LMFDB - Embedded newform 2107.4.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2107,4,Mod(1,2107)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2107, 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("2107.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2107 = 7^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2107.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:	$124.317024382$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	2107.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.80243 q^{2} +6.75863 q^{3} +6.45848 q^{4} +12.6768 q^{5} -25.6992 q^{6} +5.86153 q^{8} +18.6791 q^{9} +O(q^{10})$	\(q-3.80243 q^{2} +6.75863 q^{3} +6.45848 q^{4} +12.6768 q^{5} -25.6992 q^{6} +5.86153 q^{8} +18.6791 q^{9} -48.2025 q^{10} +28.3699 q^{11} +43.6505 q^{12} -75.4061 q^{13} +85.6775 q^{15} -73.9559 q^{16} -119.399 q^{17} -71.0259 q^{18} +123.367 q^{19} +81.8726 q^{20} -107.874 q^{22} +26.7320 q^{23} +39.6159 q^{24} +35.7002 q^{25} +286.726 q^{26} -56.2380 q^{27} -261.930 q^{29} -325.783 q^{30} +123.907 q^{31} +234.320 q^{32} +191.741 q^{33} +454.006 q^{34} +120.638 q^{36} -29.8048 q^{37} -469.093 q^{38} -509.642 q^{39} +74.3053 q^{40} +343.428 q^{41} -43.0000 q^{43} +183.226 q^{44} +236.790 q^{45} -101.647 q^{46} -368.306 q^{47} -499.841 q^{48} -135.748 q^{50} -806.974 q^{51} -487.008 q^{52} +201.519 q^{53} +213.841 q^{54} +359.638 q^{55} +833.789 q^{57} +995.972 q^{58} -145.566 q^{59} +553.346 q^{60} -351.264 q^{61} -471.148 q^{62} -299.338 q^{64} -955.905 q^{65} -729.083 q^{66} -669.904 q^{67} -771.136 q^{68} +180.672 q^{69} +673.978 q^{71} +109.488 q^{72} -685.638 q^{73} +113.331 q^{74} +241.285 q^{75} +796.760 q^{76} +1937.88 q^{78} -1112.03 q^{79} -937.521 q^{80} -884.427 q^{81} -1305.86 q^{82} +938.935 q^{83} -1513.59 q^{85} +163.505 q^{86} -1770.29 q^{87} +166.291 q^{88} -75.0452 q^{89} -900.378 q^{90} +172.648 q^{92} +837.443 q^{93} +1400.46 q^{94} +1563.89 q^{95} +1583.68 q^{96} -127.941 q^{97} +529.923 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9}+O(q^{10}) $ 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9	$ 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9} - 60 q^{11} - 96 q^{15} + 372 q^{16} - 1008 q^{18} - 234 q^{22} - 214 q^{23} + 520 q^{25} - 870 q^{29} - 12 q^{30} - 1548 q^{32} + 1142 q^{36} - 1246 q^{37} - 416 q^{39} - 1290 q^{43} - 446 q^{44} - 660 q^{46} - 278 q^{50} - 3702 q^{51} - 2960 q^{53} + 620 q^{57} - 3634 q^{58} - 898 q^{60} + 2578 q^{64} - 4848 q^{65} + 928 q^{67} - 1708 q^{71} - 7900 q^{72} + 1714 q^{74} - 138 q^{78} - 3562 q^{79} + 2210 q^{81} - 948 q^{85} + 344 q^{86} + 2502 q^{88} - 3848 q^{92} - 11986 q^{93} - 2894 q^{95} - 2804 q^{99}+O(q^{100}) $ 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9 - 60 * q^11 - 96 * q^15 + 372 * q^16 - 1008 * q^18 - 234 * q^22 - 214 * q^23 + 520 * q^25 - 870 * q^29 - 12 * q^30 - 1548 * q^32 + 1142 * q^36 - 1246 * q^37 - 416 * q^39 - 1290 * q^43 - 446 * q^44 - 660 * q^46 - 278 * q^50 - 3702 * q^51 - 2960 * q^53 + 620 * q^57 - 3634 * q^58 - 898 * q^60 + 2578 * q^64 - 4848 * q^65 + 928 * q^67 - 1708 * q^71 - 7900 * q^72 + 1714 * q^74 - 138 * q^78 - 3562 * q^79 + 2210 * q^81 - 948 * q^85 + 344 * q^86 + 2502 * q^88 - 3848 * q^92 - 11986 * q^93 - 2894 * q^95 - 2804 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−3.80243	−1.34436	−0.672181	−	0.740387i	$-0.734642\pi$
$2$	−3.80243	−1.34436	−0.672181	+	0.740387i	$0.734642\pi$
$3$	6.75863	1.30070	0.650349	−	0.759635i	$-0.274622\pi$
$3$	6.75863	1.30070	0.650349	+	0.759635i	$0.274622\pi$
$4$	6.45848	0.807310
$5$	12.6768	1.13384	0.566922	−	0.823772i	$-0.308134\pi$
$5$	12.6768	1.13384	0.566922	+	0.823772i	$0.308134\pi$
$6$	−25.6992	−1.74861
$7$	0	0
$7$	0	0
$8$	5.86153	0.259046
$9$	18.6791	0.691818
$10$	−48.2025	−1.52430
$11$	28.3699	0.777621	0.388811	−	0.921318i	$-0.372886\pi$
$11$	28.3699	0.777621	0.388811	+	0.921318i	$0.372886\pi$
$12$	43.6505	1.05007
$13$	−75.4061	−1.60876	−0.804380	−	0.594115i	$-0.797502\pi$
$13$	−75.4061	−1.60876	−0.804380	+	0.594115i	$0.797502\pi$
$14$	0	0
$15$	85.6775	1.47479
$16$	−73.9559	−1.15556
$17$	−119.399	−1.70344	−0.851721	−	0.523995i	$-0.824441\pi$
$17$	−119.399	−1.70344	−0.851721	+	0.523995i	$0.824441\pi$
$18$	−71.0259	−0.930054
$19$	123.367	1.48959	0.744796	−	0.667292i	$-0.232547\pi$
$19$	123.367	1.48959	0.744796	+	0.667292i	$0.232547\pi$
$20$	81.8726	0.915363
$21$	0	0
$22$	−107.874	−1.04540
$23$	26.7320	0.242348	0.121174	−	0.992631i	$-0.461334\pi$
$23$	26.7320	0.242348	0.121174	+	0.992631i	$0.461334\pi$
$24$	39.6159	0.336940
$25$	35.7002	0.285602
$26$	286.726	2.16276
$27$	−56.2380	−0.400852
$28$	0	0
$29$	−261.930	−1.67722	−0.838608	−	0.544735i	$-0.816630\pi$
$29$	−261.930	−1.67722	−0.838608	+	0.544735i	$0.816630\pi$
$30$	−325.783	−1.98265
$31$	123.907	0.717884	0.358942	−	0.933360i	$-0.383138\pi$
$31$	123.907	0.717884	0.358942	+	0.933360i	$0.383138\pi$
$32$	234.320	1.29445
$33$	191.741	1.01145
$34$	454.006	2.29004
$35$	0	0
$36$	120.638	0.558511
$37$	−29.8048	−0.132429	−0.0662145	−	0.997805i	$-0.521092\pi$
$37$	−29.8048	−0.132429	−0.0662145	+	0.997805i	$0.521092\pi$
$38$	−469.093	−2.00255
$39$	−509.642	−2.09251
$40$	74.3053	0.293717
$41$	343.428	1.30816	0.654078	−	0.756427i	$-0.273057\pi$
$41$	343.428	1.30816	0.654078	+	0.756427i	$0.273057\pi$
$42$	0	0
$43$	−43.0000	−0.152499
$43$	−43.0000	−0.152499
$44$	183.226	0.627781
$45$	236.790	0.784413
$46$	−101.647	−0.325804
$47$	−368.306	−1.14304	−0.571520	−	0.820588i	$-0.693646\pi$
$47$	−368.306	−1.14304	−0.571520	+	0.820588i	$0.693646\pi$
$48$	−499.841	−1.50304
