LMFDB - Embedded newform 1925.4.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,4,Mod(1,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1925.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1925.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:	$113.578676761$
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} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 $ x^5 - x^4 - 42x^3 + 18x^2 + 368*x + 352
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.44399$ of defining polynomial
Character	$\chi$	$=$	1925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.44399 q^{2} -8.26395 q^{3} +11.7491 q^{4} +36.7249 q^{6} -7.00000 q^{7} -16.6609 q^{8} +41.2928 q^{9} +11.0000 q^{11} -97.0937 q^{12} +51.5769 q^{13} +31.1080 q^{14} -19.9519 q^{16} +26.5590 q^{17} -183.505 q^{18} +99.6432 q^{19} +57.8476 q^{21} -48.8839 q^{22} -28.1455 q^{23} +137.684 q^{24} -229.207 q^{26} -118.115 q^{27} -82.2435 q^{28} -43.9369 q^{29} -83.8402 q^{31} +221.953 q^{32} -90.9034 q^{33} -118.028 q^{34} +485.152 q^{36} -306.353 q^{37} -442.814 q^{38} -426.228 q^{39} +200.991 q^{41} -257.074 q^{42} +13.7546 q^{43} +129.240 q^{44} +125.079 q^{46} +266.533 q^{47} +164.881 q^{48} +49.0000 q^{49} -219.482 q^{51} +605.980 q^{52} -308.867 q^{53} +524.903 q^{54} +116.626 q^{56} -823.446 q^{57} +195.255 q^{58} -622.446 q^{59} -87.3303 q^{61} +372.585 q^{62} -289.050 q^{63} -826.742 q^{64} +403.974 q^{66} -608.395 q^{67} +312.044 q^{68} +232.593 q^{69} -464.926 q^{71} -687.974 q^{72} +255.407 q^{73} +1361.43 q^{74} +1170.72 q^{76} -77.0000 q^{77} +1894.16 q^{78} +261.237 q^{79} -138.809 q^{81} -893.204 q^{82} -953.986 q^{83} +679.656 q^{84} -61.1255 q^{86} +363.092 q^{87} -183.269 q^{88} -839.910 q^{89} -361.038 q^{91} -330.684 q^{92} +692.851 q^{93} -1184.47 q^{94} -1834.21 q^{96} +349.146 q^{97} -217.756 q^{98} +454.221 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} - 2 q^{3} + 45 q^{4} + 4 q^{6} - 35 q^{7} - 57 q^{8} + 63 q^{9} + 55 q^{11} - 24 q^{12} + 50 q^{13} + 7 q^{14} + 433 q^{16} - 222 q^{17} - 245 q^{18} + 160 q^{19} + 14 q^{21} - 11 q^{22} - 54 q^{23}+ \cdots + 693 q^{99}+O(q^{100}) $ 5 * q - q^2 - 2 * q^3 + 45 * q^4 + 4 * q^6 - 35 * q^7 - 57 * q^8 + 63 * q^9 + 55 * q^11 - 24 * q^12 + 50 * q^13 + 7 * q^14 + 433 * q^16 - 222 * q^17 - 245 * q^18 + 160 * q^19 + 14 * q^21 - 11 * q^22 - 54 * q^23 - 220 * q^24 - 1026 * q^26 - 110 * q^27 - 315 * q^28 + 14 * q^29 - 34 * q^31 + 583 * q^32 - 22 * q^33 - 750 * q^34 - 411 * q^36 - 1044 * q^37 + 156 * q^38 - 124 * q^39 - 114 * q^41 - 28 * q^42 - 672 * q^43 + 495 * q^44 - 1224 * q^46 + 292 * q^47 + 1652 * q^48 + 245 * q^49 + 768 * q^51 + 914 * q^52 + 710 * q^53 + 1608 * q^54 + 399 * q^56 - 2012 * q^57 + 810 * q^58 + 270 * q^59 + 138 * q^61 + 480 * q^62 - 441 * q^63 + 3001 * q^64 + 44 * q^66 - 1942 * q^67 - 3130 * q^68 - 1306 * q^69 - 278 * q^71 - 3565 * q^72 + 338 * q^73 + 462 * q^74 - 1332 * q^76 - 385 * q^77 - 280 * q^78 + 576 * q^79 - 1439 * q^81 - 1386 * q^82 - 1644 * q^83 + 168 * q^84 + 2020 * q^86 + 1956 * q^87 - 627 * q^88 - 3656 * q^89 - 350 * q^91 + 748 * q^92 + 1442 * q^93 + 976 * q^94 - 4508 * q^96 - 692 * q^97 - 49 * q^98 + 693 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4.44399	−1.57119	−0.785594	−	0.618742i	$-0.787643\pi$
$2$	−4.44399	−1.57119	−0.785594	+	0.618742i	$0.787643\pi$
$3$	−8.26395	−1.59040	−0.795199	−	0.606349i	$-0.792633\pi$
$3$	−8.26395	−1.59040	−0.795199	+	0.606349i	$0.792633\pi$
$4$	11.7491	1.46863
$5$	0	0
$5$	0	0
$6$	36.7249	2.49881
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−16.6609	−0.736313
$9$	41.2928	1.52936
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−97.0937	−2.33571
$13$	51.5769	1.10037	0.550186	−	0.835042i	$-0.314557\pi$
$13$	51.5769	1.10037	0.550186	+	0.835042i	$0.314557\pi$
$14$	31.1080	0.593854
$15$	0	0
$16$	−19.9519	−0.311748
$17$	26.5590	0.378912	0.189456	−	0.981889i	$-0.439328\pi$
$17$	26.5590	0.378912	0.189456	+	0.981889i	$0.439328\pi$
$18$	−183.505	−2.40292
$19$	99.6432	1.20314	0.601571	−	0.798819i	$-0.294542\pi$
$19$	99.6432	1.20314	0.601571	+	0.798819i	$0.294542\pi$
$20$	0	0
$21$	57.8476	0.601114
$22$	−48.8839	−0.473731
$23$	−28.1455	−0.255163	−0.127582	−	0.991828i	$-0.540721\pi$
$23$	−28.1455	−0.255163	−0.127582	+	0.991828i	$0.540721\pi$
$24$	137.684	1.17103
$25$	0	0
$26$	−229.207	−1.72889
$27$	−118.115	−0.841899
$28$	−82.2435	−0.555092
$29$	−43.9369	−0.281340	−0.140670	−	0.990057i	$-0.544926\pi$
$29$	−43.9369	−0.281340	−0.140670	+	0.990057i	$0.544926\pi$
$30$	0	0
$31$	−83.8402	−0.485747	−0.242873	−	0.970058i	$-0.578090\pi$
$31$	−83.8402	−0.485747	−0.242873	+	0.970058i	$0.578090\pi$
$32$	221.953	1.22613
$33$	−90.9034	−0.479523
$34$	−118.028	−0.595342
$35$	0	0
$36$	485.152	2.24608
$37$	−306.353	−1.36119	−0.680596	−	0.732659i	$-0.738279\pi$
$37$	−306.353	−1.36119	−0.680596	+	0.732659i	$0.738279\pi$
$38$	−442.814	−1.89036
$39$	−426.228	−1.75003
$40$	0	0
$41$	200.991	0.765599	0.382800	−	0.923831i	$-0.374960\pi$
$41$	200.991	0.765599	0.382800	+	0.923831i	$0.374960\pi$
$42$	−257.074	−0.944463
$43$	13.7546	0.0487805	0.0243903	−	0.999703i	$-0.492236\pi$
$43$	13.7546	0.0487805	0.0243903	+	0.999703i	$0.492236\pi$
$44$	129.240	0.442810
$45$	0	0
$46$	125.079	0.400909
$47$	266.533	0.827189	0.413594	−	0.910461i	$-0.364273\pi$
$47$	266.533	0.827189	0.413594	+	0.910461i	$0.364273\pi$
$48$	164.881	0.495803
$49$	49.0000	0.142857
$50$	0	0
$51$	−219.482	−0.602621
$52$	605.980	1.61605
