{67} -17.4573 q^{68} -148.709 q^{69} +1168.46 q^{71} +194.915 q^{72} +404.357 q^{73} -754.963 q^{74} -189.927 q^{76} +134.105 q^{77} -42.2104 q^{78} -856.748 q^{79} +522.457 q^{81} +15.4329 q^{82} +920.898 q^{83} +17.0366 q^{84} +151.198 q^{86} +227.267 q^{87} +408.939 q^{88} -1526.04 q^{89} +34.1052 q^{91} +366.396 q^{92} -50.5579 q^{93} +373.570 q^{94} +51.9512 q^{96} +960.979 q^{97} +672.235 q^{98} +1245.44 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 7 q^{3} + 8 q^{4} - 14 q^{6} + 5 q^{7} - 16 q^{8} + 23 q^{9} - 51 q^{11} + 28 q^{12} - 26 q^{13} - 10 q^{14} + 32 q^{16} + 63 q^{17} - 46 q^{18} + 28 q^{19} + 70 q^{21} + 102 q^{22} + 9 q^{23}+ \cdots + 1251 q^{99}+O(q^{100}) $ 2 * q - 4 * q^2 + 7 * q^3 + 8 * q^4 - 14 * q^6 + 5 * q^7 - 16 * q^8 + 23 * q^9 - 51 * q^11 + 28 * q^12 - 26 * q^13 - 10 * q^14 + 32 * q^16 + 63 * q^17 - 46 * q^18 + 28 * q^19 + 70 * q^21 + 102 * q^22 + 9 * q^23 - 56 * q^24 + 52 * q^26 + 259 * q^27 + 20 * q^28 - 198 * q^29 + 175 * q^31 - 64 * q^32 + 84 * q^33 - 126 * q^34 + 92 * q^36 + 632 * q^37 - 56 * q^38 - 91 * q^39 + 210 * q^41 - 140 * q^42 + 23 * q^43 - 204 * q^44 - 18 * q^46 + 231 * q^47 + 112 * q^48 - 621 * q^49 + 588 * q^51 - 104 * q^52 - 186 * q^53 - 518 * q^54 - 40 * q^56 + 728 * q^57 + 396 * q^58 + 63 * q^59 - 182 * q^61 - 350 * q^62 + 425 * q^63 + 128 * q^64 - 168 * q^66 + 695 * q^67 + 252 * q^68 - 861 * q^69 + 390 * q^71 - 184 * q^72 + 1526 * q^73 - 1264 * q^74 + 112 * q^76 + 135 * q^77 + 182 * q^78 - 863 * q^79 + 758 * q^81 - 420 * q^82 + 315 * q^83 + 280 * q^84 - 46 * q^86 - 273 * q^87 + 408 * q^88 - 1638 * q^89 - 65 * q^91 + 36 * q^92 + 1190 * q^93 - 462 * q^94 - 224 * q^96 + 98 * q^97 + 1242 * q^98 + 1251 * 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
$3$	−1.62348	−0.312438	−0.156219	−	0.987722i	$-0.549931\pi$
$3$	−1.62348	−0.312438	−0.156219	+	0.987722i	$0.549931\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.24695	0.220927
$7$	−2.62348	−0.141654	−0.0708272	−	0.997489i	$-0.522564\pi$
$7$	−2.62348	−0.141654	−0.0708272	+	0.997489i	$0.522564\pi$
$8$	−8.00000	−0.353553
$9$	−24.3643	−0.902383
$10$	0	0
$11$	−51.1174	−1.40113	−0.700567	−	0.713587i	$-0.747069\pi$
$11$	−51.1174	−1.40113	−0.700567	+	0.713587i	$0.747069\pi$
$12$	−6.49390	−0.156219
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	5.24695	0.100165
$15$	0	0
$16$	16.0000	0.250000
$17$	−4.36433	−0.0622650	−0.0311325	−	0.999515i	$-0.509911\pi$
$17$	−4.36433	−0.0622650	−0.0311325	+	0.999515i	$0.509911\pi$
$18$	48.7287	0.638081
$19$	−47.4817	−0.573318	−0.286659	−	0.958033i	$-0.592545\pi$
$19$	−47.4817	−0.573318	−0.286659	+	0.958033i	$0.592545\pi$
$20$	0	0
$21$	4.25915	0.0442582
$22$	102.235	0.990751
$23$	91.5991	0.830423	0.415211	−	0.909725i	$-0.363708\pi$
$23$	91.5991	0.830423	0.415211	+	0.909725i	$0.363708\pi$
$24$	12.9878	0.110464
$25$	0	0
$26$	26.0000	0.196116
$27$	83.3887	0.594377
$28$	−10.4939	−0.0708272
$29$	−139.988	−0.896382	−0.448191	−	0.893938i	$-0.647932\pi$
$29$	−139.988	−0.896382	−0.448191	+	0.893938i	$0.647932\pi$
$30$	0	0
$31$	31.1418	0.180427	0.0902133	−	0.995922i	$-0.471245\pi$
$31$	31.1418	0.180427	0.0902133	+	0.995922i	$0.471245\pi$
$32$	−32.0000	−0.176777
$33$	82.9878	0.437767
$34$	8.72866	0.0440280
$35$	0	0
$36$	−97.4573	−0.451191
$37$	377.482	1.67723	0.838616	−	0.544723i	$-0.183365\pi$
$37$	377.482	1.67723	0.838616	+	0.544723i	$0.183365\pi$
$38$	94.9634	0.405397
$39$	21.1052	0.0866547
$40$	0	0
$41$	−7.71646	−0.0293929	−0.0146964	−	0.999892i	$-0.504678\pi$
$41$	−7.71646	−0.0293929	−0.0146964	+	0.999892i	$0.504678\pi$
$42$	−8.51830	−0.0312953
$43$	−75.5991	−0.268111	−0.134055	−	0.990974i	$-0.542800\pi$
$43$	−75.5991	−0.268111	−0.134055	+	0.990974i	$0.542800\pi$
$44$	−204.470	−0.700567
$45$	0	0
$46$	−183.198	−0.587198
$47$	−186.785	−0.579689	−0.289845	−	0.957074i	$-0.593604\pi$
$47$	−186.785	−0.579689	−0.289845	+	0.957074i	$0.593604\pi$
$48$	−25.9756	−0.0781095
$49$	−336.117	−0.979934
$50$	0	0
$51$	7.08538	0.0194539
$52$	−52.0000	−0.138675
$53$	−236.457	−0.612828	−0.306414	−	0.951898i	$-0.599129\pi$
$53$	−236.457	−0.612828	−0.306414	+	0.951898i	$0.599129\pi$
$54$	−166.777	−0.420288
$55$	0	0
$56$	20.9878	0.0500824
$57$	77.0854	0.179126
$58$	279.976	0.633838
$59$	−4.36433	−0.00963029	−0.00481514	−	0.999988i	$-0.501533\pi$
$59$	−4.36433	−0.00963029	−0.00481514	+	0.999988i	$0.501533\pi$
$60$	0	0
$61$	380.360	0.798362	0.399181	−	0.916872i	$-0.369295\pi$
$61$	380.360	0.798362	0.399181	+	0.916872i	$0.369295\pi$
$62$	−62.2835	−0.127581
$63$	63.9192	0.127826
$64$	64.0000	0.125000
$65$	0	0
$66$	−165.976	−0.309548
$67$	−98.2424	−0.179138	−0.0895688	−	0.995981i	$-0.528549\pi$
$67$	−98.2424	−0.179138	−0.0895688	+	0.995981i	$0.528549\pi$
$68$	−17.4573	−0.0311325
$69$	−148.709	−0.259456
$70$	0	0
$71$	1168.46	1.95311	0.976554	−	0.215271i	$-0.0690634\pi$
$71$	1168.46	1.95311	0.976554	+	0.215271i	$0.0690634\pi$
$72$	194.915	0.319040
$73$	404.357	0.648307	0.324153	−	0.946005i	$-0.394921\pi$
$73$	404.357	0.648307	0.324153	+	0.946005i	$0.394921\pi$
$74$	−754.963	−1.18598
$75$	0	0
$76$	−189.927	−0.286659
$77$	134.105	0.198477
$78$	−42.2104	−0.0612741
$79$	−856.748	−1.22015	−0.610074	−	0.792344i	$-0.708860\pi$
$79$	−856.748	−1.22015	−0.610074	+	0.792344i	$0.708860\pi$
$80$	0	0
$81$	522.457	0.716677
$82$	15.4329	0.0207839