$49$	0	0
$50$	−135.748	−0.383952
$51$	−806.974	−2.21567
$52$	−487.008	−1.29877
$53$	201.519	0.522279	0.261140	−	0.965301i	$-0.415902\pi$
$53$	201.519	0.522279	0.261140	+	0.965301i	$0.415902\pi$
$54$	213.841	0.538891
$55$	359.638	0.881701
$56$	0	0
$57$	833.789	1.93751
$58$	995.972	2.25479
$59$	−145.566	−0.321206	−0.160603	−	0.987019i	$-0.551344\pi$
$59$	−145.566	−0.321206	−0.160603	+	0.987019i	$0.551344\pi$
$60$	553.346	1.19061
$61$	−351.264	−0.737290	−0.368645	−	0.929570i	$-0.620178\pi$
$61$	−351.264	−0.737290	−0.368645	+	0.929570i	$0.620178\pi$
$62$	−471.148	−0.965095
$63$	0	0
$64$	−299.338	−0.584644
$65$	−955.905	−1.82408
$66$	−729.083	−1.35976
$67$	−669.904	−1.22152	−0.610760	−	0.791816i	$-0.709136\pi$
$67$	−669.904	−1.22152	−0.610760	+	0.791816i	$0.709136\pi$
$68$	−771.136	−1.37521
$69$	180.672	0.315222
$70$	0	0
$71$	673.978	1.12657	0.563285	−	0.826262i	$-0.309537\pi$
$71$	673.978	1.12657	0.563285	+	0.826262i	$0.309537\pi$
$72$	109.488	0.179212
$73$	−685.638	−1.09929	−0.549643	−	0.835400i	$-0.685236\pi$
$73$	−685.638	−1.09929	−0.549643	+	0.835400i	$0.685236\pi$
$74$	113.331	0.178032
$75$	241.285	0.371482
$76$	796.760	1.20256
$77$	0	0
$78$	1937.88	2.81309
$79$	−1112.03	−1.58371	−0.791854	−	0.610710i	$-0.790884\pi$
$79$	−1112.03	−1.58371	−0.791854	+	0.610710i	$0.790884\pi$
$80$	−937.521	−1.31023
$81$	−884.427	−1.21321
$82$	−1305.86	−1.75863
$83$	938.935	1.24170	0.620852	−	0.783928i	$-0.286787\pi$
$83$	938.935	1.24170	0.620852	+	0.783928i	$0.286787\pi$
$84$	0	0
$85$	−1513.59	−1.93144
$86$	163.505	0.205013
$87$	−1770.29	−2.18155
$88$	166.291	0.201439
$89$	−75.0452	−0.0893794	−0.0446897	−	0.999001i	$-0.514230\pi$
$89$	−75.0452	−0.0893794	−0.0446897	+	0.999001i	$0.514230\pi$
$90$	−900.378	−1.05454
$91$	0	0
$92$	172.648	0.195650
$93$	837.443	0.933750
$94$	1400.46	1.53666
$95$	1563.89	1.68896
$96$	1583.68	1.68369
$97$	−127.941	−0.133922	−0.0669611	−	0.997756i	$-0.521330\pi$
$97$	−127.941	−0.133922	−0.0669611	+	0.997756i	$0.521330\pi$
$98$	0	0
$99$	529.923	0.537972
$100$	230.569	0.230569
$101$	501.729	0.494296	0.247148	−	0.968978i	$-0.420507\pi$
$101$	501.729	0.494296	0.247148	+	0.968978i	$0.420507\pi$
$102$	3068.46	2.97866
$103$	−2046.08	−1.95734	−0.978669	−	0.205442i	$-0.934137\pi$
$103$	−2046.08	−1.95734	−0.978669	+	0.205442i	$0.934137\pi$
$104$	−441.995	−0.416742
$105$	0	0
$106$	−766.263	−0.702132
$107$	287.098	0.259391	0.129696	−	0.991554i	$-0.458600\pi$
$107$	287.098	0.259391	0.129696	+	0.991554i	$0.458600\pi$
$108$	−363.212	−0.323612
$109$	1744.06	1.53258	0.766288	−	0.642498i	$-0.222102\pi$
$109$	1744.06	1.53258	0.766288	+	0.642498i	$0.222102\pi$
$110$	−1367.50	−1.18533
$111$	−201.439	−0.172250
$112$	0	0
$113$	−1606.01	−1.33700	−0.668501	−	0.743712i	$-0.733064\pi$
$113$	−1606.01	−1.33700	−0.668501	+	0.743712i	$0.733064\pi$
$114$	−3170.43	−2.60472
$115$	338.876	0.274785
$116$	−1691.67	−1.35403
$117$	−1408.52	−1.11297
$118$	553.506	0.431817
$119$	0	0
$120$	502.202	0.382038
$121$	−526.151	−0.395305
$122$	1335.66	0.991185
$123$	2321.10	1.70152
$124$	800.252	0.579554
$125$	−1132.03	−0.810016
$126$	0	0
$127$	994.388	0.694784	0.347392	−	0.937720i	$-0.387067\pi$
$127$	994.388	0.694784	0.347392	+	0.937720i	$0.387067\pi$
$128$	−736.348	−0.508473
$129$	−290.621	−0.198355
$130$	3634.76	2.45223
$131$	1644.33	1.09668	0.548341	−	0.836255i	$-0.315260\pi$
$131$	1644.33	1.09668	0.548341	+	0.836255i	$0.315260\pi$
$132$	1238.36	0.816554
$133$	0	0
$134$	2547.26	1.64217
$135$	−712.916	−0.454504
$136$	−699.862	−0.441269
$137$	−3132.58	−1.95354	−0.976768	−	0.214299i	$-0.931253\pi$
$137$	−3132.58	−1.95354	−0.976768	+	0.214299i	$0.931253\pi$
$138$	−686.992	−0.423773
$139$	−2002.48	−1.22193	−0.610964	−	0.791658i	$-0.709218\pi$
$139$	−2002.48	−1.22193	−0.610964	+	0.791658i	$0.709218\pi$
$140$	0	0
$141$	−2489.24	−1.48675
$142$	−2562.76	−1.51452
$143$	−2139.26	−1.25101
$144$	−1381.43	−0.799438
$145$	−3320.43	−1.90170
$146$	2607.09	1.47784
$147$	0	0
$148$	−192.493	−0.106911
$149$	−2112.39	−1.16144	−0.580718	−	0.814105i	$-0.697228\pi$
$149$	−2112.39	−1.16144	−0.580718	+	0.814105i	$0.697228\pi$
$150$	−917.468	−0.499407
$151$	176.007	0.0948557	0.0474278	−	0.998875i	$-0.484898\pi$
$151$	176.007	0.0948557	0.0474278	+	0.998875i	$0.484898\pi$
$152$	723.118	0.385872
$153$	−2230.26	−1.17847
$154$	0	0
$155$	1570.74	0.813968
$156$	−3291.51	−1.68931
$157$	−654.484	−0.332698	−0.166349	−	0.986067i	$-0.553198\pi$
$157$	−654.484	−0.332698	−0.166349	+	0.986067i	$0.553198\pi$
$158$	4228.41	2.12908
$159$	1361.99	0.679328
$160$	2970.42	1.46770
$161$	0	0
$162$	3362.97	1.63099
$163$	−1795.08	−0.862586	−0.431293	−	0.902212i	$-0.641943\pi$
$163$	−1795.08	−0.862586	−0.431293	+	0.902212i	$0.641943\pi$
$164$	2218.02	1.05609
$165$	2430.66	1.14683
$166$	−3570.23	−1.66930
$167$	−3419.28	−1.58438	−0.792191	−	0.610273i	$-0.791060\pi$
$167$	−3419.28	−1.58438	−0.792191	+	0.610273i	$0.791060\pi$
$168$	0	0
$169$	3489.07	1.58811
$170$	5755.33	2.59655
$171$	2304.37	1.03053
$172$	−277.715	−0.123114
$173$	1274.86	0.560265	0.280133	−	0.959961i	$-0.409622\pi$
$173$	1274.86	0.560265	0.280133	+	0.959961i	$0.409622\pi$
$174$	6731.41	2.93280
$175$	0	0
$176$	−2098.12	−0.898588
$177$	−983.830	−0.417792
$178$	285.354	0.120158
$179$	475.492	0.198547	0.0992735	−	0.995060i	$-0.468348\pi$
$179$	475.492	0.198547	0.0992735	+	0.995060i	$0.468348\pi$
$180$	1529.30	0.633264
$181$	2436.29	1.00048	0.500242	−	0.865886i	$-0.333244\pi$
$181$	2436.29	1.00048	0.500242	+	0.865886i	$0.333244\pi$
$182$	0	0
$183$	−2374.06	−0.958993
$184$	156.691	0.0627793
$185$	−377.828	−0.150154
$186$	−3184.32	−1.25530