$53$	−308.867	−0.800493	−0.400247	−	0.916407i	$-0.631076\pi$
$53$	−308.867	−0.800493	−0.400247	+	0.916407i	$0.631076\pi$
$54$	524.903	1.32278
$55$	0	0
$56$	116.626	0.278300
$57$	−823.446	−1.91348
$58$	195.255	0.442039
$59$	−622.446	−1.37348	−0.686742	−	0.726901i	$-0.740960\pi$
$59$	−622.446	−1.37348	−0.686742	+	0.726901i	$0.740960\pi$
$60$	0	0
$61$	−87.3303	−0.183303	−0.0916516	−	0.995791i	$-0.529215\pi$
$61$	−87.3303	−0.183303	−0.0916516	+	0.995791i	$0.529215\pi$
$62$	372.585	0.763200
$63$	−289.050	−0.578045
$64$	−826.742	−1.61473
$65$	0	0
$66$	403.974	0.753421
$67$	−608.395	−1.10936	−0.554681	−	0.832063i	$-0.687160\pi$
$67$	−608.395	−1.10936	−0.554681	+	0.832063i	$0.687160\pi$
$68$	312.044	0.556483
$69$	232.593	0.405811
$70$	0	0
$71$	−464.926	−0.777135	−0.388567	−	0.921420i	$-0.627030\pi$
$71$	−464.926	−0.777135	−0.388567	+	0.921420i	$0.627030\pi$
$72$	−687.974	−1.12609
$73$	255.407	0.409495	0.204747	−	0.978815i	$-0.434363\pi$
$73$	255.407	0.409495	0.204747	+	0.978815i	$0.434363\pi$
$74$	1361.43	2.13869
$75$	0	0
$76$	1170.72	1.76698
$77$	−77.0000	−0.113961
$78$	1894.16	2.74963
$79$	261.237	0.372043	0.186022	−	0.982546i	$-0.440441\pi$
$79$	261.237	0.372043	0.186022	+	0.982546i	$0.440441\pi$
$80$	0	0
$81$	−138.809	−0.190410
$82$	−893.204	−1.20290
$83$	−953.986	−1.26161	−0.630804	−	0.775942i	$-0.717275\pi$
$83$	−953.986	−1.26161	−0.630804	+	0.775942i	$0.717275\pi$
$84$	679.656	0.882816
$85$	0	0
$86$	−61.1255	−0.0766434
$87$	363.092	0.447443
$88$	−183.269	−0.222007
$89$	−839.910	−1.00034	−0.500170	−	0.865927i	$-0.666729\pi$
$89$	−839.910	−1.00034	−0.500170	+	0.865927i	$0.666729\pi$
$90$	0	0
$91$	−361.038	−0.415902
$92$	−330.684	−0.374741
$93$	692.851	0.772530
$94$	−1184.47	−1.29967
$95$	0	0
$96$	−1834.21	−1.95003
$97$	349.146	0.365468	0.182734	−	0.983162i	$-0.441505\pi$
$97$	349.146	0.365468	0.182734	+	0.983162i	$0.441505\pi$
$98$	−217.756	−0.224456
$99$	454.221	0.461121
$100$	0	0
$101$	1492.44	1.47033	0.735163	−	0.677890i	$-0.237106\pi$
$101$	1492.44	1.47033	0.735163	+	0.677890i	$0.237106\pi$
$102$	975.377	0.946831
$103$	−558.687	−0.534457	−0.267228	−	0.963633i	$-0.586108\pi$
$103$	−558.687	−0.534457	−0.267228	+	0.963633i	$0.586108\pi$
$104$	−859.314	−0.810218
$105$	0	0
$106$	1372.60	1.25773
$107$	694.047	0.627066	0.313533	−	0.949577i	$-0.398487\pi$
$107$	694.047	0.627066	0.313533	+	0.949577i	$0.398487\pi$
$108$	−1387.74	−1.23644
$109$	−341.005	−0.299654	−0.149827	−	0.988712i	$-0.547872\pi$
$109$	−341.005	−0.299654	−0.149827	+	0.988712i	$0.547872\pi$
$110$	0	0
$111$	2531.68	2.16484
$112$	139.663	0.117830
$113$	990.910	0.824929	0.412464	−	0.910974i	$-0.364668\pi$
$113$	990.910	0.824929	0.412464	+	0.910974i	$0.364668\pi$
$114$	3659.39	3.00643
$115$	0	0
$116$	−516.218	−0.413186
$117$	2129.75	1.68287
$118$	2766.15	2.15800
$119$	−185.913	−0.143215
$120$	0	0
$121$	121.000	0.0909091
$122$	388.095	0.288004
$123$	−1660.98	−1.21761
$124$	−985.045	−0.713384
$125$	0	0
$126$	1284.54	0.908218
$127$	666.090	0.465401	0.232700	−	0.972548i	$-0.425244\pi$
$127$	666.090	0.465401	0.232700	+	0.972548i	$0.425244\pi$
$128$	1898.41	1.31092
$129$	−113.668	−0.0775804
$130$	0	0
$131$	30.4356	0.0202990	0.0101495	−	0.999948i	$-0.496769\pi$
$131$	30.4356	0.0202990	0.0101495	+	0.999948i	$0.496769\pi$
$132$	−1068.03	−0.704244
$133$	−697.502	−0.454745
$134$	2703.70	1.74302
$135$	0	0
$136$	−442.496	−0.278998
$137$	2810.25	1.75252	0.876262	−	0.481836i	$-0.160030\pi$
$137$	2810.25	1.75252	0.876262	+	0.481836i	$0.160030\pi$
$138$	−1033.64	−0.637605
$139$	3110.49	1.89804	0.949021	−	0.315212i	$-0.102076\pi$
$139$	3110.49	1.89804	0.949021	+	0.315212i	$0.102076\pi$
$140$	0	0
$141$	−2202.62	−1.31556
$142$	2066.13	1.22103
$143$	567.345	0.331775
$144$	−823.869	−0.476776
$145$	0	0
$146$	−1135.03	−0.643394
$147$	−404.933	−0.227200
$148$	−3599.36	−1.99909
$149$	1916.92	1.05396	0.526979	−	0.849878i	$-0.323325\pi$
$149$	1916.92	1.05396	0.526979	+	0.849878i	$0.323325\pi$
$150$	0	0
$151$	−2289.28	−1.23377	−0.616883	−	0.787055i	$-0.711605\pi$
$151$	−2289.28	−1.23377	−0.616883	+	0.787055i	$0.711605\pi$
$152$	−1660.14	−0.885889
$153$	1096.70	0.579494
$154$	342.187	0.179054
$155$	0	0
$156$	−5007.79	−2.57015
$157$	−280.036	−0.142352	−0.0711762	−	0.997464i	$-0.522675\pi$
$157$	−280.036	−0.142352	−0.0711762	+	0.997464i	$0.522675\pi$
$158$	−1160.93	−0.584550
$159$	2552.46	1.27310
$160$	0	0
$161$	197.019	0.0964426
$162$	616.865	0.299170
$163$	866.571	0.416411	0.208206	−	0.978085i	$-0.433238\pi$
$163$	866.571	0.416411	0.208206	+	0.978085i	$0.433238\pi$
$164$	2361.46	1.12439
$165$	0	0
$166$	4239.51	1.98223
$167$	1965.18	0.910600	0.455300	−	0.890338i	$-0.349532\pi$
$167$	1965.18	0.910600	0.455300	+	0.890338i	$0.349532\pi$
$168$	−963.791	−0.442608
$169$	463.173	0.210820
$170$	0	0
$171$	4114.55	1.84004
$172$	161.604	0.0716407
$173$	−3956.88	−1.73894	−0.869469	−	0.493988i	$-0.835539\pi$
$173$	−3956.88	−1.73894	−0.869469	+	0.493988i	$0.835539\pi$
$174$	−1613.58	−0.703018
$175$	0	0
$176$	−219.471	−0.0939956
$177$	5143.86	2.18439
$178$	3732.55	1.57172
$179$	−3143.58	−1.31264	−0.656318	−	0.754484i	$-0.727887\pi$
$179$	−3143.58	−1.31264	−0.656318	+	0.754484i	$0.727887\pi$
$180$	0	0
$181$	683.772	0.280798	0.140399	−	0.990095i	$-0.455162\pi$
$181$	683.772	0.280798	0.140399	+	0.990095i	$0.455162\pi$
$182$	1604.45	0.653460
$183$	721.693	0.291525
$184$	468.929	0.187880
$185$	0	0
$186$	−3079.03	−1.21379
$187$	292.149	0.114246
$188$	3131.52	1.21484