$83$	920.898	1.21785	0.608926	−	0.793227i	$-0.291601\pi$
$83$	920.898	1.21785	0.608926	+	0.793227i	$0.291601\pi$
$84$	17.0366	0.0221291
$85$	0	0
$86$	151.198	0.189583
$87$	227.267	0.280064
$88$	408.939	0.495376
$89$	−1526.04	−1.81753	−0.908763	−	0.417312i	$-0.862972\pi$
$89$	−1526.04	−1.81753	−0.908763	+	0.417312i	$0.862972\pi$
$90$	0	0
$91$	34.1052	0.0392878
$92$	366.396	0.415211
$93$	−50.5579	−0.0563721
$94$	373.570	0.409902
$95$	0	0
$96$	51.9512	0.0552318
$97$	960.979	1.00590	0.502952	−	0.864315i	$-0.332247\pi$
$97$	960.979	1.00590	0.502952	+	0.864315i	$0.332247\pi$
$98$	672.235	0.692918
$99$	1245.44	1.26436
$100$	0	0
$101$	848.854	0.836278	0.418139	−	0.908383i	$-0.362682\pi$
$101$	848.854	0.836278	0.418139	+	0.908383i	$0.362682\pi$
$102$	−14.1708	−0.0137560
$103$	1819.89	1.74096	0.870480	−	0.492203i	$-0.163808\pi$
$103$	1819.89	1.74096	0.870480	+	0.492203i	$0.163808\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	472.915	0.433335
$107$	165.008	0.149083	0.0745415	−	0.997218i	$-0.476251\pi$
$107$	165.008	0.149083	0.0745415	+	0.997218i	$0.476251\pi$
$108$	333.555	0.297188
$109$	920.719	0.809073	0.404536	−	0.914522i	$-0.367433\pi$
$109$	920.719	0.809073	0.404536	+	0.914522i	$0.367433\pi$
$110$	0	0
$111$	−612.832	−0.524031
$112$	−41.9756	−0.0354136
$113$	−965.986	−0.804180	−0.402090	−	0.915600i	$-0.631716\pi$
$113$	−965.986	−0.804180	−0.402090	+	0.915600i	$0.631716\pi$
$114$	−154.171	−0.126662
$115$	0	0
$116$	−559.951	−0.448191
$117$	316.736	0.250276
$118$	8.72866	0.00680964
$119$	11.4497	0.00882011
$120$	0	0
$121$	1281.99	0.963175
$122$	−760.719	−0.564527
$123$	12.5275	0.00918345
$124$	124.567	0.0902133
$125$	0	0
$126$	−127.838	−0.0903869
$127$	1105.46	0.772393	0.386196	−	0.922417i	$-0.373789\pi$
$127$	1105.46	0.772393	0.386196	+	0.922417i	$0.373789\pi$
$128$	−128.000	−0.0883883
$129$	122.733	0.0837679
$130$	0	0
$131$	700.020	0.466878	0.233439	−	0.972371i	$-0.425002\pi$
$131$	700.020	0.466878	0.233439	+	0.972371i	$0.425002\pi$
$132$	331.951	0.218884
$133$	124.567	0.0812131
$134$	196.485	0.126669
$135$	0	0
$136$	34.9146	0.0220140
$137$	−1230.03	−0.767070	−0.383535	−	0.923526i	$-0.625293\pi$
$137$	−1230.03	−0.767070	−0.383535	+	0.923526i	$0.625293\pi$
$138$	297.418	0.183463
$139$	−1004.05	−0.612679	−0.306340	−	0.951922i	$-0.599104\pi$
$139$	−1004.05	−0.612679	−0.306340	+	0.951922i	$0.599104\pi$
$140$	0	0
$141$	303.241	0.181117
$142$	−2336.92	−1.38106
$143$	664.526	0.388605
$144$	−389.829	−0.225596
$145$	0	0
$146$	−808.713	−0.458422
$147$	545.678	0.306169
$148$	1509.93	0.838616
$149$	2310.81	1.27053	0.635264	−	0.772295i	$-0.280891\pi$
$149$	2310.81	1.27053	0.635264	+	0.772295i	$0.280891\pi$
$150$	0	0
$151$	2268.83	1.22275	0.611374	−	0.791341i	$-0.290617\pi$
$151$	2268.83	1.22275	0.611374	+	0.791341i	$0.290617\pi$
$152$	379.854	0.202699
$153$	106.334	0.0561868
$154$	−268.210	−0.140344
$155$	0	0
$156$	84.4207	0.0433274
$157$	1943.62	0.988011	0.494006	−	0.869459i	$-0.335532\pi$
$157$	1943.62	0.988011	0.494006	+	0.869459i	$0.335532\pi$
$158$	1713.50	0.862775
$159$	383.883	0.191471
$160$	0	0
$161$	−240.308	−0.117633
$162$	−1044.91	−0.506767
$163$	−311.814	−0.149835	−0.0749177	−	0.997190i	$-0.523869\pi$
$163$	−311.814	−0.149835	−0.0749177	+	0.997190i	$0.523869\pi$
$164$	−30.8658	−0.0146964
$165$	0	0
$166$	−1841.80	−0.861151
$167$	−331.570	−0.153639	−0.0768194	−	0.997045i	$-0.524476\pi$
$167$	−331.570	−0.153639	−0.0768194	+	0.997045i	$0.524476\pi$
$168$	−34.0732	−0.0156476
$169$	169.000	0.0769231
$170$	0	0
$171$	1156.86	0.517353
$172$	−302.396	−0.134055
$173$	234.096	0.102879	0.0514393	−	0.998676i	$-0.483619\pi$
$173$	234.096	0.102879	0.0514393	+	0.998676i	$0.483619\pi$
$174$	−454.534	−0.198035
$175$	0	0
$176$	−817.878	−0.350283
$177$	7.08538	0.00300887
$178$	3052.08	1.28519
$179$	21.5807	0.00901127	0.00450564	−	0.999990i	$-0.498566\pi$
$179$	21.5807	0.00901127	0.00450564	+	0.999990i	$0.498566\pi$
$180$	0	0
$181$	1132.38	0.465024	0.232512	−	0.972594i	$-0.425306\pi$
$181$	1132.38	0.465024	0.232512	+	0.972594i	$0.425306\pi$
$182$	−68.2104	−0.0277807
$183$	−617.505	−0.249439
$184$	−732.793	−0.293599
$185$	0	0
$186$	101.116	0.0398611
$187$	223.093	0.0872416
$188$	−747.140	−0.289845
$189$	−218.768	−0.0841960
$190$	0	0
$191$	−1547.72	−0.586331	−0.293165	−	0.956062i	$-0.594709\pi$
$191$	−1547.72	−0.586331	−0.293165	+	0.956062i	$0.594709\pi$
$192$	−103.902	−0.0390547
$193$	1721.73	0.642140	0.321070	−	0.947056i	$-0.395958\pi$
$193$	1721.73	0.642140	0.321070	+	0.947056i	$0.395958\pi$
$194$	−1921.96	−0.711281
$195$	0	0
$196$	−1344.47	−0.489967
$197$	44.9786	0.0162670	0.00813349	−	0.999967i	$-0.497411\pi$
$197$	44.9786	0.0162670	0.00813349	+	0.999967i	$0.497411\pi$
$198$	−2490.88	−0.894036
$199$	2577.95	0.918323	0.459161	−	0.888353i	$-0.348150\pi$
$199$	2577.95	0.918323	0.459161	+	0.888353i	$0.348150\pi$
$200$	0	0
$201$	159.494	0.0559694
$202$	−1697.71	−0.591338
$203$	367.255	0.126976
$204$	28.3415	0.00972697
$205$	0	0
$206$	−3639.78	−1.23105
$207$	−2231.75	−0.749359
$208$	−208.000	−0.0693375
$209$	2427.14	0.803296
$210$	0	0
$211$	1434.93	0.468174	0.234087	−	0.972216i	$-0.424790\pi$
$211$	1434.93	0.468174	0.234087	+	0.972216i	$0.424790\pi$
$212$	−945.829	−0.306414
$213$	−1896.97	−0.610225
$214$	−330.015	−0.105418
$215$	0	0
$216$	−667.110	−0.210144
$217$	−81.6997	−0.0255582