$187$	−3387.33	−1.32463
$188$	−2378.69	−0.922788
$189$	0	0
$190$	−5946.58	−2.27058
$191$	1237.34	0.468749	0.234375	−	0.972146i	$-0.424696\pi$
$191$	1237.34	0.468749	0.234375	+	0.972146i	$0.424696\pi$
$192$	−2023.11	−0.760446
$193$	−2319.44	−0.865060	−0.432530	−	0.901619i	$-0.642379\pi$
$193$	−2319.44	−0.865060	−0.432530	+	0.901619i	$0.642379\pi$
$194$	486.487	0.180040
$195$	−6460.61	−2.37258
$196$	0	0
$197$	4448.32	1.60878	0.804390	−	0.594102i	$-0.202492\pi$
$197$	4448.32	1.60878	0.804390	+	0.594102i	$0.202492\pi$
$198$	−2014.99	−0.723229
$199$	−2780.33	−0.990413	−0.495206	−	0.868775i	$-0.664908\pi$
$199$	−2780.33	−0.990413	−0.495206	+	0.868775i	$0.664908\pi$
$200$	209.258	0.0739840
$201$	−4527.64	−1.58883
$202$	−1907.79	−0.664513
$203$	0	0
$204$	−5211.82	−1.78873
$205$	4353.55	1.48324
$206$	7780.06	2.63137
$207$	499.330	0.167661
$208$	5576.72	1.85902
$209$	3499.89	1.15834
$210$	0	0
$211$	913.963	0.298198	0.149099	−	0.988822i	$-0.452363\pi$
$211$	913.963	0.298198	0.149099	+	0.988822i	$0.452363\pi$
$212$	1301.51	0.421641
$213$	4555.17	1.46533
$214$	−1091.67	−0.348716
$215$	−545.101	−0.172910
$216$	−329.641	−0.103839
$217$	0	0
$218$	−6631.67	−2.06034
$219$	−4633.98	−1.42984
$220$	2322.71	0.711806
$221$	9003.41	2.74043
$222$	765.959	0.231567
$223$	−5971.30	−1.79313	−0.896564	−	0.442914i	$-0.853945\pi$
$223$	−5971.30	−1.79313	−0.896564	+	0.442914i	$0.853945\pi$
$224$	0	0
$225$	666.848	0.197585
$226$	6106.76	1.79741
$227$	3183.90	0.930939	0.465469	−	0.885064i	$-0.345886\pi$
$227$	3183.90	0.930939	0.465469	+	0.885064i	$0.345886\pi$
$228$	5385.01	1.56417
$229$	−552.331	−0.159385	−0.0796923	−	0.996820i	$-0.525394\pi$
$229$	−552.331	−0.159385	−0.0796923	+	0.996820i	$0.525394\pi$
$230$	−1288.55	−0.369411
$231$	0	0
$232$	−1535.31	−0.434476
$233$	4021.03	1.13058	0.565292	−	0.824891i	$-0.308763\pi$
$233$	4021.03	1.13058	0.565292	+	0.824891i	$0.308763\pi$
$234$	5355.78	1.49623
$235$	−4668.92	−1.29603
$236$	−940.137	−0.259312
$237$	−7515.79	−2.05993
$238$	0	0
$239$	6928.33	1.87513	0.937565	−	0.347810i	$-0.113074\pi$
$239$	6928.33	1.87513	0.937565	+	0.347810i	$0.113074\pi$
$240$	−6336.36	−1.70421
$241$	7317.59	1.95588	0.977941	−	0.208883i	$-0.0669827\pi$
$241$	7317.59	1.95588	0.977941	+	0.208883i	$0.0669827\pi$
$242$	2000.65	0.531434
$243$	−4459.09	−1.17716
$244$	−2268.63	−0.595221
$245$	0	0
$246$	−8825.82	−2.28745
$247$	−9302.59	−2.39640
$248$	726.286	0.185965
$249$	6345.91	1.61508
$250$	4304.47	1.08895
$251$	6303.78	1.58522	0.792611	−	0.609727i	$-0.208721\pi$
$251$	6303.78	1.58522	0.792611	+	0.609727i	$0.208721\pi$
$252$	0	0
$253$	758.384	0.188455
$254$	−3781.09	−0.934042
$255$	−10229.8	−2.51222
$256$	5194.61	1.26822
$257$	2437.82	0.591700	0.295850	−	0.955234i	$-0.404397\pi$
$257$	2437.82	0.591700	0.295850	+	0.955234i	$0.404397\pi$
$258$	1105.07	0.266661
$259$	0	0
$260$	−6173.69	−1.47260
$261$	−4892.62	−1.16033
$262$	−6252.43	−1.47434
$263$	−3002.93	−0.704062	−0.352031	−	0.935988i	$-0.614509\pi$
$263$	−3002.93	−0.704062	−0.352031	+	0.935988i	$0.614509\pi$
$264$	1123.90	0.262012
$265$	2554.61	0.592183
$266$	0	0
$267$	−507.202	−0.116256
$268$	−4326.56	−0.986145
$269$	−1210.71	−0.274417	−0.137209	−	0.990542i	$-0.543813\pi$
$269$	−1210.71	−0.274417	−0.137209	+	0.990542i	$0.543813\pi$
$270$	2710.81	0.611018
$271$	−2177.27	−0.488042	−0.244021	−	0.969770i	$-0.578467\pi$
$271$	−2177.27	−0.488042	−0.244021	+	0.969770i	$0.578467\pi$
$272$	8830.26	1.96843
$273$	0	0
$274$	11911.4	2.62626
$275$	1012.81	0.222090
$276$	1166.87	0.254482
$277$	7878.01	1.70882	0.854411	−	0.519597i	$-0.173918\pi$
$277$	7878.01	1.70882	0.854411	+	0.519597i	$0.173918\pi$
$278$	7614.29	1.64271
$279$	2314.47	0.496645
$280$	0	0
$281$	−2212.78	−0.469762	−0.234881	−	0.972024i	$-0.575470\pi$
$281$	−2212.78	−0.469762	−0.234881	+	0.972024i	$0.575470\pi$
$282$	9465.17	1.99873
$283$	−2748.27	−0.577270	−0.288635	−	0.957439i	$-0.593201\pi$
$283$	−2748.27	−0.577270	−0.288635	+	0.957439i	$0.593201\pi$
$284$	4352.87	0.909491
$285$	10569.7	2.19683
$286$	8134.38	1.68180
$287$	0	0
$288$	4376.88	0.895521
$289$	9343.13	1.90172
$290$	12625.7	2.55658
$291$	−864.706	−0.174192
$292$	−4428.18	−0.887464
$293$	−2266.43	−0.451898	−0.225949	−	0.974139i	$-0.572548\pi$
$293$	−2266.43	−0.451898	−0.225949	+	0.974139i	$0.572548\pi$
$294$	0	0
$295$	−1845.31	−0.364197
$296$	−174.702	−0.0343052
$297$	−1595.46	−0.311711
$298$	8032.23	1.56139
$299$	−2015.76	−0.389880
$300$	1558.33	0.299901
$301$	0	0
$302$	−669.253	−0.127520
$303$	3391.00	0.642931
$304$	−9123.69	−1.72131
$305$	−4452.88	−0.835972
$306$	8480.42	1.58429
$307$	4421.11	0.821910	0.410955	−	0.911656i	$-0.365195\pi$
$307$	4421.11	0.821910	0.410955	+	0.911656i	$0.365195\pi$
$308$	0	0
$309$	−13828.7	−2.54591
$310$	−5972.64	−1.09427
$311$	482.682	0.0880077	0.0440038	−	0.999031i	$-0.485989\pi$
$311$	482.682	0.0880077	0.0440038	+	0.999031i	$0.485989\pi$
$312$	−2987.28	−0.542056
$313$	494.067	0.0892215	0.0446108	−	0.999004i	$-0.485795\pi$
$313$	494.067	0.0892215	0.0446108	+	0.999004i	$0.485795\pi$
$314$	2488.63	0.447266
$315$	0	0
$316$	−7182.01	−1.27854
$317$	−9639.89	−1.70798	−0.853991	−	0.520288i	$-0.825824\pi$
$317$	−9639.89	−1.70798	−0.853991	+	0.520288i	$0.825824\pi$
$318$	−5178.89	−0.913263
$319$	−7430.93	−1.30424
$320$	−3794.63	−0.662895
$321$	1940.39	0.337390
$322$	0	0
$323$	−14729.9	−2.53743
$324$	−5712.05	−0.979433
$325$	−2692.01	−0.459465
$326$	6825.67	1.15963
$327$	11787.5	1.99342
$328$	2013.01	0.338872
$329$	0	0
$330$	−9242.41	−1.54175
$331$	−3810.11	−0.632697	−0.316348	−	0.948643i	$-0.602457\pi$