$189$	826.806	0.318208
$190$	0	0
$191$	2739.68	1.03789	0.518944	−	0.854809i	$-0.326325\pi$
$191$	2739.68	1.03789	0.518944	+	0.854809i	$0.326325\pi$
$192$	6832.15	2.56806
$193$	−2651.93	−0.989067	−0.494534	−	0.869159i	$-0.664661\pi$
$193$	−2651.93	−0.989067	−0.494534	+	0.869159i	$0.664661\pi$
$194$	−1551.60	−0.574219
$195$	0	0
$196$	575.705	0.209805
$197$	1879.52	0.679749	0.339874	−	0.940471i	$-0.389615\pi$
$197$	1879.52	0.679749	0.339874	+	0.940471i	$0.389615\pi$
$198$	−2018.56	−0.724507
$199$	−3119.39	−1.11119	−0.555597	−	0.831452i	$-0.687510\pi$
$199$	−3119.39	−1.11119	−0.555597	+	0.831452i	$0.687510\pi$
$200$	0	0
$201$	5027.75	1.76433
$202$	−6632.37	−2.31016
$203$	307.558	0.106337
$204$	−2578.71	−0.885029
$205$	0	0
$206$	2482.80	0.839733
$207$	−1162.21	−0.390237
$208$	−1029.05	−0.343039
$209$	1096.08	0.362761
$210$	0	0
$211$	−520.718	−0.169894	−0.0849472	−	0.996385i	$-0.527072\pi$
$211$	−520.718	−0.169894	−0.0849472	+	0.996385i	$0.527072\pi$
$212$	−3628.90	−1.17563
$213$	3842.12	1.23595
$214$	−3084.34	−0.985238
$215$	0	0
$216$	1967.90	0.619901
$217$	586.881	0.183595
$218$	1515.42	0.470814
$219$	−2110.67	−0.651260
$220$	0	0
$221$	1369.83	0.416944
$222$	−11250.8	−3.40137
$223$	−2101.08	−0.630935	−0.315467	−	0.948936i	$-0.602161\pi$
$223$	−2101.08	−0.630935	−0.315467	+	0.948936i	$0.602161\pi$
$224$	−1553.67	−0.463433
$225$	0	0
$226$	−4403.60	−1.29612
$227$	6051.96	1.76953	0.884764	−	0.466040i	$-0.154320\pi$
$227$	6051.96	1.76953	0.884764	+	0.466040i	$0.154320\pi$
$228$	−9674.73	−2.81020
$229$	−2995.73	−0.864470	−0.432235	−	0.901761i	$-0.642275\pi$
$229$	−2995.73	−0.864470	−0.432235	+	0.901761i	$0.642275\pi$
$230$	0	0
$231$	636.324	0.181243
$232$	732.026	0.207155
$233$	65.3656	0.0183787	0.00918936	−	0.999958i	$-0.497075\pi$
$233$	65.3656	0.0183787	0.00918936	+	0.999958i	$0.497075\pi$
$234$	−9464.61	−2.64411
$235$	0	0
$236$	−7313.17	−2.01715
$237$	−2158.85	−0.591697
$238$	826.196	0.225018
$239$	−1102.33	−0.298343	−0.149171	−	0.988811i	$-0.547661\pi$
$239$	−1102.33	−0.298343	−0.149171	+	0.988811i	$0.547661\pi$
$240$	0	0
$241$	5297.43	1.41592	0.707962	−	0.706250i	$-0.249615\pi$
$241$	5297.43	1.41592	0.707962	+	0.706250i	$0.249615\pi$
$242$	−537.723	−0.142835
$243$	4336.22	1.14473
$244$	−1026.05	−0.269205
$245$	0	0
$246$	7381.39	1.91309
$247$	5139.28	1.32391
$248$	1396.85	0.357661
$249$	7883.69	2.00646
$250$	0	0
$251$	177.964	0.0447530	0.0223765	−	0.999750i	$-0.492877\pi$
$251$	177.964	0.0447530	0.0223765	+	0.999750i	$0.492877\pi$
$252$	−3396.07	−0.848937
$253$	−309.601	−0.0769346
$254$	−2960.10	−0.731232
$255$	0	0
$256$	−1822.59	−0.444969
$257$	−3496.69	−0.848707	−0.424354	−	0.905497i	$-0.639499\pi$
$257$	−3496.69	−0.848707	−0.424354	+	0.905497i	$0.639499\pi$
$258$	505.138	0.121893
$259$	2144.47	0.514482
$260$	0	0
$261$	−1814.28	−0.430272
$262$	−135.256	−0.0318936
$263$	−5747.94	−1.34766	−0.673828	−	0.738889i	$-0.735351\pi$
$263$	−5747.94	−1.34766	−0.673828	+	0.738889i	$0.735351\pi$
$264$	1514.53	0.353079
$265$	0	0
$266$	3099.70	0.714491
$267$	6940.97	1.59094
$268$	−7148.08	−1.62925
$269$	235.217	0.0533140	0.0266570	−	0.999645i	$-0.491514\pi$
$269$	235.217	0.0533140	0.0266570	+	0.999645i	$0.491514\pi$
$270$	0	0
$271$	−1179.58	−0.264406	−0.132203	−	0.991223i	$-0.542205\pi$
$271$	−1179.58	−0.264406	−0.132203	+	0.991223i	$0.542205\pi$
$272$	−529.902	−0.118125
$273$	2983.60	0.661449
$274$	−12488.7	−2.75354
$275$	0	0
$276$	2732.76	0.595988
$277$	3638.98	0.789331	0.394666	−	0.918825i	$-0.370860\pi$
$277$	3638.98	0.789331	0.394666	+	0.918825i	$0.370860\pi$
$278$	−13823.0	−2.98218
$279$	−3462.00	−0.742883
$280$	0	0
$281$	3236.81	0.687160	0.343580	−	0.939123i	$-0.388360\pi$
$281$	3236.81	0.687160	0.343580	+	0.939123i	$0.388360\pi$
$282$	9788.42	2.06699
$283$	−8303.78	−1.74420	−0.872100	−	0.489328i	$-0.837242\pi$
$283$	−8303.78	−1.74420	−0.872100	+	0.489328i	$0.837242\pi$
$284$	−5462.45	−1.14133
$285$	0	0
$286$	−2521.28	−0.521281
$287$	−1406.94	−0.289369
$288$	9165.06	1.87520
$289$	−4207.62	−0.856426
$290$	0	0
$291$	−2885.32	−0.581239
$292$	3000.80	0.601398
$293$	1894.16	0.377672	0.188836	−	0.982009i	$-0.439529\pi$
$293$	1894.16	0.377672	0.188836	+	0.982009i	$0.439529\pi$
$294$	1799.52	0.356974
$295$	0	0
$296$	5104.10	1.00226
$297$	−1299.27	−0.253842
$298$	−8518.76	−1.65597
$299$	−1451.66	−0.280775
$300$	0	0
$301$	−96.2825	−0.0184373
$302$	10173.5	1.93848
$303$	−12333.4	−2.33840
$304$	−1988.07	−0.375077
$305$	0	0
$306$	−4873.71	−0.910495
$307$	−6596.30	−1.22629	−0.613144	−	0.789971i	$-0.710096\pi$
$307$	−6596.30	−1.22629	−0.613144	+	0.789971i	$0.710096\pi$
$308$	−904.679	−0.167366
$309$	4616.96	0.849999
$310$	0	0
$311$	−5242.26	−0.955824	−0.477912	−	0.878408i	$-0.658606\pi$
$311$	−5242.26	−0.955824	−0.477912	+	0.878408i	$0.658606\pi$
$312$	7101.33	1.28857
$313$	−5338.75	−0.964103	−0.482051	−	0.876143i	$-0.660108\pi$
$313$	−5338.75	−0.964103	−0.482051	+	0.876143i	$0.660108\pi$
$314$	1244.48	0.223662
$315$	0	0
$316$	3069.29	0.546395
$317$	5807.21	1.02891	0.514456	−	0.857517i	$-0.327994\pi$
$317$	5807.21	1.02891	0.514456	+	0.857517i	$0.327994\pi$
$318$	−11343.1	−2.00028
$319$	−483.306	−0.0848273
$320$	0	0
$321$	−5735.56	−0.997283
$322$	−875.550	−0.151530
$323$	2646.42	0.455885
$324$	−1630.87	−0.279642
$325$	0	0
$326$	−3851.03	−0.654261
$327$	2818.04	0.476570
$328$	−3348.69	−0.563720
$329$	−1865.73	−0.312648
$330$	0	0
$331$	−1366.51	−0.226919	−0.113460	−	0.993543i	$-0.536193\pi$