$218$	−1841.44	−0.572101
$219$	−656.463	−0.202556
$220$	0	0
$221$	56.7363	0.0172692
$222$	1225.66	0.370546
$223$	−5222.63	−1.56831	−0.784155	−	0.620565i	$-0.786903\pi$
$223$	−5222.63	−1.56831	−0.784155	+	0.620565i	$0.786903\pi$
$224$	83.9512	0.0250412
$225$	0	0
$226$	1931.97	0.568641
$227$	2349.65	0.687011	0.343506	−	0.939151i	$-0.388386\pi$
$227$	2349.65	0.687011	0.343506	+	0.939151i	$0.388386\pi$
$228$	308.342	0.0895632
$229$	317.860	0.0917238	0.0458619	−	0.998948i	$-0.485397\pi$
$229$	317.860	0.0917238	0.0458619	+	0.998948i	$0.485397\pi$
$230$	0	0
$231$	−217.716	−0.0620117
$232$	1119.90	0.316919
$233$	−1295.68	−0.364303	−0.182151	−	0.983271i	$-0.558306\pi$
$233$	−1295.68	−0.364303	−0.182151	+	0.983271i	$0.558306\pi$
$234$	−633.473	−0.176972
$235$	0	0
$236$	−17.4573	−0.00481514
$237$	1390.91	0.381221
$238$	−22.8994	−0.00623676
$239$	2149.67	0.581801	0.290900	−	0.956753i	$-0.406045\pi$
$239$	2149.67	0.581801	0.290900	+	0.956753i	$0.406045\pi$
$240$	0	0
$241$	−6687.39	−1.78744	−0.893719	−	0.448628i	$-0.851913\pi$
$241$	−6687.39	−1.78744	−0.893719	+	0.448628i	$0.851913\pi$
$242$	−2563.97	−0.681068
$243$	−3099.69	−0.818294
$244$	1521.44	0.399181
$245$	0	0
$246$	−25.0550	−0.00649368
$247$	617.262	0.159010
$248$	−249.134	−0.0637905
$249$	−1495.05	−0.380503
$250$	0	0
$251$	577.116	0.145128	0.0725642	−	0.997364i	$-0.476882\pi$
$251$	577.116	0.145128	0.0725642	+	0.997364i	$0.476882\pi$
$252$	255.677	0.0639132
$253$	−4682.30	−1.16353
$254$	−2210.92	−0.546164
$255$	0	0
$256$	256.000	0.0625000
$257$	−4004.15	−0.971876	−0.485938	−	0.873993i	$-0.661522\pi$
$257$	−4004.15	−0.971876	−0.485938	+	0.873993i	$0.661522\pi$
$258$	−245.466	−0.0592329
$259$	−990.314	−0.237587
$260$	0	0
$261$	3410.71	0.808880
$262$	−1400.04	−0.330133
$263$	290.393	0.0680852	0.0340426	−	0.999420i	$-0.489162\pi$
$263$	290.393	0.0680852	0.0340426	+	0.999420i	$0.489162\pi$
$264$	−663.902	−0.154774
$265$	0	0
$266$	−249.134	−0.0574263
$267$	2477.49	0.567864
$268$	−392.969	−0.0895688
$269$	1004.21	0.227612	0.113806	−	0.993503i	$-0.463696\pi$
$269$	1004.21	0.227612	0.113806	+	0.993503i	$0.463696\pi$
$270$	0	0
$271$	5506.55	1.23431	0.617157	−	0.786840i	$-0.288284\pi$
$271$	5506.55	1.23431	0.617157	+	0.786840i	$0.288284\pi$
$272$	−69.8292	−0.0155662
$273$	−55.3689	−0.0122750
$274$	2460.06	0.542400
$275$	0	0
$276$	−594.835	−0.129728
$277$	−4208.88	−0.912950	−0.456475	−	0.889736i	$-0.650888\pi$
$277$	−4208.88	−0.912950	−0.456475	+	0.889736i	$0.650888\pi$
$278$	2008.10	0.433230
$279$	−758.748	−0.162814
$280$	0	0
$281$	−6804.56	−1.44458	−0.722288	−	0.691592i	$-0.756910\pi$
$281$	−6804.56	−1.44458	−0.722288	+	0.691592i	$0.756910\pi$
$282$	−606.482	−0.128069
$283$	−3351.59	−0.703997	−0.351999	−	0.936001i	$-0.614498\pi$
$283$	−3351.59	−0.703997	−0.351999	+	0.936001i	$0.614498\pi$
$284$	4673.84	0.976554
$285$	0	0
$286$	−1329.05	−0.274785
$287$	20.2439	0.00416363
$288$	779.658	0.159520
$289$	−4893.95	−0.996123
$290$	0	0
$291$	−1560.13	−0.314282
$292$	1617.43	0.324153
$293$	−3486.87	−0.695239	−0.347620	−	0.937636i	$-0.613010\pi$
$293$	−3486.87	−0.695239	−0.347620	+	0.937636i	$0.613010\pi$
$294$	−1091.36	−0.216494
$295$	0	0
$296$	−3019.85	−0.592991
$297$	−4262.61	−0.832801
$298$	−4621.62	−0.898399
$299$	−1190.79	−0.230318
$300$	0	0
$301$	198.332	0.0379790
$302$	−4537.67	−0.864614
$303$	−1378.09	−0.261285
$304$	−759.707	−0.143330
$305$	0	0
$306$	−212.668	−0.0397301
$307$	8145.25	1.51425	0.757124	−	0.653271i	$-0.226604\pi$
$307$	8145.25	1.51425	0.757124	+	0.653271i	$0.226604\pi$
$308$	536.421	0.0992383
$309$	−2954.54	−0.543942
$310$	0	0
$311$	−511.038	−0.0931779	−0.0465889	−	0.998914i	$-0.514835\pi$
$311$	−511.038	−0.0931779	−0.0465889	+	0.998914i	$0.514835\pi$
$312$	−168.841	−0.0306371
$313$	−3285.96	−0.593397	−0.296699	−	0.954971i	$-0.595886\pi$
$313$	−3285.96	−0.593397	−0.296699	+	0.954971i	$0.595886\pi$
$314$	−3887.24	−0.698630
$315$	0	0
$316$	−3426.99	−0.610074
$317$	8375.89	1.48403	0.742014	−	0.670385i	$-0.233871\pi$
$317$	8375.89	1.48403	0.742014	+	0.670385i	$0.233871\pi$
$318$	−767.765	−0.135390
$319$	7155.81	1.25595
$320$	0	0
$321$	−267.886	−0.0465792
$322$	480.616	0.0831791
$323$	207.226	0.0356977
$324$	2089.83	0.358338
$325$	0	0
$326$	623.628	0.105950
$327$	−1494.77	−0.252785
$328$	61.7317	0.0103920
$329$	490.026	0.0821155
$330$	0	0
$331$	−1475.80	−0.245068	−0.122534	−	0.992464i	$-0.539102\pi$
$331$	−1475.80	−0.245068	−0.122534	+	0.992464i	$0.539102\pi$
$332$	3683.59	0.608926
$333$	−9197.09	−1.51351
$334$	663.140	0.108639
$335$	0	0
$336$	68.1464	0.0110646
$337$	9267.32	1.49799	0.748996	−	0.662575i	$-0.230536\pi$
$337$	9267.32	1.49799	0.748996	+	0.662575i	$0.230536\pi$
$338$	−338.000	−0.0543928
$339$	1568.25	0.251256
$340$	0	0
$341$	−1591.89	−0.252802
$342$	−2313.72	−0.365823
$343$	1781.65	0.280466
$344$	604.793	0.0947914
$345$	0	0
$346$	−468.192	−0.0727461
$347$	−12251.7	−1.89541	−0.947703	−	0.319152i	$-0.896602\pi$
$347$	−12251.7	−1.89541	−0.947703	+	0.319152i	$0.896602\pi$
$348$	909.067	0.140032
$349$	9704.59	1.48847	0.744233	−	0.667920i	$-0.232815\pi$
$349$	9704.59	1.48847	0.744233	+	0.667920i	$0.232815\pi$
$350$	0	0
$351$	−1084.05	−0.164850
$352$	1635.76	0.247688
$353$	−10183.3	−1.53542	−0.767711	−	0.640797i	$-0.778604\pi$
$353$	−10183.3	−1.53542	−0.767711	+	0.640797i	$0.778604\pi$