$331$	−3810.11	−0.632697	−0.316348	+	0.948643i	$0.602457\pi$
$332$	6064.09	1.00244
$333$	−556.725	−0.0916167
$334$	13001.6	2.12998
$335$	−8492.22	−1.38501
$336$	0	0
$337$	3164.18	0.511466	0.255733	−	0.966747i	$-0.417683\pi$
$337$	3164.18	0.511466	0.255733	+	0.966747i	$0.417683\pi$
$338$	−13267.0	−2.13499
$339$	−10854.5	−1.73904
$340$	−9775.50	−1.55927
$341$	3515.23	0.558241
$342$	−8762.22	−1.38540
$343$	0	0
$344$	−252.046	−0.0395041
$345$	2290.33	0.357413
$346$	−4847.57	−0.753199
$347$	−4567.09	−0.706553	−0.353277	−	0.935519i	$-0.614933\pi$
$347$	−4567.09	−0.706553	−0.353277	+	0.935519i	$0.614933\pi$
$348$	−11433.4	−1.76119
$349$	−4521.66	−0.693521	−0.346760	−	0.937954i	$-0.612718\pi$
$349$	−4521.66	−0.693521	−0.346760	+	0.937954i	$0.612718\pi$
$350$	0	0
$351$	4240.69	0.644875
$352$	6647.62	1.00659
$353$	−9420.17	−1.42035	−0.710177	−	0.704023i	$-0.751385\pi$
$353$	−9420.17	−1.42035	−0.710177	+	0.704023i	$0.751385\pi$
$354$	3740.94	0.561664
$355$	8543.86	1.27736
$356$	−484.677	−0.0721569
$357$	0	0
$358$	−1808.02	−0.266919
$359$	−2424.45	−0.356427	−0.178214	−	0.983992i	$-0.557032\pi$
$359$	−2424.45	−0.356427	−0.178214	+	0.983992i	$0.557032\pi$
$360$	1387.95	0.203199
$361$	8360.32	1.21888
$362$	−9263.81	−1.34501
$363$	−3556.06	−0.514173
$364$	0	0
$365$	−8691.67	−1.24642
$366$	9027.20	1.28923
$367$	−2289.04	−0.325577	−0.162788	−	0.986661i	$-0.552049\pi$
$367$	−2289.04	−0.325577	−0.162788	+	0.986661i	$0.552049\pi$
$368$	−1976.99	−0.280048
$369$	6414.91	0.905005
$370$	1436.66	0.201861
$371$	0	0
$372$	5408.61	0.753826
$373$	80.9887	0.0112425	0.00562123	−	0.999984i	$-0.498211\pi$
$373$	80.9887	0.0112425	0.00562123	+	0.999984i	$0.498211\pi$
$374$	12880.1	1.78079
$375$	−7650.98	−1.05359
$376$	−2158.84	−0.296100
$377$	19751.1	2.69824
$378$	0	0
$379$	1761.38	0.238722	0.119361	−	0.992851i	$-0.461915\pi$
$379$	1761.38	0.238722	0.119361	+	0.992851i	$0.461915\pi$
$380$	10100.3	1.36352
$381$	6720.70	0.903705
$382$	−4704.91	−0.630168
$383$	−1026.76	−0.136984	−0.0684918	−	0.997652i	$-0.521819\pi$
$383$	−1026.76	−0.136984	−0.0684918	+	0.997652i	$0.521819\pi$
$384$	−4976.70	−0.661370
$385$	0	0
$386$	8819.49	1.16295
$387$	−803.200	−0.105501
$388$	−826.304	−0.108117
$389$	−6777.68	−0.883398	−0.441699	−	0.897163i	$-0.645624\pi$
$389$	−6777.68	−0.883398	−0.441699	+	0.897163i	$0.645624\pi$
$390$	24566.0	3.18961
$391$	−3191.78	−0.412826
$392$	0	0
$393$	11113.4	1.42645
$394$	−16914.4	−2.16278
$395$	−14096.9	−1.79568
$396$	3422.49	0.434310
$397$	−3759.18	−0.475234	−0.237617	−	0.971359i	$-0.576366\pi$
$397$	−3759.18	−0.475234	−0.237617	+	0.971359i	$0.576366\pi$
$398$	10572.0	1.33147
$399$	0	0
$400$	−2640.24	−0.330030
$401$	−7114.34	−0.885967	−0.442984	−	0.896530i	$-0.646080\pi$
$401$	−7114.34	−0.885967	−0.442984	+	0.896530i	$0.646080\pi$
$402$	17216.0	2.13596
$403$	−9343.35	−1.15490
$404$	3240.41	0.399050
$405$	−11211.7	−1.37559
$406$	0	0
$407$	−845.557	−0.102980
$408$	−4730.11	−0.573959
$409$	4164.74	0.503504	0.251752	−	0.967792i	$-0.418993\pi$
$409$	4164.74	0.503504	0.251752	+	0.967792i	$0.418993\pi$
$410$	−16554.1	−1.99402
$411$	−21172.0	−2.54096
$412$	−13214.5	−1.58018
$413$	0	0
$414$	−1898.67	−0.225397
$415$	11902.6	1.40790
$416$	−17669.1	−2.08245
$417$	−13534.0	−1.58936
$418$	−13308.1	−1.55723
$419$	−1927.13	−0.224693	−0.112347	−	0.993669i	$-0.535837\pi$
$419$	−1927.13	−0.224693	−0.112347	+	0.993669i	$0.535837\pi$
$420$	0	0
$421$	5822.46	0.674036	0.337018	−	0.941498i	$-0.390582\pi$
$421$	5822.46	0.674036	0.337018	+	0.941498i	$0.390582\pi$
$422$	−3475.28	−0.400886
$423$	−6879.61	−0.790776
$424$	1181.21	0.135294
$425$	−4262.57	−0.486506
$426$	−17320.7	−1.96993
$427$	0	0
$428$	1854.22	0.209409
$429$	−14458.5	−1.62718
$430$	2072.71	0.232453
$431$	−10157.8	−1.13523	−0.567614	−	0.823295i	$-0.692133\pi$
$431$	−10157.8	−1.13523	−0.567614	+	0.823295i	$0.692133\pi$
$432$	4159.13	0.463209
$433$	3189.92	0.354037	0.177018	−	0.984208i	$-0.443355\pi$
$433$	3189.92	0.354037	0.177018	+	0.984208i	$0.443355\pi$
$434$	0	0
$435$	−22441.6	−2.47354
$436$	11264.0	1.23726
$437$	3297.84	0.361000
$438$	17620.4	1.92222
$439$	−8713.92	−0.947364	−0.473682	−	0.880696i	$-0.657075\pi$
$439$	−8713.92	−0.947364	−0.473682	+	0.880696i	$0.657075\pi$
$440$	2108.03	0.228401
$441$	0	0
$442$	−34234.8	−3.68413
$443$	59.2445	0.00635393	0.00317697	−	0.999995i	$-0.498989\pi$
$443$	59.2445	0.00635393	0.00317697	+	0.999995i	$0.498989\pi$
$444$	−1300.99	−0.139059
$445$	−951.329	−0.101342
$446$	22705.4	2.41061
$447$	−14276.9	−1.51068
$448$	0	0
$449$	−7370.83	−0.774724	−0.387362	−	0.921928i	$-0.626614\pi$
$449$	−7370.83	−0.774724	−0.387362	+	0.921928i	$0.626614\pi$
$450$	−2535.64	−0.265625
$451$	9742.99	1.01725
$452$	−10372.4	−1.07937
$453$	1189.56	0.123379
$454$	−12106.6	−1.25152
$455$	0	0
$456$	4887.28	0.501904
$457$	9912.81	1.01466	0.507332	−	0.861751i	$-0.330632\pi$
$457$	9912.81	1.01466	0.507332	+	0.861751i	$0.330632\pi$
$458$	2100.20	0.214271
$459$	6714.76	0.682829
$460$	2188.62	0.221837
$461$	1260.97	0.127395	0.0636976	−	0.997969i	$-0.479711\pi$
$461$	1260.97	0.127395	0.0636976	+	0.997969i	$0.479711\pi$
$462$	0	0
$463$	−3831.19	−0.384558	−0.192279	−	0.981340i	$-0.561588\pi$
$463$	−3831.19	−0.384558	−0.192279	+	0.981340i	$0.561588\pi$
$464$	19371.3	1.93813
$465$	10616.1	1.05873
$466$	−15289.7	−1.51991
$467$	−11916.4	−1.18078	−0.590391	−	0.807117i	$-0.701027\pi$
$467$	−11916.4	−1.18078	−0.590391	+	0.807117i	$0.701027\pi$
$468$	−9096.87	−0.898510
$469$	0	0
$470$	17753.3	1.74233
$471$	−4423.42	−0.432739
$472$	−853.243	−0.0832069
$473$	−1219.90	−0.118586
$474$	28578.3	2.76929
$475$	4404.22	0.425430