$331$	−1366.51	−0.226919	−0.113460	+	0.993543i	$0.536193\pi$
$332$	−11208.4	−1.85284
$333$	−12650.2	−2.08176
$334$	−8733.25	−1.43072
$335$	0	0
$336$	−1154.17	−0.187396
$337$	3363.75	0.543724	0.271862	−	0.962336i	$-0.412361\pi$
$337$	3363.75	0.543724	0.271862	+	0.962336i	$0.412361\pi$
$338$	−2058.34	−0.331239
$339$	−8188.83	−1.31196
$340$	0	0
$341$	−922.242	−0.146458
$342$	−18285.0	−2.89106
$343$	−343.000	−0.0539949
$344$	−229.164	−0.0359177
$345$	0	0
$346$	17584.4	2.73220
$347$	−2984.97	−0.461791	−0.230896	−	0.972979i	$-0.574166\pi$
$347$	−2984.97	−0.461791	−0.230896	+	0.972979i	$0.574166\pi$
$348$	4265.99	0.657130
$349$	1286.08	0.197255	0.0986276	−	0.995124i	$-0.468555\pi$
$349$	1286.08	0.197255	0.0986276	+	0.995124i	$0.468555\pi$
$350$	0	0
$351$	−6092.01	−0.926403
$352$	2441.48	0.369691
$353$	−8417.60	−1.26919	−0.634594	−	0.772846i	$-0.718833\pi$
$353$	−8417.60	−1.26919	−0.634594	+	0.772846i	$0.718833\pi$
$354$	−22859.3	−3.43208
$355$	0	0
$356$	−9868.16	−1.46913
$357$	1536.38	0.227769
$358$	13970.0	2.06240
$359$	7483.47	1.10017	0.550087	−	0.835108i	$-0.314595\pi$
$359$	7483.47	1.10017	0.550087	+	0.835108i	$0.314595\pi$
$360$	0	0
$361$	3069.76	0.447553
$362$	−3038.68	−0.441186
$363$	−999.938	−0.144582
$364$	−4241.86	−0.610808
$365$	0	0
$366$	−3207.20	−0.458041
$367$	−8588.73	−1.22160	−0.610801	−	0.791784i	$-0.709153\pi$
$367$	−8588.73	−1.22160	−0.610801	+	0.791784i	$0.709153\pi$
$368$	561.556	0.0795466
$369$	8299.50	1.17088
$370$	0	0
$371$	2162.07	0.302558
$372$	8140.36	1.13456
$373$	−11833.0	−1.64260	−0.821298	−	0.570500i	$-0.806749\pi$
$373$	−11833.0	−1.64260	−0.821298	+	0.570500i	$0.806749\pi$
$374$	−1298.31	−0.179502
$375$	0	0
$376$	−4440.67	−0.609070
$377$	−2266.13	−0.309579
$378$	−3674.32	−0.499965
$379$	5056.39	0.685301	0.342651	−	0.939463i	$-0.388675\pi$
$379$	5056.39	0.685301	0.342651	+	0.939463i	$0.388675\pi$
$380$	0	0
$381$	−5504.53	−0.740172
$382$	−12175.1	−1.63072
$383$	−6457.09	−0.861467	−0.430733	−	0.902479i	$-0.641745\pi$
$383$	−6457.09	−0.861467	−0.430733	+	0.902479i	$0.641745\pi$
$384$	−15688.4	−2.08488
$385$	0	0
$386$	11785.1	1.55401
$387$	567.968	0.0746032
$388$	4102.14	0.536739
$389$	12444.5	1.62201	0.811004	−	0.585040i	$-0.198921\pi$
$389$	12444.5	1.62201	0.811004	+	0.585040i	$0.198921\pi$
$390$	0	0
$391$	−747.518	−0.0966844
$392$	−816.382	−0.105188
$393$	−251.518	−0.0322835
$394$	−8352.59	−1.06801
$395$	0	0
$396$	5336.68	0.677217
$397$	619.207	0.0782799	0.0391400	−	0.999234i	$-0.487538\pi$
$397$	619.207	0.0782799	0.0391400	+	0.999234i	$0.487538\pi$
$398$	13862.5	1.74589
$399$	5764.12	0.723226
$400$	0	0
$401$	9731.89	1.21194	0.605969	−	0.795488i	$-0.292785\pi$
$401$	9731.89	1.21194	0.605969	+	0.795488i	$0.292785\pi$
$402$	−22343.3	−2.77209
$403$	−4324.22	−0.534502
$404$	17534.7	2.15937
$405$	0	0
$406$	−1366.79	−0.167075
$407$	−3369.88	−0.410415
$408$	3656.76	0.443717
$409$	4621.43	0.558717	0.279358	−	0.960187i	$-0.409878\pi$
$409$	4621.43	0.558717	0.279358	+	0.960187i	$0.409878\pi$
$410$	0	0
$411$	−23223.7	−2.78721
$412$	−6564.05	−0.784922
$413$	4357.12	0.519128
$414$	5164.85	0.613137
$415$	0	0
$416$	11447.6	1.34920
$417$	−25704.9	−3.01864
$418$	−4870.95	−0.569966
$419$	186.428	0.0217365	0.0108682	−	0.999941i	$-0.496540\pi$
$419$	186.428	0.0217365	0.0108682	+	0.999941i	$0.496540\pi$
$420$	0	0
$421$	2670.29	0.309126	0.154563	−	0.987983i	$-0.450603\pi$
$421$	2670.29	0.309126	0.154563	+	0.987983i	$0.450603\pi$
$422$	2314.07	0.266936
$423$	11005.9	1.26507
$424$	5145.99	0.589413
$425$	0	0
$426$	−17074.4	−1.94192
$427$	611.312	0.0692821
$428$	8154.40	0.920930
$429$	−4688.51	−0.527654
$430$	0	0
$431$	−12514.9	−1.39866	−0.699328	−	0.714801i	$-0.746517\pi$
$431$	−12514.9	−1.39866	−0.699328	+	0.714801i	$0.746517\pi$
$432$	2356.62	0.262460
$433$	16651.2	1.84805	0.924025	−	0.382332i	$-0.124879\pi$
$433$	16651.2	1.84805	0.924025	+	0.382332i	$0.124879\pi$
$434$	−2608.10	−0.288462
$435$	0	0
$436$	−4006.49	−0.440083
$437$	−2804.51	−0.306998
$438$	9379.80	1.02325
$439$	6033.38	0.655940	0.327970	−	0.944688i	$-0.393636\pi$
$439$	6033.38	0.655940	0.327970	+	0.944688i	$0.393636\pi$
$440$	0	0
$441$	2023.35	0.218481
$442$	−6087.51	−0.655098
$443$	−6320.03	−0.677819	−0.338910	−	0.940819i	$-0.610058\pi$
$443$	−6320.03	−0.677819	−0.338910	+	0.940819i	$0.610058\pi$
$444$	29744.9	3.17935
$445$	0	0
$446$	9337.17	0.991318
$447$	−15841.3	−1.67621
$448$	5787.19	0.610311
$449$	−17893.6	−1.88074	−0.940368	−	0.340159i	$-0.889519\pi$
$449$	−17893.6	−1.88074	−0.940368	+	0.340159i	$0.889519\pi$
$450$	0	0
$451$	2210.90	0.230837
$452$	11642.3	1.21152
$453$	18918.5	1.96218
$454$	−26894.9	−2.78026
$455$	0	0
$456$	13719.3	1.40892
$457$	−6208.00	−0.635444	−0.317722	−	0.948184i	$-0.602918\pi$
$457$	−6208.00	−0.635444	−0.317722	+	0.948184i	$0.602918\pi$
$458$	13313.0	1.35825
$459$	−3137.02	−0.319006
$460$	0	0
$461$	7981.28	0.806346	0.403173	−	0.915124i	$-0.367907\pi$
$461$	7981.28	0.806346	0.403173	+	0.915124i	$0.367907\pi$
$462$	−2827.82	−0.284766
$463$	7495.19	0.752334	0.376167	−	0.926552i	$-0.377242\pi$
$463$	7495.19	0.752334	0.376167	+	0.926552i	$0.377242\pi$
$464$	876.623	0.0877073
$465$	0	0
$466$	−290.484	−0.0288765
$467$	−1519.39	−0.150555	−0.0752773	−	0.997163i	$-0.523984\pi$
$467$	−1519.39	−0.150555	−0.0752773	+	0.997163i	$0.523984\pi$
$468$	25022.6	2.47152
$469$	4258.77	0.419300
$470$	0	0
$471$	2314.20	0.226397
$472$	10370.5	1.01131
$473$	151.301	0.0147079
$474$	9593.90	0.929667
$475$	0	0
$476$	−2184.31	−0.210331