$354$	−14.1708	−0.00212759
$355$	0	0
$356$	−6104.16	−0.908763
$357$	−18.5883	−0.00275574
$358$	−43.1614	−0.00637193
$359$	8393.64	1.23398	0.616991	−	0.786970i	$-0.288351\pi$
$359$	8393.64	1.23398	0.616991	+	0.786970i	$0.288351\pi$
$360$	0	0
$361$	−4604.49	−0.671306
$362$	−2264.77	−0.328822
$363$	−2081.27	−0.300933
$364$	136.421	0.0196439
$365$	0	0
$366$	1235.01	0.176380
$367$	−1694.93	−0.241075	−0.120538	−	0.992709i	$-0.538462\pi$
$367$	−1694.93	−0.241075	−0.120538	+	0.992709i	$0.538462\pi$
$368$	1465.59	0.207606
$369$	188.006	0.0265236
$370$	0	0
$371$	620.340	0.0868098
$372$	−202.232	−0.0281861
$373$	4925.86	0.683784	0.341892	−	0.939739i	$-0.388932\pi$
$373$	4925.86	0.683784	0.341892	+	0.939739i	$0.388932\pi$
$374$	−446.186	−0.0616891
$375$	0	0
$376$	1494.28	0.204951
$377$	1819.84	0.248612
$378$	437.537	0.0595356
$379$	4563.89	0.618552	0.309276	−	0.950972i	$-0.399913\pi$
$379$	4563.89	0.618552	0.309276	+	0.950972i	$0.399913\pi$
$380$	0	0
$381$	−1794.69	−0.241325
$382$	3095.44	0.414598
$383$	6385.74	0.851948	0.425974	−	0.904735i	$-0.359932\pi$
$383$	6385.74	0.851948	0.425974	+	0.904735i	$0.359932\pi$
$384$	207.805	0.0276159
$385$	0	0
$386$	−3443.46	−0.454061
$387$	1841.92	0.241938
$388$	3843.91	0.502952
$389$	−12304.3	−1.60374	−0.801870	−	0.597498i	$-0.796162\pi$
$389$	−12304.3	−1.60374	−0.801870	+	0.597498i	$0.796162\pi$
$390$	0	0
$391$	−399.768	−0.0517063
$392$	2688.94	0.346459
$393$	−1136.46	−0.145870
$394$	−89.9572	−0.0115025
$395$	0	0
$396$	4981.76	0.632179
$397$	3593.57	0.454298	0.227149	−	0.973860i	$-0.427060\pi$
$397$	3593.57	0.454298	0.227149	+	0.973860i	$0.427060\pi$
$398$	−5155.91	−0.649352
$399$	−202.232	−0.0253740
$400$	0	0
$401$	5943.15	0.740116	0.370058	−	0.929009i	$-0.379338\pi$
$401$	5943.15	0.740116	0.370058	+	0.929009i	$0.379338\pi$
$402$	−318.988	−0.0395763
$403$	−404.843	−0.0500414
$404$	3395.41	0.418139
$405$	0	0
$406$	−734.509	−0.0897859
$407$	−19295.9	−2.35003
$408$	−56.6830	−0.00687801
$409$	−3673.76	−0.444146	−0.222073	−	0.975030i	$-0.571282\pi$
$409$	−3673.76	−0.444146	−0.222073	+	0.975030i	$0.571282\pi$
$410$	0	0
$411$	1996.92	0.239662
$412$	7279.55	0.870480
$413$	11.4497	0.00136417
$414$	4463.50	0.529877
$415$	0	0
$416$	416.000	0.0490290
$417$	1630.05	0.191424
$418$	−4854.28	−0.568016
$419$	−6367.72	−0.742443	−0.371221	−	0.928544i	$-0.621061\pi$
$419$	−6367.72	−0.742443	−0.371221	+	0.928544i	$0.621061\pi$
$420$	0	0
$421$	−5184.48	−0.600180	−0.300090	−	0.953911i	$-0.597017\pi$
$421$	−5184.48	−0.600180	−0.300090	+	0.953911i	$0.597017\pi$
$422$	−2869.86	−0.331049
$423$	4550.89	0.523102
$424$	1891.66	0.216668
$425$	0	0
$426$	3793.93	0.431495
$427$	−997.864	−0.113091
$428$	660.030	0.0745415
$429$	−1078.84	−0.121415
$430$	0	0
$431$	6252.06	0.698727	0.349363	−	0.936987i	$-0.386398\pi$
$431$	6252.06	0.698727	0.349363	+	0.936987i	$0.386398\pi$
$432$	1334.22	0.148594
$433$	13563.3	1.50533	0.752667	−	0.658402i	$-0.228767\pi$
$433$	13563.3	1.50533	0.752667	+	0.658402i	$0.228767\pi$
$434$	163.399	0.0180724
$435$	0	0
$436$	3682.88	0.404536
$437$	−4349.28	−0.476097
$438$	1312.93	0.143228
$439$	4955.82	0.538789	0.269394	−	0.963030i	$-0.413176\pi$
$439$	4955.82	0.538789	0.269394	+	0.963030i	$0.413176\pi$
$440$	0	0
$441$	8189.27	0.884275
$442$	−113.473	−0.0122112
$443$	10721.6	1.14988	0.574942	−	0.818194i	$-0.305025\pi$
$443$	10721.6	1.14988	0.574942	+	0.818194i	$0.305025\pi$
$444$	−2451.33	−0.262016
$445$	0	0
$446$	10445.3	1.10896
$447$	−3751.54	−0.396961
$448$	−167.902	−0.0177068
$449$	−3704.31	−0.389348	−0.194674	−	0.980868i	$-0.562365\pi$
$449$	−3704.31	−0.389348	−0.194674	+	0.980868i	$0.562365\pi$
$450$	0	0
$451$	394.445	0.0411834
$452$	−3863.94	−0.402090
$453$	−3683.40	−0.382033
$454$	−4699.30	−0.485790
$455$	0	0
$456$	−616.683	−0.0633308
$457$	−6710.74	−0.686904	−0.343452	−	0.939170i	$-0.611596\pi$
$457$	−6710.74	−0.686904	−0.343452	+	0.939170i	$0.611596\pi$
$458$	−635.719	−0.0648585
$459$	−363.936	−0.0370089
$460$	0	0
$461$	−977.491	−0.0987555	−0.0493777	−	0.998780i	$-0.515724\pi$
$461$	−977.491	−0.0987555	−0.0493777	+	0.998780i	$0.515724\pi$
$462$	435.433	0.0438489
$463$	14489.1	1.45435	0.727176	−	0.686452i	$-0.240833\pi$
$463$	14489.1	1.45435	0.727176	+	0.686452i	$0.240833\pi$
$464$	−2239.80	−0.224096
$465$	0	0
$466$	2591.35	0.257601
$467$	−14849.7	−1.47144	−0.735721	−	0.677285i	$-0.763156\pi$
$467$	−14849.7	−1.47144	−0.735721	+	0.677285i	$0.763156\pi$
$468$	1266.95	0.125138
$469$	257.736	0.0253756
$470$	0	0
$471$	−3155.42	−0.308692
$472$	34.9146	0.00340482
$473$	3864.43	0.375659
$474$	−2781.82	−0.269564
$475$	0	0
$476$	45.7988	0.00441005
$477$	5761.12	0.553006
$478$	−4299.33	−0.411395
$479$	8109.85	0.773588	0.386794	−	0.922166i	$-0.373582\pi$
$479$	8109.85	0.773588	0.386794	+	0.922166i	$0.373582\pi$
$480$	0	0
$481$	−4907.26	−0.465181
$482$	13374.8	1.26391
$483$	390.134	0.0367530
$484$	5127.94	0.481588
$485$	0	0
$486$	6199.38	0.578621
$487$	−1215.92	−0.113138	−0.0565692	−	0.998399i	$-0.518016\pi$
$487$	−1215.92	−0.113138	−0.0565692	+	0.998399i	$0.518016\pi$
$488$	−3042.88	−0.282264
$489$	506.222	0.0468143
$490$	0	0
$491$	−12639.4	−1.16173	−0.580866	−	0.814000i	$-0.697286\pi$
$491$	−12639.4	−1.16173	−0.580866	+	0.814000i	$0.697286\pi$
$492$	50.1099	0.00459173
$493$	610.953	0.0558132
$494$	−1234.52	−0.112437
$495$	0	0