$476$	0	0
$477$	3764.19	0.361322
$478$	−26344.5	−2.52085
$479$	−8818.14	−0.841150	−0.420575	−	0.907258i	$-0.638172\pi$
$479$	−8818.14	−0.841150	−0.420575	+	0.907258i	$0.638172\pi$
$480$	20075.9	1.90904
$481$	2247.46	0.213046
$482$	−27824.6	−2.62941
$483$	0	0
$484$	−3398.14	−0.319134
$485$	−1621.88	−0.151847
$486$	16955.4	1.58253
$487$	−12066.9	−1.12280	−0.561399	−	0.827545i	$-0.689737\pi$
$487$	−12066.9	−1.12280	−0.561399	+	0.827545i	$0.689737\pi$
$488$	−2058.94	−0.190992
$489$	−12132.3	−1.12197
$490$	0	0
$491$	−2060.88	−0.189422	−0.0947109	−	0.995505i	$-0.530193\pi$
$491$	−2060.88	−0.189422	−0.0947109	+	0.995505i	$0.530193\pi$
$492$	14990.8	1.37365
$493$	31274.2	2.85704
$494$	35372.4	3.22162
$495$	6717.70	0.609976
$496$	−9163.67	−0.829558
$497$	0	0
$498$	−24129.9	−2.17126
$499$	11618.0	1.04227	0.521136	−	0.853473i	$-0.325508\pi$
$499$	11618.0	1.04227	0.521136	+	0.853473i	$0.325508\pi$
$500$	−7311.20	−0.653934
$501$	−23109.7	−2.06081
$502$	−23969.7	−2.13111
$503$	17340.4	1.53711	0.768557	−	0.639782i	$-0.220975\pi$
$503$	17340.4	1.53711	0.768557	+	0.639782i	$0.220975\pi$
$504$	0	0
$505$	6360.30	0.560455
$506$	−2883.70	−0.253352
$507$	23581.4	2.06565
$508$	6422.23	0.560906
$509$	−527.062	−0.0458971	−0.0229485	−	0.999737i	$-0.507305\pi$
$509$	−527.062	−0.0458971	−0.0229485	+	0.999737i	$0.507305\pi$
$510$	38898.2	3.37733
$511$	0	0
$512$	−13861.4	−1.19647
$513$	−6937.89	−0.597106
$514$	−9269.63	−0.795459
$515$	−25937.6	−2.21932
$516$	−1876.97	−0.160134
$517$	−10448.8	−0.888853
$518$	0	0
$519$	8616.31	0.728736
$520$	−5603.07	−0.472521
$521$	9343.00	0.785651	0.392826	−	0.919613i	$-0.371498\pi$
$521$	9343.00	0.785651	0.392826	+	0.919613i	$0.371498\pi$
$522$	18603.9	1.55990
$523$	−1789.26	−0.149596	−0.0747981	−	0.997199i	$-0.523831\pi$
$523$	−1789.26	−0.149596	−0.0747981	+	0.997199i	$0.523831\pi$
$524$	10619.8	0.885362
$525$	0	0
$526$	11418.4	0.946514
$527$	−14794.4	−1.22287
$528$	−14180.4	−1.16879
$529$	−11452.4	−0.941267
$530$	−9713.73	−0.796108
$531$	−2719.05	−0.222216
$532$	0	0
$533$	−25896.5	−2.10451
$534$	1928.60	0.156290
$535$	3639.48	0.294109
$536$	−3926.67	−0.316429
$537$	3213.67	0.258250
$538$	4603.64	0.368916
$539$	0	0
$540$	−4604.35	−0.366925
$541$	3114.78	0.247532	0.123766	−	0.992311i	$-0.460503\pi$
$541$	3114.78	0.247532	0.123766	+	0.992311i	$0.460503\pi$
$542$	8278.90	0.656105
$543$	16466.0	1.30133
$544$	−27977.6	−2.20501
$545$	22109.0	1.73770
$546$	0	0
$547$	5471.36	0.427675	0.213838	−	0.976869i	$-0.431404\pi$
$547$	5471.36	0.427675	0.213838	+	0.976869i	$0.431404\pi$
$548$	−20231.7	−1.57711
$549$	−6561.28	−0.510070
$550$	−3851.14	−0.298570
$551$	−32313.5	−2.49837
$552$	1059.01	0.0816570
$553$	0	0
$554$	−29955.6	−2.29728
$555$	−2553.60	−0.195305
$556$	−12933.0	−0.986475
$557$	−3674.54	−0.279524	−0.139762	−	0.990185i	$-0.544634\pi$
$557$	−3674.54	−0.279524	−0.139762	+	0.990185i	$0.544634\pi$
$558$	−8800.62	−0.667670
$559$	3242.46	0.245334
$560$	0	0
$561$	−22893.7	−1.72295
$562$	8413.93	0.631530
$563$	−16269.5	−1.21790	−0.608952	−	0.793207i	$-0.708410\pi$
$563$	−16269.5	−1.21790	−0.608952	+	0.793207i	$0.708410\pi$
$564$	−16076.7	−1.20027
$565$	−20359.1	−1.51595
$566$	10450.1	0.776060
$567$	0	0
$568$	3950.55	0.291833
$569$	−11561.7	−0.851830	−0.425915	−	0.904763i	$-0.640048\pi$
$569$	−11561.7	−0.851830	−0.425915	+	0.904763i	$0.640048\pi$
$570$	−40190.7	−2.95334
$571$	−13255.7	−0.971512	−0.485756	−	0.874095i	$-0.661456\pi$
$571$	−13255.7	−0.971512	−0.485756	+	0.874095i	$0.661456\pi$
$572$	−13816.4	−1.00995
$573$	8362.75	0.609701
$574$	0	0
$575$	954.340	0.0692152
$576$	−5591.35	−0.404467
$577$	7370.94	0.531813	0.265907	−	0.963999i	$-0.414329\pi$
$577$	7370.94	0.531813	0.265907	+	0.963999i	$0.414329\pi$
$578$	−35526.6	−2.55659
$579$	−15676.2	−1.12518
$580$	−21444.9	−1.53526
$581$	0	0
$582$	3287.99	0.234178
$583$	5717.07	0.406135
$584$	−4018.89	−0.284765
$585$	−17855.4	−1.26193
$586$	8617.94	0.607515
$587$	8585.10	0.603654	0.301827	−	0.953363i	$-0.402404\pi$
$587$	8585.10	0.603654	0.301827	+	0.953363i	$0.402404\pi$
$588$	0	0
$589$	15286.0	1.06935
$590$	7016.66	0.489613
$591$	30064.5	2.09254
$592$	2204.24	0.153030
$593$	−20328.8	−1.40776	−0.703881	−	0.710318i	$-0.748551\pi$
$593$	−20328.8	−1.40776	−0.703881	+	0.710318i	$0.748551\pi$
$594$	6066.64	0.419053
$595$	0	0
$596$	−13642.8	−0.937638
$597$	−18791.2	−1.28823
$598$	7664.78	0.524140
$599$	−18556.0	−1.26574	−0.632870	−	0.774258i	$-0.718123\pi$
$599$	−18556.0	−1.26574	−0.632870	+	0.774258i	$0.718123\pi$
$600$	1414.30	0.0962309
$601$	−4313.52	−0.292766	−0.146383	−	0.989228i	$-0.546763\pi$
$601$	−4313.52	−0.292766	−0.146383	+	0.989228i	$0.546763\pi$
$602$	0	0
$603$	−12513.2	−0.845069
$604$	1136.73	0.0765779
$605$	−6669.90	−0.448215
$606$	−12894.1	−0.864332
$607$	−2860.25	−0.191258	−0.0956292	−	0.995417i	$-0.530486\pi$
$607$	−2860.25	−0.191258	−0.0956292	+	0.995417i	$0.530486\pi$
$608$	28907.2	1.92820
$609$	0	0
$610$	16931.8	1.12385
$611$	27772.5	1.83888
$612$	−14404.1	−0.951391
$613$	22155.6	1.45980	0.729898	−	0.683556i	$-0.239568\pi$
$613$	22155.6	1.45980	0.729898	+	0.683556i	$0.239568\pi$
$614$	−16811.0	−1.10494
$615$	29424.0	1.92925
$616$	0	0
$617$	−408.357	−0.0266448	−0.0133224	−	0.999911i	$-0.504241\pi$
$617$	−408.357	−0.0266448	−0.0133224	+	0.999911i	$0.504241\pi$
$618$	52582.6	3.42262
$619$	6553.03	0.425507	0.212753	−	0.977106i	$-0.431757\pi$
$619$	6553.03	0.425507	0.212753	+	0.977106i	$0.431757\pi$
$620$	10144.6	0.657124
$621$	−1503.36	−0.0971459
$622$	−1835.36	−0.118314
$623$	0	0
$624$	37691.0	2.41803
$625$	−18813.0	−1.20403