$477$	−12754.0	−1.22425
$478$	4898.75	0.468753
$479$	−16394.2	−1.56382	−0.781909	−	0.623393i	$-0.785754\pi$
$479$	−16394.2	−1.56382	−0.781909	+	0.623393i	$0.785754\pi$
$480$	0	0
$481$	−15800.7	−1.49782
$482$	−23541.8	−2.22468
$483$	−1628.15	−0.153382
$484$	1421.64	0.133512
$485$	0	0
$486$	−19270.1	−1.79858
$487$	−2275.01	−0.211685	−0.105843	−	0.994383i	$-0.533754\pi$
$487$	−2275.01	−0.211685	−0.105843	+	0.994383i	$0.533754\pi$
$488$	1455.00	0.134968
$489$	−7161.29	−0.662259
$490$	0	0
$491$	−14629.2	−1.34462	−0.672308	−	0.740272i	$-0.734697\pi$
$491$	−14629.2	−1.34462	−0.672308	+	0.740272i	$0.734697\pi$
$492$	−19515.0	−1.78822
$493$	−1166.92	−0.106603
$494$	−22838.9	−2.08011
$495$	0	0
$496$	1672.77	0.151431
$497$	3254.48	0.293729
$498$	−35035.1	−3.15253
$499$	3534.01	0.317042	0.158521	−	0.987356i	$-0.449327\pi$
$499$	3534.01	0.317042	0.158521	+	0.987356i	$0.449327\pi$
$500$	0	0
$501$	−16240.1	−1.44822
$502$	−790.872	−0.0703155
$503$	9233.35	0.818479	0.409239	−	0.912427i	$-0.365794\pi$
$503$	9233.35	0.818479	0.409239	+	0.912427i	$0.365794\pi$
$504$	4815.82	0.425622
$505$	0	0
$506$	1375.86	0.120879
$507$	−3827.63	−0.335288
$508$	7825.93	0.683503
$509$	11565.3	1.00712	0.503560	−	0.863960i	$-0.332023\pi$
$509$	11565.3	1.00712	0.503560	+	0.863960i	$0.332023\pi$
$510$	0	0
$511$	−1787.85	−0.154775
$512$	−7087.70	−0.611787
$513$	−11769.4	−1.01292
$514$	15539.3	1.33348
$515$	0	0
$516$	−1335.49	−0.113937
$517$	2931.87	0.249407
$518$	−9530.01	−0.808349
$519$	32699.5	2.76560
$520$	0	0
$521$	−2440.24	−0.205200	−0.102600	−	0.994723i	$-0.532716\pi$
$521$	−2440.24	−0.205200	−0.102600	+	0.994723i	$0.532716\pi$
$522$	8062.64	0.676038
$523$	911.213	0.0761847	0.0380923	−	0.999274i	$-0.487872\pi$
$523$	911.213	0.0761847	0.0380923	+	0.999274i	$0.487872\pi$
$524$	357.590	0.0298119
$525$	0	0
$526$	25543.8	2.11742
$527$	−2226.71	−0.184055
$528$	1813.69	0.149490
$529$	−11374.8	−0.934892
$530$	0	0
$531$	−25702.6	−2.10056
$532$	−8195.01	−0.667854
$533$	10366.5	0.842445
$534$	−30845.6	−2.49966
$535$	0	0
$536$	10136.4	0.816838
$537$	25978.3	2.08761
$538$	−1045.30	−0.0837663
$539$	539.000	0.0430730
$540$	0	0
$541$	12277.5	0.975692	0.487846	−	0.872930i	$-0.337783\pi$
$541$	12277.5	0.975692	0.487846	+	0.872930i	$0.337783\pi$
$542$	5242.02	0.415432
$543$	−5650.65	−0.446580
$544$	5894.84	0.464594
$545$	0	0
$546$	−13259.1	−1.03926
$547$	12539.2	0.980141	0.490071	−	0.871683i	$-0.336971\pi$
$547$	12539.2	0.980141	0.490071	+	0.871683i	$0.336971\pi$
$548$	33017.8	2.57382
$549$	−3606.11	−0.280337
$550$	0	0
$551$	−4378.01	−0.338493
$552$	−3875.20	−0.298804
$553$	−1828.66	−0.140619
$554$	−16171.6	−1.24019
$555$	0	0
$556$	36545.3	2.78753
$557$	14212.8	1.08118	0.540588	−	0.841287i	$-0.318202\pi$
$557$	14212.8	1.08118	0.540588	+	0.841287i	$0.318202\pi$
$558$	15385.1	1.16721
$559$	709.421	0.0536768
$560$	0	0
$561$	−2414.30	−0.181697
$562$	−14384.4	−1.07966
$563$	−4446.83	−0.332880	−0.166440	−	0.986052i	$-0.553227\pi$
$563$	−4446.83	−0.332880	−0.166440	+	0.986052i	$0.553227\pi$
$564$	−25878.7	−1.93207
$565$	0	0
$566$	36901.9	2.74047
$567$	971.661	0.0719681
$568$	7746.07	0.572214
$569$	11258.8	0.829511	0.414756	−	0.909933i	$-0.363867\pi$
$569$	11258.8	0.829511	0.414756	+	0.909933i	$0.363867\pi$
$570$	0	0
$571$	−16450.6	−1.20567	−0.602835	−	0.797866i	$-0.705962\pi$
$571$	−16450.6	−1.20567	−0.602835	+	0.797866i	$0.705962\pi$
$572$	6665.78	0.487256
$573$	−22640.6	−1.65065
$574$	6252.43	0.454654
$575$	0	0
$576$	−34138.5	−2.46951
$577$	−10175.6	−0.734173	−0.367086	−	0.930187i	$-0.619645\pi$
$577$	−10175.6	−0.734173	−0.367086	+	0.930187i	$0.619645\pi$
$578$	18698.6	1.34561
$579$	21915.4	1.57301
$580$	0	0
$581$	6677.90	0.476843
$582$	12822.4	0.913237
$583$	−3397.54	−0.241358
$584$	−4255.30	−0.301516
$585$	0	0
$586$	−8417.62	−0.593394
$587$	5123.98	0.360289	0.180144	−	0.983640i	$-0.442344\pi$
$587$	5123.98	0.360289	0.180144	+	0.983640i	$0.442344\pi$
$588$	−4757.59	−0.333673
$589$	−8354.11	−0.584423
$590$	0	0
$591$	−15532.3	−1.08107
$592$	6112.31	0.424349
$593$	23816.7	1.64930	0.824650	−	0.565643i	$-0.191372\pi$
$593$	23816.7	1.64930	0.824650	+	0.565643i	$0.191372\pi$
$594$	5773.93	0.398834
$595$	0	0
$596$	22522.0	1.54788
$597$	25778.4	1.76724
$598$	6451.16	0.441150
$599$	11801.0	0.804965	0.402482	−	0.915428i	$-0.368147\pi$
$599$	11801.0	0.804965	0.402482	+	0.915428i	$0.368147\pi$
$600$	0	0
$601$	−10944.5	−0.742820	−0.371410	−	0.928469i	$-0.621125\pi$
$601$	−10944.5	−0.742820	−0.371410	+	0.928469i	$0.621125\pi$
$602$	427.879	0.0289685
$603$	−25122.4	−1.69662
$604$	−26896.9	−1.81195
$605$	0	0
$606$	54809.6	3.67407
$607$	1280.36	0.0856150	0.0428075	−	0.999083i	$-0.486370\pi$
$607$	1280.36	0.0856150	0.0428075	+	0.999083i	$0.486370\pi$
$608$	22116.1	1.47521
$609$	−2541.64	−0.169118
$610$	0	0
$611$	13747.0	0.910216
$612$	12885.2	0.851065
$613$	11029.9	0.726744	0.363372	−	0.931644i	$-0.381625\pi$
$613$	11029.9	0.726744	0.363372	+	0.931644i	$0.381625\pi$
$614$	29313.9	1.92673
$615$	0	0
$616$	1282.89	0.0839106
$617$	20861.3	1.36117	0.680586	−	0.732668i	$-0.261725\pi$
$617$	20861.3	1.36117	0.680586	+	0.732668i	$0.261725\pi$
$618$	−20517.7	−1.33551
$619$	−16877.9	−1.09593	−0.547966	−	0.836501i	$-0.684598\pi$
$619$	−16877.9	−1.09593	−0.547966	+	0.836501i	$0.684598\pi$
$620$	0	0
$621$	3324.42	0.214822
$622$	23296.6	1.50178
$623$	5879.37	0.378093
$624$	8504.06	0.545568
$625$	0	0
$626$	23725.4	1.51479
$627$	−9057.91	−0.576935
$628$	−3290.17	−0.209064