$496$	498.268	0.0451067
$497$	−3065.43	−0.276666
$498$	2990.11	0.269056
$499$	2658.21	0.238473	0.119236	−	0.992866i	$-0.461955\pi$
$499$	2658.21	0.238473	0.119236	+	0.992866i	$0.461955\pi$
$500$	0	0
$501$	538.296	0.0480026
$502$	−1154.23	−0.102621
$503$	18698.8	1.65753	0.828764	−	0.559599i	$-0.189045\pi$
$503$	18698.8	1.65753	0.828764	+	0.559599i	$0.189045\pi$
$504$	−511.354	−0.0451935
$505$	0	0
$506$	9364.61	0.822742
$507$	−274.367	−0.0240337
$508$	4421.85	0.386196
$509$	−20005.8	−1.74212	−0.871062	−	0.491173i	$-0.836568\pi$
$509$	−20005.8	−1.74212	−0.871062	+	0.491173i	$0.836568\pi$
$510$	0	0
$511$	−1060.82	−0.0918354
$512$	−512.000	−0.0441942
$513$	−3959.44	−0.340767
$514$	8008.30	0.687220
$515$	0	0
$516$	490.933	0.0418840
$517$	9547.96	0.812222
$518$	1980.63	0.168000
$519$	−380.049	−0.0321432
$520$	0	0
$521$	−9981.25	−0.839321	−0.419661	−	0.907681i	$-0.637851\pi$
$521$	−9981.25	−0.839321	−0.419661	+	0.907681i	$0.637851\pi$
$522$	−6821.42	−0.571964
$523$	398.152	0.0332887	0.0166443	−	0.999861i	$-0.494702\pi$
$523$	398.152	0.0332887	0.0166443	+	0.999861i	$0.494702\pi$
$524$	2800.08	0.233439
$525$	0	0
$526$	−580.786	−0.0481435
$527$	−135.913	−0.0112343
$528$	1327.80	0.109442
$529$	−3776.61	−0.310398
$530$	0	0
$531$	106.334	0.00869020
$532$	498.268	0.0406065
$533$	100.314	0.00815212
$534$	−4954.98	−0.401541
$535$	0	0
$536$	785.939	0.0633347
$537$	−35.0358	−0.00281546
$538$	−2008.41	−0.160946
$539$	17181.4	1.37302
$540$	0	0
$541$	7263.34	0.577219	0.288609	−	0.957447i	$-0.406807\pi$
$541$	7263.34	0.577219	0.288609	+	0.957447i	$0.406807\pi$
$542$	−11013.1	−0.872792
$543$	−1838.40	−0.145291
$544$	139.658	0.0110070
$545$	0	0
$546$	110.738	0.00867975
$547$	16741.6	1.30863	0.654315	−	0.756222i	$-0.272957\pi$
$547$	16741.6	1.30863	0.654315	+	0.756222i	$0.272957\pi$
$548$	−4920.12	−0.383535
$549$	−9267.21	−0.720428
$550$	0	0
$551$	6646.86	0.513912
$552$	1189.67	0.0917314
$553$	2247.66	0.172839
$554$	8417.76	0.645553
$555$	0	0
$556$	−4016.20	−0.306340
$557$	−6000.76	−0.456482	−0.228241	−	0.973605i	$-0.573297\pi$
$557$	−6000.76	−0.456482	−0.228241	+	0.973605i	$0.573297\pi$
$558$	1517.50	0.115127
$559$	982.788	0.0743605
$560$	0	0
$561$	−362.186	−0.0272576
$562$	13609.1	1.02147
$563$	−147.660	−0.0110535	−0.00552674	−	0.999985i	$-0.501759\pi$
$563$	−147.660	−0.0110535	−0.00552674	+	0.999985i	$0.501759\pi$
$564$	1212.96	0.0905585
$565$	0	0
$566$	6703.18	0.497801
$567$	−1370.65	−0.101520
$568$	−9347.68	−0.690528
$569$	−19161.2	−1.41174	−0.705868	−	0.708343i	$-0.749443\pi$
$569$	−19161.2	−1.41174	−0.705868	+	0.708343i	$0.749443\pi$
$570$	0	0
$571$	20439.8	1.49804	0.749019	−	0.662549i	$-0.230525\pi$
$571$	20439.8	1.49804	0.749019	+	0.662549i	$0.230525\pi$
$572$	2658.10	0.194302
$573$	2512.69	0.183192
$574$	−40.4879	−0.00294413
$575$	0	0
$576$	−1559.32	−0.112798
$577$	22889.0	1.65144	0.825720	−	0.564081i	$-0.190769\pi$
$577$	22889.0	1.65144	0.825720	+	0.564081i	$0.190769\pi$
$578$	9787.91	0.704365
$579$	−2795.19	−0.200629
$580$	0	0
$581$	−2415.95	−0.172514
$582$	3120.25	0.222231
$583$	12087.1	0.858655
$584$	−3234.85	−0.229211
$585$	0	0
$586$	6973.74	0.491608
$587$	−1492.73	−0.104960	−0.0524799	−	0.998622i	$-0.516713\pi$
$587$	−1492.73	−0.104960	−0.0524799	+	0.998622i	$0.516713\pi$
$588$	2182.71	0.153084
$589$	−1478.66	−0.103442
$590$	0	0
$591$	−73.0217	−0.00508242
$592$	6039.71	0.419308
$593$	23920.6	1.65650	0.828249	−	0.560360i	$-0.189337\pi$
$593$	23920.6	1.65650	0.828249	+	0.560360i	$0.189337\pi$
$594$	8525.23	0.588879
$595$	0	0
$596$	9243.23	0.635264
$597$	−4185.24	−0.286919
$598$	2381.58	0.162859
$599$	−23867.0	−1.62801	−0.814005	−	0.580857i	$-0.802717\pi$
$599$	−23867.0	−1.62801	−0.814005	+	0.580857i	$0.802717\pi$
$600$	0	0
$601$	−6618.71	−0.449222	−0.224611	−	0.974448i	$-0.572111\pi$
$601$	−6618.71	−0.449222	−0.224611	+	0.974448i	$0.572111\pi$
$602$	−396.665	−0.0268552
$603$	2393.61	0.161651
$604$	9075.34	0.611374
$605$	0	0
$606$	2756.19	0.184756
$607$	13982.7	0.934994	0.467497	−	0.883995i	$-0.345156\pi$
$607$	13982.7	0.934994	0.467497	+	0.883995i	$0.345156\pi$
$608$	1519.41	0.101349
$609$	−596.229	−0.0396723
$610$	0	0
$611$	2428.21	0.160777
$612$	425.336	0.0280934
$613$	−4043.90	−0.266446	−0.133223	−	0.991086i	$-0.542533\pi$
$613$	−4043.90	−0.266446	−0.133223	+	0.991086i	$0.542533\pi$
$614$	−16290.5	−1.07074
$615$	0	0
$616$	−1072.84	−0.0701721
$617$	−2871.83	−0.187384	−0.0936918	−	0.995601i	$-0.529867\pi$
$617$	−2871.83	−0.187384	−0.0936918	+	0.995601i	$0.529867\pi$
$618$	5909.09	0.384625
$619$	26324.3	1.70931	0.854656	−	0.519195i	$-0.173768\pi$
$619$	26324.3	1.70931	0.854656	+	0.519195i	$0.173768\pi$
$620$	0	0
$621$	7638.33	0.493584
$622$	1022.08	0.0658867
$623$	4003.53	0.257461
$624$	337.683	0.0216637
$625$	0	0
$626$	6571.91	0.419595
$627$	−3940.40	−0.250980
$628$	7774.48	0.494006
$629$	−1647.45	−0.104433
$630$	0	0
$631$	−17428.2	−1.09953	−0.549766	−	0.835319i	$-0.685283\pi$
$631$	−17428.2	−1.09953	−0.549766	+	0.835319i	$0.685283\pi$
$632$	6853.99	0.431388
$633$	−2329.57	−0.146275
$634$	−16751.8	−1.04937
$635$	0	0
$636$	1535.53	0.0957354
$637$	4369.53	0.271785
$638$	−14311.6	−0.888092
$639$	−28468.8	−1.76245
$640$	0	0
$641$	22541.7	1.38899	0.694496	−	0.719496i	$-0.255627\pi$
$641$	22541.7	1.38899	0.694496	+	0.719496i	$0.255627\pi$
$642$	535.772	0.0329365