$626$	−1878.66	−0.119946
$627$	23654.5	1.50665
$628$	−4226.97	−0.268590
$629$	3558.66	0.225585
$630$	0	0
$631$	26332.1	1.66128	0.830639	−	0.556812i	$-0.187975\pi$
$631$	26332.1	1.66128	0.830639	+	0.556812i	$0.187975\pi$
$632$	−6518.19	−0.410253
$633$	6177.14	0.387866
$634$	36655.0	2.29615
$635$	12605.6	0.787777
$636$	8796.40	0.548428
$637$	0	0
$638$	28255.6	1.75337
$639$	12589.3	0.779382
$640$	−9334.50	−0.576529
$641$	29858.9	1.83987	0.919935	−	0.392072i	$-0.128242\pi$
$641$	29858.9	1.83987	0.919935	+	0.392072i	$0.128242\pi$
$642$	−7378.21	−0.453574
$643$	−13613.0	−0.834903	−0.417452	−	0.908699i	$-0.637077\pi$
$643$	−13613.0	−0.834903	−0.417452	+	0.908699i	$0.637077\pi$
$644$	0	0
$645$	−3684.13	−0.224903
$646$	56009.2	3.41123
$647$	−13260.7	−0.805768	−0.402884	−	0.915251i	$-0.631992\pi$
$647$	−13260.7	−0.805768	−0.402884	+	0.915251i	$0.631992\pi$
$648$	−5184.10	−0.314276
$649$	−4129.70	−0.249776
$650$	10236.2	0.617687
$651$	0	0
$652$	−11593.5	−0.696374
$653$	−28250.3	−1.69299	−0.846494	−	0.532399i	$-0.821291\pi$
$653$	−28250.3	−1.69299	−0.846494	+	0.532399i	$0.821291\pi$
$654$	−44821.0	−2.67988
$655$	20844.7	1.24347
$656$	−25398.5	−1.51165
$657$	−12807.1	−0.760506
$658$	0	0
$659$	−4996.39	−0.295344	−0.147672	−	0.989036i	$-0.547178\pi$
$659$	−4996.39	−0.295344	−0.147672	+	0.989036i	$0.547178\pi$
$660$	15698.4	0.925845
$661$	22499.1	1.32393	0.661963	−	0.749537i	$-0.269724\pi$
$661$	22499.1	1.32393	0.661963	+	0.749537i	$0.269724\pi$
$662$	14487.7	0.850574
$663$	60850.7	3.56447
$664$	5503.60	0.321658
$665$	0	0
$666$	2116.91	0.123166
$667$	−7001.93	−0.406471
$668$	−22083.4	−1.27909
$669$	−40357.8	−2.33232
$670$	32291.1	1.86196
$671$	−9965.30	−0.573332
$672$	0	0
$673$	31378.1	1.79723	0.898616	−	0.438736i	$-0.144574\pi$
$673$	31378.1	1.79723	0.898616	+	0.438736i	$0.144574\pi$
$674$	−12031.6	−0.687595
$675$	−2007.71	−0.114484
$676$	22534.1	1.28210
$677$	−1589.71	−0.0902476	−0.0451238	−	0.998981i	$-0.514368\pi$
$677$	−1589.71	−0.0902476	−0.0451238	+	0.998981i	$0.514368\pi$
$678$	41273.3	2.33789
$679$	0	0
$680$	−8871.98	−0.500331
$681$	21518.8	1.21087
$682$	−13366.4	−0.750479
$683$	19232.4	1.07747	0.538733	−	0.842477i	$-0.318903\pi$
$683$	19232.4	1.07747	0.538733	+	0.842477i	$0.318903\pi$
$684$	14882.8	0.831954
$685$	−39711.0	−2.21501
$686$	0	0
$687$	−3733.00	−0.207311
$688$	3180.10	0.176221
$689$	−15195.8	−0.840222
$690$	−8708.84	−0.480492
$691$	20393.1	1.12271	0.561353	−	0.827577i	$-0.310281\pi$
$691$	20393.1	1.12271	0.561353	+	0.827577i	$0.310281\pi$
$692$	8233.66	0.452307
$693$	0	0
$694$	17366.0	0.949864
$695$	−25385.0	−1.38548
$696$	−10376.6	−0.565122
$697$	−41004.9	−2.22837
$698$	17193.3	0.932343
$699$	27176.6	1.47055
$700$	0	0
$701$	−3875.69	−0.208820	−0.104410	−	0.994534i	$-0.533295\pi$
$701$	−3875.69	−0.208820	−0.104410	+	0.994534i	$0.533295\pi$
$702$	−16124.9	−0.866946
$703$	−3676.91	−0.197265
$704$	−8492.17	−0.454632
$705$	−31555.5	−1.68574
$706$	35819.5	1.90947
$707$	0	0
$708$	−6354.04	−0.337287
$709$	8690.16	0.460318	0.230159	−	0.973153i	$-0.426075\pi$
$709$	8690.16	0.460318	0.230159	+	0.973153i	$0.426075\pi$
$710$	−32487.4	−1.71723
$711$	−20771.7	−1.09564
$712$	−439.880	−0.0231534
$713$	3312.29	0.173978
$714$	0	0
$715$	−27118.9	−1.41845
$716$	3070.95	0.160289
$717$	46826.0	2.43898
$718$	9218.79	0.479167
$719$	21251.0	1.10227	0.551133	−	0.834417i	$-0.314196\pi$
$719$	21251.0	1.10227	0.551133	+	0.834417i	$0.314196\pi$
$720$	−17512.0	−0.906437
$721$	0	0
$722$	−31789.5	−1.63862
$723$	49456.9	2.54401
$724$	15734.7	0.807701
$725$	−9350.98	−0.479016
$726$	13521.7	0.691235
$727$	21295.6	1.08640	0.543199	−	0.839604i	$-0.317213\pi$
$727$	21295.6	1.08640	0.543199	+	0.839604i	$0.317213\pi$
$728$	0	0
$729$	−6257.80	−0.317929
$730$	33049.5	1.67564
$731$	5134.16	0.259772
$732$	−15332.8	−0.774204
$733$	−10692.2	−0.538777	−0.269389	−	0.963032i	$-0.586822\pi$
$733$	−10692.2	−0.538777	−0.269389	+	0.963032i	$0.586822\pi$
$734$	8703.90	0.437693
$735$	0	0
$736$	6263.84	0.313707
$737$	−19005.1	−0.949880
$738$	−24392.3	−1.21665
$739$	−35167.5	−1.75055	−0.875276	−	0.483625i	$-0.839320\pi$
$739$	−35167.5	−1.75055	−0.875276	+	0.483625i	$0.839320\pi$
$740$	−2440.19	−0.121221
$741$	−62872.8	−3.11699
$742$	0	0
$743$	12367.0	0.610636	0.305318	−	0.952250i	$-0.401237\pi$
$743$	12367.0	0.610636	0.305318	+	0.952250i	$0.401237\pi$
$744$	4908.70	0.241884
$745$	−26778.3	−1.31689
$746$	−307.954	−0.0151139
$747$	17538.4	0.859033
$748$	−21877.0	−1.06939
$749$	0	0
$750$	29092.3	1.41640
$751$	−26740.0	−1.29928	−0.649639	−	0.760243i	$-0.725080\pi$
$751$	−26740.0	−1.29928	−0.649639	+	0.760243i	$0.725080\pi$
$752$	27238.4	1.32085
$753$	42604.9	2.06190
$754$	−75102.4	−3.62741
$755$	2231.19	0.107552
$756$	0	0
$757$	−14163.8	−0.680042	−0.340021	−	0.940418i	$-0.610434\pi$
$757$	−14163.8	−0.680042	−0.340021	+	0.940418i	$0.610434\pi$
$758$	−6697.51	−0.320929
$759$	5125.63	0.245124
$760$	9166.79	0.437519
$761$	18647.2	0.888252	0.444126	−	0.895964i	$-0.353514\pi$
$761$	18647.2	0.888252	0.444126	+	0.895964i	$0.353514\pi$
$762$	−25555.0	−1.21491
$763$	0	0
$764$	7991.36	0.378426
$765$	−28272.5	−1.33620
$766$	3904.17	0.184156
$767$	10976.6	0.516743
$768$	35108.5	1.64957
$769$	−33696.9	−1.58016	−0.790078	−	0.613006i	$-0.789960\pi$
$769$	−33696.9	−1.58016	−0.790078	+	0.613006i	$0.789960\pi$
$770$	0	0
$771$	16476.3	0.769624
$772$	−14980.0	−0.698372
$773$	33982.6	1.58120	0.790602	−	0.612331i	$-0.209768\pi$
$773$	33982.6	1.58120	0.790602	+	0.612331i	$0.209768\pi$
$774$	3054.11	0.141832
$775$	4423.52	0.205029
$776$	−749.931	−0.0346919
$777$	0	0
$778$	25771.6	1.18761