$629$	−8136.43	−0.515772
$630$	0	0
$631$	−1332.58	−0.0840719	−0.0420359	−	0.999116i	$-0.513384\pi$
$631$	−1332.58	−0.0840719	−0.0420359	+	0.999116i	$0.513384\pi$
$632$	−4352.42	−0.273940
$633$	4303.19	0.270200
$634$	−25807.2	−1.61662
$635$	0	0
$636$	29989.0	1.86972
$637$	2527.27	0.157196
$638$	2147.81	0.133280
$639$	−19198.1	−1.18852
$640$	0	0
$641$	20472.2	1.26147	0.630736	−	0.775998i	$-0.282753\pi$
$641$	20472.2	1.26147	0.630736	+	0.775998i	$0.282753\pi$
$642$	25488.8	1.56692
$643$	−27140.3	−1.66455	−0.832276	−	0.554361i	$-0.812963\pi$
$643$	−27140.3	−1.66455	−0.832276	+	0.554361i	$0.812963\pi$
$644$	2314.79	0.141639
$645$	0	0
$646$	−11760.7	−0.716282
$647$	−13662.1	−0.830159	−0.415079	−	0.909785i	$-0.636246\pi$
$647$	−13662.1	−0.830159	−0.415079	+	0.909785i	$0.636246\pi$
$648$	2312.67	0.140201
$649$	−6846.91	−0.414121
$650$	0	0
$651$	−4849.96	−0.291989
$652$	10181.4	0.611556
$653$	−1607.56	−0.0963379	−0.0481689	−	0.998839i	$-0.515339\pi$
$653$	−1607.56	−0.0963379	−0.0481689	+	0.998839i	$0.515339\pi$
$654$	−12523.4	−0.748781
$655$	0	0
$656$	−4010.15	−0.238674
$657$	10546.5	0.626267
$658$	8291.30	0.491229
$659$	27361.6	1.61738	0.808692	−	0.588233i	$-0.200176\pi$
$659$	27361.6	1.61738	0.808692	+	0.588233i	$0.200176\pi$
$660$	0	0
$661$	−5117.29	−0.301119	−0.150559	−	0.988601i	$-0.548107\pi$
$661$	−5117.29	−0.301119	−0.150559	+	0.988601i	$0.548107\pi$
$662$	6072.76	0.356533
$663$	−11320.2	−0.663107
$664$	15894.2	0.928939
$665$	0	0
$666$	56217.3	3.27083
$667$	1236.63	0.0717877
$668$	23089.0	1.33734
$669$	17363.2	1.00344
$670$	0	0
$671$	−960.633	−0.0552680
$672$	12839.4	0.737042
$673$	−11605.6	−0.664729	−0.332365	−	0.943151i	$-0.607846\pi$
$673$	−11605.6	−0.664729	−0.332365	+	0.943151i	$0.607846\pi$
$674$	−14948.5	−0.854292
$675$	0	0
$676$	5441.85	0.309618
$677$	−32514.6	−1.84584	−0.922922	−	0.384986i	$-0.874206\pi$
$677$	−32514.6	−1.84584	−0.922922	+	0.384986i	$0.874206\pi$
$678$	36391.1	2.06134
$679$	−2444.02	−0.138134
$680$	0	0
$681$	−50013.1	−2.81425
$682$	4098.44	0.230113
$683$	−7201.06	−0.403427	−0.201714	−	0.979445i	$-0.564651\pi$
$683$	−7201.06	−0.403427	−0.201714	+	0.979445i	$0.564651\pi$
$684$	48342.1	2.70235
$685$	0	0
$686$	1524.29	0.0848362
$687$	24756.6	1.37485
$688$	−274.431	−0.0152072
$689$	−15930.4	−0.880841
$690$	0	0
$691$	−32357.7	−1.78140	−0.890698	−	0.454596i	$-0.849784\pi$
$691$	−32357.7	−1.78140	−0.890698	+	0.454596i	$0.849784\pi$
$692$	−46489.7	−2.55386
$693$	−3179.55	−0.174287
$694$	13265.2	0.725562
$695$	0	0
$696$	−6049.42	−0.329458
$697$	5338.13	0.290095
$698$	−5715.31	−0.309925
$699$	−540.178	−0.0292295
$700$	0	0
$701$	11077.3	0.596838	0.298419	−	0.954435i	$-0.403541\pi$
$701$	11077.3	0.596838	0.298419	+	0.954435i	$0.403541\pi$
$702$	27072.8	1.45555
$703$	−30526.0	−1.63771
$704$	−9094.16	−0.486859
$705$	0	0
$706$	37407.7	1.99413
$707$	−10447.1	−0.555731
$708$	60435.6	3.20806
$709$	−28594.0	−1.51463	−0.757314	−	0.653051i	$-0.773489\pi$
$709$	−28594.0	−1.51463	−0.757314	+	0.653051i	$0.773489\pi$
$710$	0	0
$711$	10787.2	0.568990
$712$	13993.6	0.736563
$713$	2359.73	0.123945
$714$	−6827.64	−0.357868
$715$	0	0
$716$	−36934.1	−1.92778
$717$	9109.61	0.474483
$718$	−33256.5	−1.72858
$719$	18240.5	0.946114	0.473057	−	0.881032i	$-0.343150\pi$
$719$	18240.5	0.946114	0.473057	+	0.881032i	$0.343150\pi$
$720$	0	0
$721$	3910.81	0.202006
$722$	−13642.0	−0.703190
$723$	−43777.7	−2.25188
$724$	8033.68	0.412389
$725$	0	0
$726$	4443.72	0.227165
$727$	9792.53	0.499566	0.249783	−	0.968302i	$-0.419641\pi$
$727$	9792.53	0.499566	0.249783	+	0.968302i	$0.419641\pi$
$728$	6015.20	0.306234
$729$	−32086.4	−1.63016
$730$	0	0
$731$	365.309	0.0184835
$732$	8479.22	0.428143
$733$	22723.8	1.14505	0.572525	−	0.819887i	$-0.305964\pi$
$733$	22723.8	1.14505	0.572525	+	0.819887i	$0.305964\pi$
$734$	38168.3	1.91937
$735$	0	0
$736$	−6246.98	−0.312863
$737$	−6692.35	−0.334485
$738$	−36882.9	−1.83967
$739$	−24063.6	−1.19783	−0.598913	−	0.800814i	$-0.704401\pi$
$739$	−24063.6	−1.19783	−0.598913	+	0.800814i	$0.704401\pi$
$740$	0	0
$741$	−42470.8	−2.10554
$742$	−9608.22	−0.475376
$743$	29211.6	1.44235	0.721177	−	0.692751i	$-0.243601\pi$
$743$	29211.6	1.44235	0.721177	+	0.692751i	$0.243601\pi$
$744$	−11543.5	−0.568824
$745$	0	0
$746$	52585.6	2.58083
$747$	−39392.8	−1.92946
$748$	3432.48	0.167786
$749$	−4858.33	−0.237008
$750$	0	0
$751$	1880.93	0.0913931	0.0456965	−	0.998955i	$-0.485449\pi$
$751$	1880.93	0.0913931	0.0456965	+	0.998955i	$0.485449\pi$
$752$	−5317.84	−0.257874
$753$	−1470.69	−0.0711751
$754$	10070.6	0.486408
$755$	0	0
$756$	9714.21	0.467331
$757$	36218.7	1.73896	0.869480	−	0.493968i	$-0.164454\pi$
$757$	36218.7	1.73896	0.869480	+	0.493968i	$0.164454\pi$
$758$	−22470.5	−1.07674
$759$	2558.53	0.122357
$760$	0	0
$761$	36966.4	1.76088	0.880441	−	0.474156i	$-0.157247\pi$
$761$	36966.4	1.76088	0.880441	+	0.474156i	$0.157247\pi$
$762$	24462.1	1.16295
$763$	2387.03	0.113259
$764$	32188.7	1.52428
$765$	0	0
$766$	28695.2	1.35353
$767$	−32103.8	−1.51135
$768$	15061.8	0.707678
$769$	−38975.5	−1.82769	−0.913845	−	0.406062i	$-0.866902\pi$
$769$	−38975.5	−1.82769	−0.913845	+	0.406062i	$0.866902\pi$
$770$	0	0
$771$	28896.5	1.34978
$772$	−31157.7	−1.45258
$773$	−27341.9	−1.27221	−0.636105	−	0.771603i	$-0.719455\pi$
$773$	−27341.9	−1.27221	−0.636105	+	0.771603i	$0.719455\pi$
$774$	−2524.05	−0.117216
$775$	0	0
$776$	−5817.07	−0.269099
$777$	−17721.8	−0.818231
$778$	−55303.3	−2.54848
$779$	20027.4	0.921125
$780$	0	0