$643$	21498.3	1.31852	0.659262	−	0.751913i	$-0.270869\pi$
$643$	21498.3	1.31852	0.659262	+	0.751913i	$0.270869\pi$
$644$	−961.232	−0.0588165
$645$	0	0
$646$	−414.451	−0.0252421
$647$	8290.26	0.503746	0.251873	−	0.967760i	$-0.418953\pi$
$647$	8290.26	0.503746	0.251873	+	0.967760i	$0.418953\pi$
$648$	−4179.66	−0.253383
$649$	223.093	0.0134933
$650$	0	0
$651$	132.637	0.00798536
$652$	−1247.26	−0.0749177
$653$	8650.10	0.518383	0.259192	−	0.965826i	$-0.416544\pi$
$653$	8650.10	0.518383	0.259192	+	0.965826i	$0.416544\pi$
$654$	2989.53	0.178746
$655$	0	0
$656$	−123.463	−0.00734822
$657$	−9851.88	−0.585020
$658$	−980.052	−0.0580644
$659$	−29128.3	−1.72182	−0.860908	−	0.508761i	$-0.830104\pi$
$659$	−29128.3	−1.72182	−0.860908	+	0.508761i	$0.830104\pi$
$660$	0	0
$661$	−14221.5	−0.836839	−0.418420	−	0.908254i	$-0.637416\pi$
$661$	−14221.5	−0.836839	−0.418420	+	0.908254i	$0.637416\pi$
$662$	2951.61	0.173289
$663$	−92.1099	−0.00539555
$664$	−7367.18	−0.430576
$665$	0	0
$666$	18394.2	1.07021
$667$	−12822.8	−0.744376
$668$	−1326.28	−0.0768194
$669$	8478.81	0.490000
$670$	0	0
$671$	−19443.0	−1.11861
$672$	−136.293	−0.00782382
$673$	−33065.1	−1.89386	−0.946929	−	0.321444i	$-0.895832\pi$
$673$	−33065.1	−1.89386	−0.946929	+	0.321444i	$0.895832\pi$
$674$	−18534.6	−1.05924
$675$	0	0
$676$	676.000	0.0384615
$677$	24785.0	1.40704	0.703520	−	0.710675i	$-0.251610\pi$
$677$	24785.0	1.40704	0.703520	+	0.710675i	$0.251610\pi$
$678$	−3136.51	−0.177665
$679$	−2521.10	−0.142491
$680$	0	0
$681$	−3814.60	−0.214648
$682$	3183.77	0.178758
$683$	8376.31	0.469269	0.234634	−	0.972084i	$-0.424611\pi$
$683$	8376.31	0.469269	0.234634	+	0.972084i	$0.424611\pi$
$684$	4627.44	0.258676
$685$	0	0
$686$	−3563.30	−0.198320
$687$	−516.037	−0.0286580
$688$	−1209.59	−0.0670276
$689$	3073.95	0.169968
$690$	0	0
$691$	−1387.54	−0.0763883	−0.0381942	−	0.999270i	$-0.512161\pi$
$691$	−1387.54	−0.0763883	−0.0381942	+	0.999270i	$0.512161\pi$
$692$	936.384	0.0514393
$693$	−3267.38	−0.179102
$694$	24503.4	1.34026
$695$	0	0
$696$	−1818.13	−0.0990175
$697$	33.6772	0.00183015
$698$	−19409.2	−1.05250
$699$	2103.50	0.113822
$700$	0	0
$701$	−23975.7	−1.29180	−0.645898	−	0.763424i	$-0.723517\pi$
$701$	−23975.7	−1.29180	−0.645898	+	0.763424i	$0.723517\pi$
$702$	2168.11	0.116567
$703$	−17923.5	−0.961588
$704$	−3271.51	−0.175142
$705$	0	0
$706$	20366.7	1.08571
$707$	−2226.95	−0.118462
$708$	28.3415	0.00150443
$709$	−6875.84	−0.364214	−0.182107	−	0.983279i	$-0.558292\pi$
$709$	−6875.84	−0.364214	−0.182107	+	0.983279i	$0.558292\pi$
$710$	0	0
$711$	20874.1	1.10104
$712$	12208.3	0.642593
$713$	2852.56	0.149830
$714$	37.1766	0.00194860
$715$	0	0
$716$	86.3228	0.00450564
$717$	−3489.93	−0.181777
$718$	−16787.3	−0.872557
$719$	35132.8	1.82230	0.911149	−	0.412078i	$-0.135197\pi$
$719$	35132.8	1.82230	0.911149	+	0.412078i	$0.135197\pi$
$720$	0	0
$721$	−4774.43	−0.246615
$722$	9208.98	0.474685
$723$	10856.8	0.558463
$724$	4529.53	0.232512
$725$	0	0
$726$	4162.55	0.212791
$727$	−21820.6	−1.11318	−0.556589	−	0.830788i	$-0.687890\pi$
$727$	−21820.6	−1.11318	−0.556589	+	0.830788i	$0.687890\pi$
$728$	−272.841	−0.0138904
$729$	−9074.07	−0.461011
$730$	0	0
$731$	329.939	0.0166939
$732$	−2470.02	−0.124719
$733$	39336.5	1.98216	0.991081	−	0.133258i	$-0.0425440\pi$
$733$	39336.5	1.98216	0.991081	+	0.133258i	$0.0425440\pi$
$734$	3389.86	0.170466
$735$	0	0
$736$	−2931.17	−0.146799
$737$	5021.89	0.250996
$738$	−376.013	−0.0187550
$739$	16015.2	0.797198	0.398599	−	0.917125i	$-0.369497\pi$
$739$	16015.2	0.797198	0.398599	+	0.917125i	$0.369497\pi$
$740$	0	0
$741$	−1002.11	−0.0496807
$742$	−1240.68	−0.0613838
$743$	−7885.18	−0.389340	−0.194670	−	0.980869i	$-0.562364\pi$
$743$	−7885.18	−0.389340	−0.194670	+	0.980869i	$0.562364\pi$
$744$	404.463	0.0199306
$745$	0	0
$746$	−9851.72	−0.483508
$747$	−22437.1	−1.09897
$748$	892.372	0.0436208
$749$	−432.893	−0.0211183
$750$	0	0
$751$	−22509.9	−1.09374	−0.546870	−	0.837218i	$-0.684181\pi$
$751$	−22509.9	−1.09374	−0.546870	+	0.837218i	$0.684181\pi$
$752$	−2988.56	−0.144922
$753$	−936.933	−0.0453436
$754$	−3639.68	−0.175795
$755$	0	0
$756$	−875.073	−0.0420980
$757$	11376.9	0.546234	0.273117	−	0.961981i	$-0.411945\pi$
$757$	11376.9	0.546234	0.273117	+	0.961981i	$0.411945\pi$
$758$	−9127.78	−0.437382
$759$	7601.61	0.363532
$760$	0	0
$761$	−1967.75	−0.0937329	−0.0468665	−	0.998901i	$-0.514924\pi$
$761$	−1967.75	−0.0937329	−0.0468665	+	0.998901i	$0.514924\pi$
$762$	3589.38	0.170642
$763$	−2415.48	−0.114609
$764$	−6190.88	−0.293165
$765$	0	0
$766$	−12771.5	−0.602418
$767$	56.7363	0.00267096
$768$	−415.610	−0.0195274
$769$	−29055.2	−1.36250	−0.681248	−	0.732053i	$-0.738562\pi$
$769$	−29055.2	−1.36250	−0.681248	+	0.732053i	$0.738562\pi$
$770$	0	0
$771$	6500.64	0.303651
$772$	6886.93	0.321070
$773$	−17339.6	−0.806809	−0.403404	−	0.915022i	$-0.632173\pi$
$773$	−17339.6	−0.806809	−0.403404	+	0.915022i	$0.632173\pi$
$774$	−3683.84	−0.171076
$775$	0	0
$776$	−7687.83	−0.355640
$777$	1607.75	0.0742313
$778$	24608.7	1.13402
$779$	366.391	0.0168515
$780$	0	0
$781$	−59728.6	−2.73657
$782$	799.537	0.0365619
$783$	−11673.4	−0.532789
$784$	−5377.88	−0.244984
$785$	0	0
$786$	2272.93	0.103146
$787$	20499.1	0.928482	0.464241	−	0.885709i	$-0.346327\pi$
$787$	20499.1	0.928482	0.464241	+	0.885709i	$0.346327\pi$