$779$	42367.5	1.94862
$780$	−41725.7	−1.91541
$781$	19120.7	0.876045
$782$	12136.5	0.554988
$783$	14730.5	0.672316
$784$	0	0
$785$	−8296.74	−0.377227
$786$	−42257.9	−1.91767
$787$	24142.5	1.09351	0.546753	−	0.837294i	$-0.315864\pi$
$787$	24142.5	1.09351	0.546753	+	0.837294i	$0.315864\pi$
$788$	28729.4	1.29878
$789$	−20295.7	−0.915773
$790$	53602.6	2.41404
$791$	0	0
$792$	3106.16	0.139359
$793$	26487.4	1.18612
$794$	14294.0	0.638886
$795$	17265.7	0.770252
$796$	−17956.7	−0.799570
$797$	−4633.35	−0.205924	−0.102962	−	0.994685i	$-0.532832\pi$
$797$	−4633.35	−0.205924	−0.102962	+	0.994685i	$0.532832\pi$
$798$	0	0
$799$	43975.3	1.94710
$800$	8365.28	0.369696
$801$	−1401.77	−0.0618343
$802$	27051.8	1.19106
$803$	−19451.5	−0.854828
$804$	−29241.6	−1.28268
$805$	0	0
$806$	35527.5	1.55261
$807$	−8182.74	−0.356934
$808$	2940.90	0.128045
$809$	−14896.5	−0.647385	−0.323693	−	0.946162i	$-0.604924\pi$
$809$	−14896.5	−0.647385	−0.323693	+	0.946162i	$0.604924\pi$
$810$	42631.6	1.84929
$811$	−127.801	−0.00553356	−0.00276678	−	0.999996i	$-0.500881\pi$
$811$	−127.801	−0.00553356	−0.00276678	+	0.999996i	$0.500881\pi$
$812$	0	0
$813$	−14715.3	−0.634796
$814$	3215.17	0.138442
$815$	−22755.8	−0.978038
$816$	59680.5	2.56034
$817$	−5304.76	−0.227161
$818$	−15836.1	−0.676891
$819$	0	0
$820$	28117.3	1.19744
$821$	32985.2	1.40218	0.701092	−	0.713071i	$-0.252696\pi$
$821$	32985.2	1.40218	0.701092	+	0.713071i	$0.252696\pi$
$822$	80504.9	3.41597
$823$	−19957.6	−0.845294	−0.422647	−	0.906294i	$-0.638899\pi$
$823$	−19957.6	−0.845294	−0.422647	+	0.906294i	$0.638899\pi$
$824$	−11993.1	−0.507040
$825$	6845.21	0.288872
$826$	0	0
$827$	9682.05	0.407108	0.203554	−	0.979064i	$-0.434751\pi$
$827$	9682.05	0.407108	0.203554	+	0.979064i	$0.434751\pi$
$828$	3224.91	0.135354
$829$	−35422.3	−1.48404	−0.742018	−	0.670380i	$-0.766131\pi$
$829$	−35422.3	−1.48404	−0.742018	+	0.670380i	$0.766131\pi$
$830$	−45259.0	−1.89273
$831$	53244.6	2.22266
$832$	22571.9	0.940552
$833$	0	0
$834$	51462.2	2.13668
$835$	−43345.4	−1.79644
$836$	22604.0	0.935137
$837$	−6968.29	−0.287765
$838$	7327.78	0.302069
$839$	−24543.0	−1.00991	−0.504957	−	0.863145i	$-0.668491\pi$
$839$	−24543.0	−1.00991	−0.504957	+	0.863145i	$0.668491\pi$
$840$	0	0
$841$	44218.6	1.81305
$842$	−22139.5	−0.906149
$843$	−14955.3	−0.611019
$844$	5902.81	0.240738
$845$	44230.2	1.80067
$846$	26159.2	1.06309
$847$	0	0
$848$	−14903.5	−0.603525
$849$	−18574.5	−0.750855
$850$	16208.1	0.654041
$851$	−796.742	−0.0320939
$852$	29419.5	1.18297
$853$	26601.9	1.06780	0.533898	−	0.845549i	$-0.320727\pi$
$853$	26601.9	1.06780	0.533898	+	0.845549i	$0.320727\pi$
$854$	0	0
$855$	29212.0	1.16846
$856$	1682.84	0.0671942
$857$	32418.1	1.29216	0.646081	−	0.763269i	$-0.276407\pi$
$857$	32418.1	1.29216	0.646081	+	0.763269i	$0.276407\pi$
$858$	54977.3	2.18752
$859$	−21270.7	−0.844872	−0.422436	−	0.906393i	$-0.638825\pi$
$859$	−21270.7	−0.844872	−0.422436	+	0.906393i	$0.638825\pi$
$860$	−3520.52	−0.139592
$861$	0	0
$862$	38624.3	1.52616
$863$	8035.35	0.316948	0.158474	−	0.987363i	$-0.449343\pi$
$863$	8035.35	0.316948	0.158474	+	0.987363i	$0.449343\pi$
$864$	−13177.7	−0.518882
$865$	16161.1	0.635253
$866$	−12129.5	−0.475953
$867$	63146.7	2.47356
$868$	0	0
$869$	−31548.1	−1.23153
$870$	85332.5	3.32533
$871$	50514.8	1.96513
$872$	10222.9	0.397007
$873$	−2389.82	−0.0926497
$874$	−12539.8	−0.485315
$875$	0	0
$876$	−29928.4	−1.15432
$877$	7302.32	0.281165	0.140583	−	0.990069i	$-0.455102\pi$
$877$	7302.32	0.281165	0.140583	+	0.990069i	$0.455102\pi$
$878$	33134.1	1.27360
$879$	−15318.0	−0.587784
$880$	−26597.3	−1.01886
$881$	−6156.38	−0.235430	−0.117715	−	0.993047i	$-0.537557\pi$
$881$	−6156.38	−0.235430	−0.117715	+	0.993047i	$0.537557\pi$
$882$	0	0
$883$	−12281.9	−0.468083	−0.234041	−	0.972227i	$-0.575195\pi$
$883$	−12281.9	−0.468083	−0.234041	+	0.972227i	$0.575195\pi$
$884$	58148.3	2.21237
$885$	−12471.8	−0.473711
$886$	−225.273	−0.00854198
$887$	12943.7	0.489975	0.244988	−	0.969526i	$-0.421216\pi$
$887$	12943.7	0.489975	0.244988	+	0.969526i	$0.421216\pi$
$888$	−1180.74	−0.0446207
$889$	0	0
$890$	3617.36	0.136241
$891$	−25091.1	−0.943415
$892$	−38565.5	−1.44761
$893$	−45436.6	−1.70266
$894$	54286.8	2.03090
$895$	6027.69	0.225121
$896$	0	0
$897$	−13623.8	−0.507117
$898$	28027.1	1.04151
$899$	−32455.1	−1.20405
$900$	4306.82	0.159512
$901$	−24061.2	−0.889672
$902$	−37047.0	−1.36755
$903$	0	0
$904$	−9413.71	−0.346344
$905$	30884.2	1.13439
$906$	−4523.23	−0.165866
$907$	35691.6	1.30664	0.653318	−	0.757084i	$-0.273376\pi$
$907$	35691.6	1.30664	0.653318	+	0.757084i	$0.273376\pi$
$908$	20563.2	0.751556
$909$	9371.84	0.341963
$910$	0	0
$911$	−4631.90	−0.168454	−0.0842270	−	0.996447i	$-0.526842\pi$
$911$	−4631.90	−0.168454	−0.0842270	+	0.996447i	$0.526842\pi$
$912$	−61663.6	−2.23891
$913$	26637.4	0.965576
$914$	−37692.8	−1.36408
$915$	−30095.4	−1.08735
$916$	−3567.22	−0.128673
$917$	0	0
$918$	−25532.4	−0.917969
$919$	−424.153	−0.0152247	−0.00761236	−	0.999971i	$-0.502423\pi$
$919$	−424.153	−0.0152247	−0.00761236	+	0.999971i	$0.502423\pi$
$920$	1986.33	0.0711819
$921$	29880.7	1.06906
$922$	−4794.75	−0.171265
$923$	−50822.1	−1.81238
$924$	0	0
$925$	−1064.04	−0.0378220
$926$	14567.8	0.516985
$927$	−38218.8	−1.35412
$928$	−61375.5	−2.17107
$929$	−42668.1	−1.50688	−0.753442	−	0.657515i	$-0.771608\pi$
$929$	−42668.1	−1.50688	−0.753442	+	0.657515i	$0.771608\pi$
$930$	−40366.8	−1.42331
$931$	0	0
$932$	25969.7	0.912732
$933$	3262.27	0.114471
$934$	45311.3	1.58740
$935$	−42940.4	−1.50193
$936$	−8256.06	−0.288310
$937$	32095.7	1.11902	0.559509	−	0.828824i	$-0.310990\pi$