$781$	−5114.19	−0.234315
$782$	3321.96	0.151909
$783$	5189.61	0.236860
$784$	−977.642	−0.0445354
$785$	0	0
$786$	1117.75	0.0507235
$787$	18268.5	0.827446	0.413723	−	0.910403i	$-0.364228\pi$
$787$	18268.5	0.827446	0.413723	+	0.910403i	$0.364228\pi$
$788$	22082.7	0.998302
$789$	47500.7	2.14331
$790$	0	0
$791$	−6936.37	−0.311794
$792$	−7567.71	−0.339529
$793$	−4504.22	−0.201702
$794$	−2751.75	−0.122993
$795$	0	0
$796$	−36649.9	−1.63194
$797$	−12717.6	−0.565219	−0.282610	−	0.959235i	$-0.591200\pi$
$797$	−12717.6	−0.565219	−0.282610	+	0.959235i	$0.591200\pi$
$798$	−25615.7	−1.13632
$799$	7078.86	0.313432
$800$	0	0
$801$	−34682.2	−1.52988
$802$	−43248.4	−1.90418
$803$	2809.48	0.123467
$804$	59071.4	2.59115
$805$	0	0
$806$	19216.8	0.839804
$807$	−1943.82	−0.0847904
$808$	−24865.3	−1.08262
$809$	12502.0	0.543322	0.271661	−	0.962393i	$-0.412427\pi$
$809$	12502.0	0.543322	0.271661	+	0.962393i	$0.412427\pi$
$810$	0	0
$811$	−23431.3	−1.01453	−0.507264	−	0.861791i	$-0.669343\pi$
$811$	−23431.3	−1.01453	−0.507264	+	0.861791i	$0.669343\pi$
$812$	3613.52	0.156170
$813$	9747.95	0.420511
$814$	14975.7	0.644839
$815$	0	0
$816$	4379.08	0.187866
$817$	1370.56	0.0586899
$818$	−20537.6	−0.877850
$819$	−14908.3	−0.636065
$820$	0	0
$821$	33116.5	1.40776	0.703881	−	0.710317i	$-0.251449\pi$
$821$	33116.5	1.40776	0.703881	+	0.710317i	$0.251449\pi$
$822$	103206.	4.37923
$823$	6383.39	0.270366	0.135183	−	0.990821i	$-0.456838\pi$
$823$	6383.39	0.270366	0.135183	+	0.990821i	$0.456838\pi$
$824$	9308.20	0.393527
$825$	0	0
$826$	−19363.0	−0.815649
$827$	27701.3	1.16477	0.582386	−	0.812912i	$-0.302119\pi$
$827$	27701.3	1.16477	0.582386	+	0.812912i	$0.302119\pi$
$828$	−13654.9	−0.573116
$829$	−13160.2	−0.551353	−0.275676	−	0.961251i	$-0.588902\pi$
$829$	−13160.2	−0.551353	−0.275676	+	0.961251i	$0.588902\pi$
$830$	0	0
$831$	−30072.3	−1.25535
$832$	−42640.7	−1.77680
$833$	1301.39	0.0541303
$834$	114232.	4.74286
$835$	0	0
$836$	12877.9	0.532763
$837$	9902.80	0.408950
$838$	−828.483	−0.0341521
$839$	−24842.9	−1.02226	−0.511128	−	0.859505i	$-0.670772\pi$
$839$	−24842.9	−1.02226	−0.511128	+	0.859505i	$0.670772\pi$
$840$	0	0
$841$	−22458.6	−0.920848
$842$	−11866.8	−0.485696
$843$	−26748.8	−1.09286
$844$	−6117.95	−0.249513
$845$	0	0
$846$	−48910.2	−1.98767
$847$	−847.000	−0.0343604
$848$	6162.48	0.249552
$849$	68622.0	2.77397
$850$	0	0
$851$	8622.47	0.347326
$852$	45141.4	1.81516
$853$	10131.6	0.406681	0.203340	−	0.979108i	$-0.434820\pi$
$853$	10131.6	0.406681	0.203340	+	0.979108i	$0.434820\pi$
$854$	−2716.67	−0.108855
$855$	0	0
$856$	−11563.4	−0.461716
$857$	−10115.6	−0.403199	−0.201599	−	0.979468i	$-0.564614\pi$
$857$	−10115.6	−0.403199	−0.201599	+	0.979468i	$0.564614\pi$
$858$	20835.7	0.829044
$859$	−27491.4	−1.09196	−0.545980	−	0.837798i	$-0.683843\pi$
$859$	−27491.4	−1.09196	−0.545980	+	0.837798i	$0.683843\pi$
$860$	0	0
$861$	11626.9	0.460212
$862$	55616.1	2.19755
$863$	117.276	0.00462588	0.00231294	−	0.999997i	$-0.499264\pi$
$863$	117.276	0.00462588	0.00231294	+	0.999997i	$0.499264\pi$
$864$	−26216.0	−1.03228
$865$	0	0
$866$	−73997.8	−2.90364
$867$	34771.5	1.36206
$868$	6895.31	0.269634
$869$	2873.60	0.112175
$870$	0	0
$871$	−31379.1	−1.22071
$872$	5681.43	0.220639
$873$	14417.2	0.558933
$874$	12463.2	0.482351
$875$	0	0
$876$	−24798.4	−0.956462
$877$	−11597.4	−0.446543	−0.223271	−	0.974756i	$-0.571674\pi$
$877$	−11597.4	−0.446543	−0.223271	+	0.974756i	$0.571674\pi$
$878$	−26812.3	−1.03061
$879$	−15653.2	−0.600648
$880$	0	0
$881$	7524.18	0.287737	0.143868	−	0.989597i	$-0.454046\pi$
$881$	7524.18	0.287737	0.143868	+	0.989597i	$0.454046\pi$
$882$	−8991.75	−0.343274
$883$	−13467.4	−0.513266	−0.256633	−	0.966509i	$-0.582613\pi$
$883$	−13467.4	−0.513266	−0.256633	+	0.966509i	$0.582613\pi$
$884$	16094.2	0.612339
$885$	0	0
$886$	28086.2	1.06498
$887$	12955.7	0.490427	0.245214	−	0.969469i	$-0.421142\pi$
$887$	12955.7	0.490427	0.245214	+	0.969469i	$0.421142\pi$
$888$	−42180.0	−1.59400
$889$	−4662.63	−0.175905
$890$	0	0
$891$	−1526.90	−0.0574107
$892$	−24685.7	−0.926612
$893$	26558.2	0.995226
$894$	70398.6	2.63365
$895$	0	0
$896$	−13288.9	−0.495480
$897$	11996.4	0.446543
$898$	79518.9	2.95499
$899$	3683.68	0.136660
$900$	0	0
$901$	−8203.20	−0.303317
$902$	−9825.25	−0.362688
$903$	795.673	0.0293226
$904$	−16509.4	−0.607405
$905$	0	0
$906$	−84073.5	−3.08295
$907$	−47843.5	−1.75151	−0.875753	−	0.482759i	$-0.839635\pi$
$907$	−47843.5	−1.75151	−0.875753	+	0.482759i	$0.839635\pi$
$908$	71104.9	2.59879
$909$	61626.9	2.24866
$910$	0	0
$911$	16969.8	0.617162	0.308581	−	0.951198i	$-0.400146\pi$
$911$	16969.8	0.617162	0.308581	+	0.951198i	$0.400146\pi$
$912$	16429.3	0.596522
$913$	−10493.8	−0.380389
$914$	27588.3	0.998402
$915$	0	0
$916$	−35197.1	−1.26959
$917$	−213.049	−0.00767231
$918$	13940.9	0.501218
$919$	12095.2	0.434149	0.217075	−	0.976155i	$-0.430348\pi$
$919$	12095.2	0.434149	0.217075	+	0.976155i	$0.430348\pi$
$920$	0	0
$921$	54511.4	1.95029
$922$	−35468.8	−1.26692
$923$	−23979.4	−0.855138
$924$	7476.22	0.266179
$925$	0	0
$926$	−33308.6	−1.18206
$927$	−23069.8	−0.817379
$928$	−9751.91	−0.344959
$929$	−44544.3	−1.57314	−0.786571	−	0.617499i	$-0.788146\pi$
$929$	−44544.3	−1.57314	−0.786571	+	0.617499i	$0.788146\pi$
$930$	0	0
$931$	4882.52	0.171878
$932$	767.985	0.0269916
$933$	43321.8	1.52014
$934$	6752.16	0.236550
$935$	0	0
$936$	−35483.5	−1.23912
$937$	49265.8	1.71766	0.858828	−	0.512264i	$-0.171193\pi$