$788$	179.914	0.00813349
$789$	−471.446	−0.0212724
$790$	0	0
$791$	2534.24	0.113916
$792$	−9963.52	−0.447018
$793$	−4944.68	−0.221426
$794$	−7187.14	−0.321237
$795$	0	0
$796$	10311.8	0.459161
$797$	−20639.2	−0.917289	−0.458644	−	0.888620i	$-0.651665\pi$
$797$	−20639.2	−0.917289	−0.458644	+	0.888620i	$0.651665\pi$
$798$	404.463	0.0179422
$799$	815.191	0.0360944
$800$	0	0
$801$	37180.9	1.64010
$802$	−11886.3	−0.523341
$803$	−20669.7	−0.908364
$804$	637.976	0.0279847
$805$	0	0
$806$	809.686	0.0353846
$807$	−1630.31	−0.0711146
$808$	−6790.83	−0.295669
$809$	−24472.6	−1.06355	−0.531773	−	0.846887i	$-0.678474\pi$
$809$	−24472.6	−1.06355	−0.531773	+	0.846887i	$0.678474\pi$
$810$	0	0
$811$	−928.998	−0.0402238	−0.0201119	−	0.999798i	$-0.506402\pi$
$811$	−928.998	−0.0402238	−0.0201119	+	0.999798i	$0.506402\pi$
$812$	1469.02	0.0634882
$813$	−8939.75	−0.385647
$814$	38591.7	1.66172
$815$	0	0
$816$	113.366	0.00486349
$817$	3589.57	0.153713
$818$	7347.52	0.314059
$819$	−830.950	−0.0354527
$820$	0	0
$821$	3761.43	0.159896	0.0799481	−	0.996799i	$-0.474525\pi$
$821$	3761.43	0.159896	0.0799481	+	0.996799i	$0.474525\pi$
$822$	−3993.85	−0.169466
$823$	5718.32	0.242197	0.121098	−	0.992640i	$-0.461358\pi$
$823$	5718.32	0.242197	0.121098	+	0.992640i	$0.461358\pi$
$824$	−14559.1	−0.615523
$825$	0	0
$826$	−22.8994	−0.000964616 0
$827$	−30541.0	−1.28418	−0.642089	−	0.766630i	$-0.721932\pi$
$827$	−30541.0	−1.28418	−0.642089	+	0.766630i	$0.721932\pi$
$828$	−8927.00	−0.374680
$829$	−41273.1	−1.72916	−0.864581	−	0.502493i	$-0.832416\pi$
$829$	−41273.1	−1.72916	−0.864581	+	0.502493i	$0.832416\pi$
$830$	0	0
$831$	6833.02	0.285240
$832$	−832.000	−0.0346688
$833$	1466.93	0.0610156
$834$	−3260.10	−0.135357
$835$	0	0
$836$	9708.56	0.401648
$837$	2596.87	0.107241
$838$	12735.4	0.524986
$839$	38391.2	1.57975	0.789875	−	0.613268i	$-0.210145\pi$
$839$	38391.2	1.57975	0.789875	+	0.613268i	$0.210145\pi$
$840$	0	0
$841$	−4792.41	−0.196499
$842$	10369.0	0.424392
$843$	11047.0	0.451341
$844$	5739.72	0.234087
$845$	0	0
$846$	−9101.78	−0.369889
$847$	−3363.26	−0.136438
$848$	−3783.32	−0.153207
$849$	5441.22	0.219955
$850$	0	0
$851$	34577.0	1.39281
$852$	−7587.87	−0.305113
$853$	−10176.9	−0.408498	−0.204249	−	0.978919i	$-0.565475\pi$
$853$	−10176.9	−0.408498	−0.204249	+	0.978919i	$0.565475\pi$
$854$	1995.73	0.0799677
$855$	0	0
$856$	−1320.06	−0.0527088
$857$	−20605.8	−0.821331	−0.410666	−	0.911786i	$-0.634704\pi$
$857$	−20605.8	−0.821331	−0.410666	+	0.911786i	$0.634704\pi$
$858$	2157.68	0.0858532
$859$	23472.9	0.932345	0.466173	−	0.884694i	$-0.345633\pi$
$859$	23472.9	0.932345	0.466173	+	0.884694i	$0.345633\pi$
$860$	0	0
$861$	−32.8655	−0.00130088
$862$	−12504.1	−0.494074
$863$	−22922.3	−0.904153	−0.452076	−	0.891979i	$-0.649317\pi$
$863$	−22922.3	−0.904153	−0.452076	+	0.891979i	$0.649317\pi$
$864$	−2668.44	−0.105072
$865$	0	0
$866$	−27126.6	−1.06443
$867$	7945.21	0.311227
$868$	−326.799	−0.0127791
$869$	43794.7	1.70959
$870$	0	0
$871$	1277.15	0.0496838
$872$	−7365.76	−0.286050
$873$	−23413.6	−0.907709
$874$	8698.56	0.336651
$875$	0	0
$876$	−2625.85	−0.101278
$877$	−12732.4	−0.490243	−0.245122	−	0.969492i	$-0.578828\pi$
$877$	−12732.4	−0.490243	−0.245122	+	0.969492i	$0.578828\pi$
$878$	−9911.63	−0.380981
$879$	5660.85	0.217219
$880$	0	0
$881$	32910.4	1.25855	0.629273	−	0.777184i	$-0.283353\pi$
$881$	32910.4	1.25855	0.629273	+	0.777184i	$0.283353\pi$
$882$	−16378.5	−0.625277
$883$	−17053.9	−0.649952	−0.324976	−	0.945722i	$-0.605356\pi$
$883$	−17053.9	−0.649952	−0.324976	+	0.945722i	$0.605356\pi$
$884$	226.945	0.00863460
$885$	0	0
$886$	−21443.2	−0.813090
$887$	−7601.23	−0.287739	−0.143869	−	0.989597i	$-0.545954\pi$
$887$	−7601.23	−0.287739	−0.143869	+	0.989597i	$0.545954\pi$
$888$	4902.66	0.185273
$889$	−2900.15	−0.109413
$890$	0	0
$891$	−26706.6	−1.00416
$892$	−20890.5	−0.784155
$893$	8868.87	0.332347
$894$	7503.08	0.280694
$895$	0	0
$896$	335.805	0.0125206
$897$	1933.22	0.0719601
$898$	7408.63	0.275311
$899$	−4359.47	−0.161731
$900$	0	0
$901$	1031.98	0.0381578
$902$	−788.890	−0.0291210
$903$	−321.988	−0.0118661
$904$	7727.89	0.284321
$905$	0	0
$906$	7366.79	0.270138
$907$	49076.2	1.79664	0.898318	−	0.439347i	$-0.144790\pi$
$907$	49076.2	1.79664	0.898318	+	0.439347i	$0.144790\pi$
$908$	9398.59	0.343506
$909$	−20681.7	−0.754643
$910$	0	0
$911$	−22427.3	−0.815642	−0.407821	−	0.913062i	$-0.633711\pi$
$911$	−22427.3	−0.815642	−0.407821	+	0.913062i	$0.633711\pi$
$912$	1233.37	0.0447816
$913$	−47073.9	−1.70637
$914$	13421.5	0.485715
$915$	0	0
$916$	1271.44	0.0458619
$917$	−1836.48	−0.0661353
$918$	727.871	0.0261692
$919$	47062.8	1.68929	0.844645	−	0.535327i	$-0.179812\pi$
$919$	47062.8	1.68929	0.844645	+	0.535327i	$0.179812\pi$
$920$	0	0
$921$	−13223.6	−0.473109
$922$	1954.98	0.0698307
$923$	−15190.0	−0.541695
$924$	−870.866	−0.0310058
$925$	0	0
$926$	−28978.2	−1.02838
$927$	−44340.4	−1.57101
$928$	4479.61	0.158459
$929$	18074.8	0.638338	0.319169	−	0.947698i	$-0.396596\pi$
$929$	18074.8	0.638338	0.319169	+	0.947698i	$0.396596\pi$
$930$	0	0
$931$	15959.4	0.561814
$932$	−5182.70	−0.182151
$933$	829.658	0.0291123
$934$	29699.4	1.04047
$935$	0	0
$936$	−2533.89	−0.0884859
$937$	39939.1	1.39248	0.696241	−	0.717809i	$-0.254855\pi$
$937$	39939.1	1.39248	0.696241	+	0.717809i	$0.254855\pi$
$938$	−515.473	−0.0179433