$937$	32095.7	1.11902	0.559509	+	0.828824i	$0.310990\pi$
$938$	0	0
$939$	3339.22	0.116050
$940$	−30154.1	−1.04630
$941$	−3598.42	−0.124660	−0.0623301	−	0.998056i	$-0.519853\pi$
$941$	−3598.42	−0.124660	−0.0623301	+	0.998056i	$0.519853\pi$
$942$	16819.7	0.581758
$943$	9180.52	0.317029
$944$	10765.5	0.371173
$945$	0	0
$946$	4638.60	0.159423
$947$	32209.7	1.10525	0.552626	−	0.833429i	$-0.313626\pi$
$947$	32209.7	1.10525	0.552626	+	0.833429i	$0.313626\pi$
$948$	−48540.6	−1.66300
$949$	51701.3	1.76849
$950$	−16746.7	−0.571932
$951$	−65152.4	−2.22157
$952$	0	0
$953$	37673.7	1.28056	0.640279	−	0.768142i	$-0.278819\pi$
$953$	37673.7	1.28056	0.640279	+	0.768142i	$0.278819\pi$
$954$	−14313.1	−0.485748
$955$	15685.5	0.531488
$956$	44746.4	1.51381
$957$	−50222.9	−1.69642
$958$	33530.4	1.13081
$959$	0	0
$960$	−25646.5	−0.862227
$961$	−14438.0	−0.484643
$962$	−8545.81	−0.286412
$963$	5362.74	0.179451
$964$	47260.5	1.57900
$965$	−29402.9	−0.980843
$966$	0	0
$967$	−24517.2	−0.815326	−0.407663	−	0.913132i	$-0.633656\pi$
$967$	−24517.2	−0.815326	−0.407663	+	0.913132i	$0.633656\pi$
$968$	−3084.06	−0.102402
$969$	−99553.6	−3.30044
$970$	6167.08	0.204137
$971$	−3476.89	−0.114911	−0.0574556	−	0.998348i	$-0.518299\pi$
$971$	−3476.89	−0.114911	−0.0574556	+	0.998348i	$0.518299\pi$
$972$	−28798.9	−0.950335
$973$	0	0
$974$	45883.5	1.50945
$975$	−18194.3	−0.597626
$976$	25978.0	0.851984
$977$	13916.3	0.455704	0.227852	−	0.973696i	$-0.426830\pi$
$977$	13916.3	0.455704	0.227852	+	0.973696i	$0.426830\pi$
$978$	46132.2	1.50833
$979$	−2129.02	−0.0695033
$980$	0	0
$981$	32577.5	1.06026
$982$	7836.35	0.254652
$983$	−45815.4	−1.48656	−0.743279	−	0.668982i	$-0.766730\pi$
$983$	−45815.4	−1.48656	−0.743279	+	0.668982i	$0.766730\pi$
$984$	13605.2	0.440771
$985$	56390.3	1.82411
$986$	−118918.	−3.84090
$987$	0	0
$988$	−60080.6	−1.93463
$989$	−1149.48	−0.0369578
$990$	−25543.6	−0.820029
$991$	−23660.2	−0.758418	−0.379209	−	0.925311i	$-0.623804\pi$
$991$	−23660.2	−0.758418	−0.379209	+	0.925311i	$0.623804\pi$
$992$	29033.9	0.929262
$993$	−25751.1	−0.822948
$994$	0	0
$995$	−35245.5	−1.12297
$996$	40984.9	1.30387
$997$	−23352.1	−0.741793	−0.370896	−	0.928674i	$-0.620950\pi$
$997$	−23352.1	−0.741793	−0.370896	+	0.928674i	$0.620950\pi$
$998$	−44176.7	−1.40119
$999$	1676.16	0.0530844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2107.4.a.h.1.6	yes	30
7.6	odd	2	inner	2107.4.a.h.1.5	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2107.4.a.h.1.5	✓	30	7.6	odd	2	inner
2107.4.a.h.1.6	yes	30	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:	\( N \)	\(=\)	\( 2107 = 7^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2107.a (trivial)

\(f(q)\)	\(=\)	\(q-3.80243 q^{2} +6.75863 q^{3} +6.45848 q^{4} +12.6768 q^{5} -25.6992 q^{6} +5.86153 q^{8} +18.6791 q^{9} +O(q^{10})\)	\(q-3.80243 q^{2} +6.75863 q^{3} +6.45848 q^{4} +12.6768 q^{5} -25.6992 q^{6} +5.86153 q^{8} +18.6791 q^{9} -48.2025 q^{10} +28.3699 q^{11} +43.6505 q^{12} -75.4061 q^{13} +85.6775 q^{15} -73.9559 q^{16} -119.399 q^{17} -71.0259 q^{18} +123.367 q^{19} +81.8726 q^{20} -107.874 q^{22} +26.7320 q^{23} +39.6159 q^{24} +35.7002 q^{25} +286.726 q^{26} -56.2380 q^{27} -261.930 q^{29} -325.783 q^{30} +123.907 q^{31} +234.320 q^{32} +191.741 q^{33} +454.006 q^{34} +120.638 q^{36} -29.8048 q^{37} -469.093 q^{38} -509.642 q^{39} +74.3053 q^{40} +343.428 q^{41} -43.0000 q^{43} +183.226 q^{44} +236.790 q^{45} -101.647 q^{46} -368.306 q^{47} -499.841 q^{48} -135.748 q^{50} -806.974 q^{51} -487.008 q^{52} +201.519 q^{53} +213.841 q^{54} +359.638 q^{55} +833.789 q^{57} +995.972 q^{58} -145.566 q^{59} +553.346 q^{60} -351.264 q^{61} -471.148 q^{62} -299.338 q^{64} -955.905 q^{65} -729.083 q^{66} -669.904 q^{67} -771.136 q^{68} +180.672 q^{69} +673.978 q^{71} +109.488 q^{72} -685.638 q^{73} +113.331 q^{74} +241.285 q^{75} +796.760 q^{76} +1937.88 q^{78} -1112.03 q^{79} -937.521 q^{80} -884.427 q^{81} -1305.86 q^{82} +938.935 q^{83} -1513.59 q^{85} +163.505 q^{86} -1770.29 q^{87} +166.291 q^{88} -75.0452 q^{89} -900.378 q^{90} +172.648 q^{92} +837.443 q^{93} +1400.46 q^{94} +1563.89 q^{95} +1583.68 q^{96} -127.941 q^{97} +529.923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9}+O(q^{10}) \) 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9	\( 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9} - 60 q^{11} - 96 q^{15} + 372 q^{16} - 1008 q^{18} - 234 q^{22} - 214 q^{23} + 520 q^{25} - 870 q^{29} - 12 q^{30} - 1548 q^{32} + 1142 q^{36} - 1246 q^{37} - 416 q^{39} - 1290 q^{43} - 446 q^{44} - 660 q^{46} - 278 q^{50} - 3702 q^{51} - 2960 q^{53} + 620 q^{57} - 3634 q^{58} - 898 q^{60} + 2578 q^{64} - 4848 q^{65} + 928 q^{67} - 1708 q^{71} - 7900 q^{72} + 1714 q^{74} - 138 q^{78} - 3562 q^{79} + 2210 q^{81} - 948 q^{85} + 344 q^{86} + 2502 q^{88} - 3848 q^{92} - 11986 q^{93} - 2894 q^{95} - 2804 q^{99}+O(q^{100}) \) 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9 - 60 * q^11 - 96 * q^15 + 372 * q^16 - 1008 * q^18 - 234 * q^22 - 214 * q^23 + 520 * q^25 - 870 * q^29 - 12 * q^30 - 1548 * q^32 + 1142 * q^36 - 1246 * q^37 - 416 * q^39 - 1290 * q^43 - 446 * q^44 - 660 * q^46 - 278 * q^50 - 3702 * q^51 - 2960 * q^53 + 620 * q^57 - 3634 * q^58 - 898 * q^60 + 2578 * q^64 - 4848 * q^65 + 928 * q^67 - 1708 * q^71 - 7900 * q^72 + 1714 * q^74 - 138 * q^78 - 3562 * q^79 + 2210 * q^81 - 948 * q^85 + 344 * q^86 + 2502 * q^88 - 3848 * q^92 - 11986 * q^93 - 2894 * q^95 - 2804 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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