$937$	49265.8	1.71766	0.858828	+	0.512264i	$0.171193\pi$
$938$	−18925.9	−0.658799
$939$	44119.2	1.53331
$940$	0	0
$941$	−18403.1	−0.637538	−0.318769	−	0.947832i	$-0.603269\pi$
$941$	−18403.1	−0.637538	−0.318769	+	0.947832i	$0.603269\pi$
$942$	−10284.3	−0.355712
$943$	−5657.01	−0.195353
$944$	12419.0	0.428181
$945$	0	0
$946$	−672.381	−0.0231089
$947$	−17689.3	−0.606996	−0.303498	−	0.952832i	$-0.598155\pi$
$947$	−17689.3	−0.606996	−0.303498	+	0.952832i	$0.598155\pi$
$948$	−25364.4	−0.868986
$949$	13173.1	0.450597
$950$	0	0
$951$	−47990.5	−1.63638
$952$	3097.47	0.105451
$953$	5298.19	0.180090	0.0900448	−	0.995938i	$-0.471299\pi$
$953$	5298.19	0.180090	0.0900448	+	0.995938i	$0.471299\pi$
$954$	56678.7	1.92352
$955$	0	0
$956$	−12951.4	−0.438156
$957$	3994.01	0.134909
$958$	72855.6	2.45705
$959$	−19671.7	−0.662392
$960$	0	0
$961$	−22761.8	−0.764050
$962$	70218.3	2.35336
$963$	28659.1	0.959011
$964$	62239.9	2.07947
$965$	0	0
$966$	7235.50	0.240992
$967$	−33990.6	−1.13037	−0.565184	−	0.824965i	$-0.691195\pi$
$967$	−33990.6	−1.13037	−0.565184	+	0.824965i	$0.691195\pi$
$968$	−2015.96	−0.0669375
$969$	−21869.9	−0.725039
$970$	0	0
$971$	41991.0	1.38780	0.693900	−	0.720071i	$-0.255891\pi$
$971$	41991.0	1.38780	0.693900	+	0.720071i	$0.255891\pi$
$972$	50946.5	1.68118
$973$	−21773.4	−0.717393
$974$	10110.1	0.332597
$975$	0	0
$976$	1742.40	0.0571444
$977$	31233.4	1.02277	0.511384	−	0.859352i	$-0.329133\pi$
$977$	31233.4	1.02277	0.511384	+	0.859352i	$0.329133\pi$
$978$	31824.7	1.04053
$979$	−9239.00	−0.301614
$980$	0	0
$981$	−14081.0	−0.458281
$982$	65012.0	2.11264
$983$	45702.5	1.48289	0.741447	−	0.671012i	$-0.234140\pi$
$983$	45702.5	1.48289	0.741447	+	0.671012i	$0.234140\pi$
$984$	27673.4	0.896540
$985$	0	0
$986$	5185.78	0.167494
$987$	15418.3	0.497235
$988$	60381.8	1.94433
$989$	−387.132	−0.0124470
$990$	0	0
$991$	−4310.72	−0.138178	−0.0690890	−	0.997611i	$-0.522009\pi$
$991$	−4310.72	−0.138178	−0.0690890	+	0.997611i	$0.522009\pi$
$992$	−18608.6	−0.595587
$993$	11292.8	0.360892
$994$	−14462.9	−0.461504
$995$	0	0
$996$	92626.0	2.94675
$997$	6103.28	0.193875	0.0969373	−	0.995290i	$-0.469095\pi$
$997$	6103.28	0.193875	0.0969373	+	0.995290i	$0.469095\pi$
$998$	−15705.1	−0.498133
$999$	36184.9	1.14599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.4.a.r.1.2		5
5.4	even	2		77.4.a.e.1.4	✓	5
15.14	odd	2		693.4.a.o.1.2		5
20.19	odd	2		1232.4.a.y.1.1		5
35.34	odd	2		539.4.a.h.1.4		5
55.54	odd	2		847.4.a.f.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.e.1.4	✓	5	5.4	even	2
539.4.a.h.1.4		5	35.34	odd	2
693.4.a.o.1.2		5	15.14	odd	2
847.4.a.f.1.2		5	55.54	odd	2
1232.4.a.y.1.1		5	20.19	odd	2
1925.4.a.r.1.2		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:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q-4.44399 q^{2} -8.26395 q^{3} +11.7491 q^{4} +36.7249 q^{6} -7.00000 q^{7} -16.6609 q^{8} +41.2928 q^{9} +11.0000 q^{11} -97.0937 q^{12} +51.5769 q^{13} +31.1080 q^{14} -19.9519 q^{16} +26.5590 q^{17} -183.505 q^{18} +99.6432 q^{19} +57.8476 q^{21} -48.8839 q^{22} -28.1455 q^{23} +137.684 q^{24} -229.207 q^{26} -118.115 q^{27} -82.2435 q^{28} -43.9369 q^{29} -83.8402 q^{31} +221.953 q^{32} -90.9034 q^{33} -118.028 q^{34} +485.152 q^{36} -306.353 q^{37} -442.814 q^{38} -426.228 q^{39} +200.991 q^{41} -257.074 q^{42} +13.7546 q^{43} +129.240 q^{44} +125.079 q^{46} +266.533 q^{47} +164.881 q^{48} +49.0000 q^{49} -219.482 q^{51} +605.980 q^{52} -308.867 q^{53} +524.903 q^{54} +116.626 q^{56} -823.446 q^{57} +195.255 q^{58} -622.446 q^{59} -87.3303 q^{61} +372.585 q^{62} -289.050 q^{63} -826.742 q^{64} +403.974 q^{66} -608.395 q^{67} +312.044 q^{68} +232.593 q^{69} -464.926 q^{71} -687.974 q^{72} +255.407 q^{73} +1361.43 q^{74} +1170.72 q^{76} -77.0000 q^{77} +1894.16 q^{78} +261.237 q^{79} -138.809 q^{81} -893.204 q^{82} -953.986 q^{83} +679.656 q^{84} -61.1255 q^{86} +363.092 q^{87} -183.269 q^{88} -839.910 q^{89} -361.038 q^{91} -330.684 q^{92} +692.851 q^{93} -1184.47 q^{94} -1834.21 q^{96} +349.146 q^{97} -217.756 q^{98} +454.221 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} - 2 q^{3} + 45 q^{4} + 4 q^{6} - 35 q^{7} - 57 q^{8} + 63 q^{9} + 55 q^{11} - 24 q^{12} + 50 q^{13} + 7 q^{14} + 433 q^{16} - 222 q^{17} - 245 q^{18} + 160 q^{19} + 14 q^{21} - 11 q^{22} - 54 q^{23}+ \cdots + 693 q^{99}+O(q^{100}) \) 5 * q - q^2 - 2 * q^3 + 45 * q^4 + 4 * q^6 - 35 * q^7 - 57 * q^8 + 63 * q^9 + 55 * q^11 - 24 * q^12 + 50 * q^13 + 7 * q^14 + 433 * q^16 - 222 * q^17 - 245 * q^18 + 160 * q^19 + 14 * q^21 - 11 * q^22 - 54 * q^23 - 220 * q^24 - 1026 * q^26 - 110 * q^27 - 315 * q^28 + 14 * q^29 - 34 * q^31 + 583 * q^32 - 22 * q^33 - 750 * q^34 - 411 * q^36 - 1044 * q^37 + 156 * q^38 - 124 * q^39 - 114 * q^41 - 28 * q^42 - 672 * q^43 + 495 * q^44 - 1224 * q^46 + 292 * q^47 + 1652 * q^48 + 245 * q^49 + 768 * q^51 + 914 * q^52 + 710 * q^53 + 1608 * q^54 + 399 * q^56 - 2012 * q^57 + 810 * q^58 + 270 * q^59 + 138 * q^61 + 480 * q^62 - 441 * q^63 + 3001 * q^64 + 44 * q^66 - 1942 * q^67 - 3130 * q^68 - 1306 * q^69 - 278 * q^71 - 3565 * q^72 + 338 * q^73 + 462 * q^74 - 1332 * q^76 - 385 * q^77 - 280 * q^78 + 576 * q^79 - 1439 * q^81 - 1386 * q^82 - 1644 * q^83 + 168 * q^84 + 2020 * q^86 + 1956 * q^87 - 627 * q^88 - 3656 * q^89 - 350 * q^91 + 748 * q^92 + 1442 * q^93 + 976 * q^94 - 4508 * q^96 - 692 * q^97 - 49 * q^98 + 693 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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