$939$	5334.67	0.185400
$940$	0	0
$941$	−12975.7	−0.449518	−0.224759	−	0.974414i	$-0.572159\pi$
$941$	−12975.7	−0.449518	−0.224759	+	0.974414i	$0.572159\pi$
$942$	6310.84	0.218278
$943$	−706.821	−0.0244085
$944$	−69.8292	−0.00240757
$945$	0	0
$946$	−7728.85	−0.265631
$947$	−43713.9	−1.50001	−0.750005	−	0.661432i	$-0.769949\pi$
$947$	−43713.9	−1.50001	−0.750005	+	0.661432i	$0.769949\pi$
$948$	5563.64	0.190610
$949$	−5256.64	−0.179808
$950$	0	0
$951$	−13598.0	−0.463667
$952$	−91.5976	−0.00311838
$953$	−13116.8	−0.445850	−0.222925	−	0.974836i	$-0.571561\pi$
$953$	−13116.8	−0.445850	−0.222925	+	0.974836i	$0.571561\pi$
$954$	−11522.2	−0.391034
$955$	0	0
$956$	8598.66	0.290900
$957$	−11617.3	−0.392407
$958$	−16219.7	−0.547009
$959$	3226.95	0.108659
$960$	0	0
$961$	−28821.2	−0.967446
$962$	9814.52	0.328932
$963$	−4020.30	−0.134530
$964$	−26749.5	−0.893719
$965$	0	0
$966$	−780.268	−0.0259883
$967$	−20837.0	−0.692941	−0.346471	−	0.938061i	$-0.612620\pi$
$967$	−20837.0	−0.692941	−0.346471	+	0.938061i	$0.612620\pi$
$968$	−10255.9	−0.340534
$969$	−336.426	−0.0111533
$970$	0	0
$971$	36214.0	1.19687	0.598436	−	0.801170i	$-0.295789\pi$
$971$	36214.0	1.19687	0.598436	+	0.801170i	$0.295789\pi$
$972$	−12398.8	−0.409147
$973$	2634.10	0.0867887
$974$	2431.83	0.0800010
$975$	0	0
$976$	6085.76	0.199590
$977$	4171.16	0.136589	0.0682943	−	0.997665i	$-0.478244\pi$
$977$	4171.16	0.136589	0.0682943	+	0.997665i	$0.478244\pi$
$978$	−1012.44	−0.0331027
$979$	78007.1	2.54660
$980$	0	0
$981$	−22432.7	−0.730093
$982$	25278.9	0.821468
$983$	45275.4	1.46903	0.734517	−	0.678590i	$-0.237409\pi$
$983$	45275.4	1.46903	0.734517	+	0.678590i	$0.237409\pi$
$984$	−100.220	−0.00324684
$985$	0	0
$986$	−1221.91	−0.0394659
$987$	−795.545	−0.0256560
$988$	2469.05	0.0795050
$989$	−6924.81	−0.222645
$990$	0	0
$991$	−2463.81	−0.0789764	−0.0394882	−	0.999220i	$-0.512573\pi$
$991$	−2463.81	−0.0789764	−0.0394882	+	0.999220i	$0.512573\pi$
$992$	−996.537	−0.0318952
$993$	2395.93	0.0765686
$994$	6130.85	0.195633
$995$	0	0
$996$	−5980.22	−0.190252
$997$	29662.7	0.942253	0.471127	−	0.882066i	$-0.343848\pi$
$997$	29662.7	0.942253	0.471127	+	0.882066i	$0.343848\pi$
$998$	−5316.42	−0.168626
$999$	31477.7	0.996908

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.4.a.m.1.1	✓	2
5.2	odd	4		650.4.b.i.599.2		4
5.3	odd	4		650.4.b.i.599.3		4
5.4	even	2		650.4.a.p.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.4.a.m.1.1	✓	2	1.1	even	1	trivial
650.4.a.p.1.2	yes	2	5.4	even	2
650.4.b.i.599.2		4	5.2	odd	4
650.4.b.i.599.3		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	650.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -1.62348 q^{3} +4.00000 q^{4} +3.24695 q^{6} -2.62348 q^{7} -8.00000 q^{8} -24.3643 q^{9} -51.1174 q^{11} -6.49390 q^{12} -13.0000 q^{13} +5.24695 q^{14} +16.0000 q^{16} -4.36433 q^{17} +48.7287 q^{18} -47.4817 q^{19} +4.25915 q^{21} +102.235 q^{22} +91.5991 q^{23} +12.9878 q^{24} +26.0000 q^{26} +83.3887 q^{27} -10.4939 q^{28} -139.988 q^{29} +31.1418 q^{31} -32.0000 q^{32} +82.9878 q^{33} +8.72866 q^{34} -97.4573 q^{36} +377.482 q^{37} +94.9634 q^{38} +21.1052 q^{39} -7.71646 q^{41} -8.51830 q^{42} -75.5991 q^{43} -204.470 q^{44} -183.198 q^{46} -186.785 q^{47} -25.9756 q^{48} -336.117 q^{49} +7.08538 q^{51} -52.0000 q^{52} -236.457 q^{53} -166.777 q^{54} +20.9878 q^{56} +77.0854 q^{57} +279.976 q^{58} -4.36433 q^{59} +380.360 q^{61} -62.2835 q^{62} +63.9192 q^{63} +64.0000 q^{64} -165.976 q^{66} -98.2424 q^{67} -17.4573 q^{68} -148.709 q^{69} +1168.46 q^{71} +194.915 q^{72} +404.357 q^{73} -754.963 q^{74} -189.927 q^{76} +134.105 q^{77} -42.2104 q^{78} -856.748 q^{79} +522.457 q^{81} +15.4329 q^{82} +920.898 q^{83} +17.0366 q^{84} +151.198 q^{86} +227.267 q^{87} +408.939 q^{88} -1526.04 q^{89} +34.1052 q^{91} +366.396 q^{92} -50.5579 q^{93} +373.570 q^{94} +51.9512 q^{96} +960.979 q^{97} +672.235 q^{98} +1245.44 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 7 q^{3} + 8 q^{4} - 14 q^{6} + 5 q^{7} - 16 q^{8} + 23 q^{9} - 51 q^{11} + 28 q^{12} - 26 q^{13} - 10 q^{14} + 32 q^{16} + 63 q^{17} - 46 q^{18} + 28 q^{19} + 70 q^{21} + 102 q^{22} + 9 q^{23}+ \cdots + 1251 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 7 * q^3 + 8 * q^4 - 14 * q^6 + 5 * q^7 - 16 * q^8 + 23 * q^9 - 51 * q^11 + 28 * q^12 - 26 * q^13 - 10 * q^14 + 32 * q^16 + 63 * q^17 - 46 * q^18 + 28 * q^19 + 70 * q^21 + 102 * q^22 + 9 * q^23 - 56 * q^24 + 52 * q^26 + 259 * q^27 + 20 * q^28 - 198 * q^29 + 175 * q^31 - 64 * q^32 + 84 * q^33 - 126 * q^34 + 92 * q^36 + 632 * q^37 - 56 * q^38 - 91 * q^39 + 210 * q^41 - 140 * q^42 + 23 * q^43 - 204 * q^44 - 18 * q^46 + 231 * q^47 + 112 * q^48 - 621 * q^49 + 588 * q^51 - 104 * q^52 - 186 * q^53 - 518 * q^54 - 40 * q^56 + 728 * q^57 + 396 * q^58 + 63 * q^59 - 182 * q^61 - 350 * q^62 + 425 * q^63 + 128 * q^64 - 168 * q^66 + 695 * q^67 + 252 * q^68 - 861 * q^69 + 390 * q^71 - 184 * q^72 + 1526 * q^73 - 1264 * q^74 + 112 * q^76 + 135 * q^77 + 182 * q^78 - 863 * q^79 + 758 * q^81 - 420 * q^82 + 315 * q^83 + 280 * q^84 - 46 * q^86 - 273 * q^87 + 408 * q^88 - 1638 * q^89 - 65 * q^91 + 36 * q^92 + 1190 * q^93 - 462 * q^94 - 224 * q^96 + 98 * q^97 + 1242 * q^98 + 1251 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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